The wavefunction as an oscillation of spacetime

Post scriptum note added on 11 July 2016: This is one of the more speculative posts which led to my e-publication analyzing the wavefunction as an energy propagation. With the benefit of hindsight, I would recommend you to immediately the more recent exposé on the matter that is being presented here, which you can find by clicking on the provided link.

Original post:

You probably heard about the experimental confirmation of the existence of gravitational waves by Caltech’s LIGO Lab. Besides further confirming our understanding of the Universe, I also like to think it confirms that the elementary wavefunction represents a propagation mechanism that is common to all forces. However, the fundamental question remains: what is the wavefunction? What are those real and imaginary parts of those complex-valued wavefunctions describing particles and/or photons? [In case you think photons have no wavefunction, see my post on it: it’s fairly straightforward to re-formulate the usual description of an electromagnetic wave (i.e. the description in terms of the electric and magnetic field vectors) in terms of a complex-valued wavefunction. To be precise, in the mentioned post, I showed an electromagnetic wave can be represented as the  sum of two wavefunctions whose components reflect each other through a rotation by 90 degrees.]

So what? Well… I’ve started to think that the wavefunction may not only describe some oscillation in spacetime. I’ve started to think the wavefunction—any wavefunction, really (so I am not talking gravitational waves only)—is nothing but an oscillation of spacetime. What makes them different is the geometry of those wavefunctions, and the coefficient(s) representing their amplitude, which must be related to their relative strength—somehow, although I still have to figure out how exactly.

Huh? Yes. Maxwell, after jotting down his equations for the electric and magnetic field vectors, wrote the following back in 1862: “The velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments of MM. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”

We now know there is no medium – no aether – but physicists still haven’t answered the most fundamental question: what is it that is oscillating? No one has gone beyond the abstract math. I dare to say now that it must be spacetime itself. In order to prove this, I’ll have to study Einstein’s general theory of relativity. But this post will already cover some basics.

The quantum of action and natural units

We can re-write the quantum of action in natural units, which we’ll call Planck units for the time being. They may or may not be the Planck units you’ve heard about, so just think of them as being fundamental, or natural—for the time being, that is. You’ll wonder: what’s natural? What’s fundamental? Well… That’s the question we’re trying to explore in this post, so… Well… Just be patient… 🙂 We’ll denote those natural units as FP, lP, and tP, i.e. the Planck force, Planck distance and Planck time unit respectively. Hence, we write:

ħ = FPlP∙tP

Note that FP, lP, and tP are expressed in our old-fashioned SI units here, i.e. in newton (N), meter (m) and seconds (s) respectively. So FP, lP, and tP have a numerical value as well as a dimension, just like ħ. They’re not just numbers. If we’d want to be very explicit, we could write: FP = FP [force], or FP = FP N, and you could do the same for land tP. However, it’s rather tedious to mention those dimensions all the time, so I’ll just assume you understand the symbols we’re using do not represent some dimensionless number. In fact, that’s what distinguishes physical constants from mathematical constants.

Dimensions are also distinguishes physics equations from purely mathematical ones: an equation in physics will always relate some physical quantities and, hence, when you’re dealing with physics equations, you always need to wonder about the dimensions. [Note that the term ‘dimension’ has many definitions… But… Well… I suppose you know what I am talking about here, and I need to move on. So let’s do that.] Let’s re-write that ħ = FPlP∙tP formula as follows: ħ/tP = FPlP.

FPlP is, obviously, a force times a distance, so that’s energy. Please do check the dimensions on the left-hand side as well: [ħ/tP] = [[ħ]/[tP] = (N·m·s)/s = N·m. In short, we can think of EP = FPlP = ħ/tP as being some ‘natural’ unit as well. But what would it correspond to—physically? What is its meaning? We may be tempted to call it the quantum of energy that’s associated with our quantum of action, but… Well… No. While it’s referred to as the Planck energy, it’s actually a rather large unit, and so… Well… No. We should not think of it as the quantum of energy. We have a quantum of action but no quantum of energy. Sorry. Let’s move on.

In the same vein, we can re-write the ħ = FPlP∙tP as ħ/lP = FP∙tP. Same thing with the dimensions—or ‘same-same but different’, as they say in Asia: [ħ/lP] = [FP∙tP] = N·m·s)/m = N·s. Force times time is momentum and, hence, we may now be tempted to think of pP = FP∙tP = ħ/lP as the quantum of momentum that’s associated with ħ, but… Well… No. There’s no such thing as a quantum of momentum. Not now in any case. Maybe later. 🙂 But, for now, we only have a quantum of action. So we’ll just call ħ/lP = FP∙tP the Planck momentum for the time being.

So now we have two ways of looking at the dimension of Planck’s constant:

  1. [Planck’s constant] = N∙m∙s = (N∙m)∙s = [energy]∙[time]
  2. [Planck’s constant] = N∙m∙s = (N∙s)∙m = [momentum]∙[distance]

In case you didn’t get this from what I wrote above: the brackets here, i.e. the [ and ] symbols, mean: ‘the dimension of what’s between the brackets’. OK. So far so good. It may all look like kids stuff – it actually is kids stuff so far – but the idea is quite fundamental: we’re thinking here of some amount of action (h or ħ, to be precise, i.e. the quantum of action) expressing itself in time or, alternatively, expressing itself in spaceIn the former case, some amount of energy is expended during some time. In the latter case, some momentum is expended over some distance.

Of course, ideally, we should try to think of action expressing itself in space and time simultaneously, so we should think of it as expressing itself in spacetime. In fact, that’s what the so-called Principle of Least Action in physics is all about—but I won’t dwell on that here, because… Well… It’s not an easy topic, and the diversion would lead us astray. 🙂 What we will do, however, is apply the idea above to the two de Broglie relations: E = ħω and p = ħk. I assume you know these relations by now. If not, just check one of my many posts on them. Let’s see what we can do with them.

The de Broglie relations

We can re-write the two de Broglie relations as ħ = E/ω and ħ = p/k. We can immediately derive an interesting property here:

ħ/ħ = 1 = (E/ω)/(p/k) ⇔ E/p = ω/k

So the ratio of the energy and the momentum is equal to the wave velocity. What wave velocity? The group of the phase velocity? We’re talking an elementary wave here, so both are the same: we have only one E and p, and, hence, only one ω and k. The E/p = ω/k identity underscores the following point: the de Broglie equations are a pair of equations here, and one of the key things to learn when trying to understand quantum mechanics is to think of them as an inseparable pair—like an inseparable twin really—as the quantum of action incorporates both a spatial as well as a temporal dimension. Just think of what Minkowski wrote back in 1907, shortly after he had re-formulated Einstein’s special relativity theory in terms of four-dimensional spacetime, and just two years before he died—unexpectely—from an appendicitis: “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

So we should try to think of what that union might represent—and that surely includes looking at the de Broglie equations as a pair of matter-wave equations. Likewise, we should also think of the Uncertainty Principle as a pair of equations: ΔpΔx ≥ ħ/2 and ΔEΔt ≥ ħ/2—but I’ll come back to those later.

The ω in the E = ħω equation and the argument (θ = kx – ωt) of the wavefunction is a frequency in time (or temporal frequency). It’s a frequency expressed in radians per second. You get one radian by dividing one cycle by 2π. In other words, we have 2π radians in one cycle. So ω is related the frequency you’re used to, i.e. f—the frequency expressed in cycles per second (i.e. hertz): we multiply f by 2π to get ω. So we can write: E = ħω = ħ∙2π∙f = h∙f, with h = ħ∙2π (or ħ = h/2π).

Likewise, the k in the p = ħk equation and the argument (θ = kx – ωt) of the wavefunction is a frequency in space (or spatial frequency). Unsurprisingly, it’s expressed in radians per meter.

At this point, it’s good to properly define the radian as a unit in quantum mechanics. We often think of a radian as some distance measured along the circumference, because of the way the unit is constructed (see the illustration below) but that’s right and wrong at the same time. In fact, it’s more wrong than right: the radian is an angle that’s defined using the length of the radius of the unit circle but, when everything is said and done, it’s a unit used to measure some anglenot a distance. That should be obvious from the 2π rad = 360 degrees identity. The angle here is the argument of our wavefunction in quantum mechanics, and so that argument combines both time (t) as well as distance (x): θ = kx – ωt = k(x – c∙t). So our angle (the argument of the wavefunction) integrates both dimensions: space as well as time. If you’re not convinced, just do the dimensional analysis of the kx – ωt expression: both the kx and ωt yield a dimensionless number—or… Well… To be precise, I should say: the kx and ωt products both yield an angle expressed in radians. That angle connects the real and imaginary part of the argument of the wavefunction. Hence, it’s a dimensionless number—but that does not imply it is just some meaningless number. It’s not meaningless at all—obviously!

Circle_radians

Let me try to present what I wrote above in yet another way. The θ = kx – ωt = (p/ħ)·x − (E/ħ)·t equation suggests a fungibility: the wavefunction itself also expresses itself in time and/or in space, so to speak—just like the quantum of action. Let me be precise: the p·x factor in the (p/ħ)·x term represents momentum (whose dimension is N·s) being expended over a distance, while the E·t factor in the (E/ħ)·t term represents energy (expressed in N·m) being expended over some time. [As for the minus sign in front of the (E/ħ)·t term, that’s got to do with the fact that the arrow of time points in one direction only while, in space, we can go in either direction: forward or backwards.] Hence, the expression for the argument tells us that both are essentially fungible—which suggests they’re aspects of one and the same thing. So that‘s what Minkowski intuition is all about: spacetime is one, and the wavefunction just connects the physical properties of whatever it is that we are observing – an electron, or a photon, or whatever other elementary particle – to it.

Of course, the corollary to thinking of unified spacetime is thinking of the real and imaginary part of the wavefunction as one—which we’re supposed to do as a complex number is… Well… One complex number. But that’s easier said than actually done, of course. One way of thinking about the connection between the two spacetime ‘dimensions’ – i.e. t and x, with x actually incorporating three spatial dimensions in space in its own right (see how confusing the term ‘dimension’ is?) – and the two ‘dimensions’ of a complex number is going from Cartesian to polar coordinates, and vice versa. You now think of Euler’s formula, of course – if not, you should – but let me insert something more interesting here. 🙂 I took it from Wikipedia. It illustrates how a simple sinusoidal function transforms as we go from Cartesian to polar coordinates.

Cartesian_to_polar

Interesting, isn’t it? Think of the horizontal and vertical axis in the Cartesian space as representing time and… Well… Space indeed. 🙂 The function connects the space and time dimension and might, for example, represent the trajectory of some object in spacetime. Admittedly, it’s a rather weird trajectory, as the object goes back and forth in some box in space, and accelerates and decelerates all of the time, reversing its direction in the process… But… Well… Isn’t that how we think of a an electron moving in some orbital? 🙂 With that in mind, look at how the same movement in spacetime looks like in polar coordinates. It’s also some movement in a box—but both the ‘horizontal’ and ‘vertical’ axis (think of these axes as the real and imaginary part of a complex number) are now delineating our box. So, whereas our box is a one-dimensional box in spacetime only (our object is constrained in space, but time keeps ticking), it’s a two-dimensional box in our ‘complex’ space. Isn’t it just nice to think about stuff this way?

As far as I am concerned, it triggers the same profound thoughts as that E/p = ω/k relation. The  left-hand side is a ratio between energy and momentum. Now, one way to look at energy is that it’s action per time unit. Likewise, momentum is action per distance unit. Of course, ω is expressed as some quantity (expressed in radians, to be precise) per time unit, and k is some quantity (again, expressed in radians) per distance unit. Because this is a physical equation, the dimension of both sides of the equation has to be the same—and, of course, it is the same: the action dimension in the numerator and denominator of the ratio on the left-hand side of the E/p = ω/k equation cancel each other. But… What? Well… Wouldn’t it be nice to think of the dimension of the argument of our wavefunction as being the dimension of action, rather than thinking of it as just some mathematical thing, i.e. an angle. I like to think the argument of our wavefunction is more than just an angle. When everything is said and done, it has to be something physical—if onlyh because the wavefunction describes something physical. But… Well… I need to do some more thinking on this, so I’ll just move on here. Otherwise this post risks becoming a book in its own right. 🙂

Let’s get back to the topic we were discussing here. We were talking about natural units. More in particular, we were wondering: what’s natural? What does it mean?

Back to Planck units

Let’s start with time and distance. We may want to think of lP and tP as the smallest distance and time units possible—so small, in fact, that both distance and time become countable variables at that scale.

Huh? Yes. I am sure you’ve never thought of time and distance as countable variables but I’ll come back to this rather particular interpretation of the Planck length and time unit later. So don’t worry about it now: just make a mental note of it. The thing is: if tP and lP are the smallest time and distance units possible, then the smallest cycle we can possibly imagine will be associated with those two units: we write: ωP = 1/tP and kP = 1/lP. What’s the ‘smallest possible’ cycle? Well… Not sure. You should not think of some oscillation in spacetime as for now. Just think of a cycle. Whatever cycle. So, as for now, the smallest cycle is just the cycle you’d associate with the smallest time and distance units possible—so we cannot write ωP = 2/tP, for example, because that would imply we can imagine a time unit that’s smaller than tP, as we can associate two cycles with tP now.

OK. Next step. We can now define the Planck energy and the Planck momentum using the de Broglie relations:

EP = ħ∙ωP = ħ/tP  and pP = ħ∙kP = ħ/lP

You’ll say that I am just repeating myself here, as I’ve given you those two equations already. Well… Yes and no. At this point, you should raise the following question: why are we using the angular frequency (ωP = 2π·fP) and the reduced Planck constant (ħ = h/2π), rather than fP or h?

That’s a great question. In fact, it begs the question: what’s the physical constant really? We have two mathematical constants – ħ and h – but they must represent the same physical reality. So is one of the two constants more real than the other? The answer is unambiguously: yes! The Planck energy is defined as EP = ħ/tP =(h/2π)/tP, so we cannot write this as EP = h/tP. The difference is that 1/2π factor, and it’s quite fundamental, as it implies we’re actually not associating a full cycle with tP and lP but a radian of that cycle only.

Huh? Yes. It’s a rather strange observation, and I must admit I haven’t quite sorted out what this actually means. The fundamental idea remains the same, however: we have a quantum of action, ħ (not h!), that can express itself as energy over the smallest distance unit possible or, alternatively, that expresses itself as momentum over the smallest time unit possible. In the former case, we write it as EP = FPlP = ħ/tP. In the latter, we write it as pP = FP∙tP = ħ/lP. Both are aspects of the same reality, though, as our particle moves in space as well as in time, i.e. it moves in spacetime. Hence, one step in space, or in time, corresponds to one radian. Well… Sort of… Not sure how to further explain this. I probably shouldn’t try anyway. 🙂

The more fundamental question is: with what speed is is moving? That question brings us to the next point. The objective is to get some specific value for lP and tP, so how do we do that? How can we determine these two values? Well… That’s another great question. 🙂

The first step is to relate the natural time and distance units to the wave velocity. Now, we do not want to complicate the analysis and so we’re not going to throw in some rest mass or potential energy here. No. We’ll be talking a theoretical zero-mass particle. So we’re not going to consider some electron moving in spacetime, or some other elementary particle. No. We’re going to think about some zero-mass particle here, or a photon. [Note that a photon is not just a zero-mass particle. It’s similar but different: in one of my previous posts, I showed a photon packs more energy, as you get two wavefunctions for the price of one, so to speak. However, don’t worry about the difference here.]

Now, you know that the wave velocity for a zero-mass particle and/or a photon is equal to the speed of light. To be precise, the wave velocity of a photon is the speed of light and, hence, the speed of any zero-mass particle must be the same—as per the definition of mass in Einstein’s special relativity theory. So we write: lP/tP = c ⇔ lP = c∙tP and tP = lP/c. In fact, we also get this from dividing EP by pP, because we know that E/p = c, for any photon (and for any zero-mass particle, really). So we know that EP/pP must also equal c. We can easily double-check that by writing: EP/pP = (ħ/tP)/(ħ/lP) = lP/tP = c. Substitution in ħ = FPlP∙tP yields ħ = c∙FP∙tP2 or, alternatively, ħ = FPlP2/c. So we can now write FP as a function of lP and/or tP:

FP = ħ∙c/lP2 = ħ/(c∙tP2)

We can quickly double-check this by dividing FP = ħ∙c/lP2 by FP = ħ/(c∙tP2). We get: 1 = c2∙tP2/lP2 ⇔ lP2/tP2 = c2 ⇔ lP/tP = c.

Nice. However, this does not uniquely define FP, lP, and tP. The problem is that we’ve got only two equations (ħ = FPlP∙tP and lP/tP = c) for three unknowns (FP, lP, and tP). Can we throw in one or both of the de Broglie equations to get some final answer?

I wish that would help, but it doesn’t—because we get the same ħ = FPlP∙tP equation. Indeed, we’re just re-defining the Planck energy (and the Planck momentum) by that EP = ħ/tP (and pP = ħ/lP) equation here, and so that does not give us a value for EP (and pP). So we’re stuck. We need some other formula so we can calculate the third unknown, which is the Planck force unit (FP). What formula could we possibly choose?

Well… We got a relationship by imposing the condition that lP/tP = c, which implies that if we’d measure the velocity of a photon in Planck time and distance units, we’d find that its velocity is one, so c = 1. Can we think of some similar condition involving ħ? The answer is: we can and we can’t. It’s not so simple. Remember we were thinking of the smallest cycle possible? We said it was small because tP and lP were the smallest units we could imagine. But how do we define that? The idea is as follows: the smallest possible cycle will pack the smallest amount of action, i.e. h (or, expressed per radian rather than per cycle, ħ).

Now, we usually think of packing energy, or momentum, instead of packing action, but that’s because… Well… Because we’re not good at thinking the way Minkowski wanted us to think: we’re not good at thinking of some kind of union of space and time. We tend to think of something moving in space, or, alternatively, of something moving in time—rather than something moving in spacetime. In short, we tend to separate dimensions. So that’s why we’d say the smallest possible cycle would pack an amount of energy that’s equal to EP = ħ∙ωP = ħ/tP, or an amount of momentum that’s equal to pP = ħ∙kP = ħ/lP. But both are part and parcel of the same reality, as evidenced by the E = m∙c2 = m∙cc = p∙c equality. [This equation only holds for a zero-mass particle (and a photon), of course. It’s a bit more complicated when we’d throw in some rest mass, but we can do that later. Also note I keep repeating my idea of the smallest cycle, but we’re talking radians of a cycle, really.]

So we have that mass-energy equivalence, which is also a mass-momentum equivalence according to that E = m∙c2 = m∙cc = p∙c formula. And so now the gravitational force comes into play: there’s a limit to the amount of energy we can pack into a tiny space. Or… Well… Perhaps there’s no limit—but if we pack an awful lot of energy into a really tiny speck of space, then we get a black hole.

However, we’re getting a bit ahead of ourselves here, so let’s first try something else. Let’s throw in the Uncertainty Principle.

The Uncertainty Principle

As mentioned above, we can think of some amount of action expressing itself over some time or, alternatively, over some distance. In the former case, some amount of energy is expended over some time. In the latter case, some momentum is expended over some distance. That’s why the energy and time variables, and the momentum and distance variables, are referred to as complementary. It’s hard to think of both things happening simultaneously (whatever that means in spacetime), but we should try! Let’s now look at the Uncertainty relations once again (I am writing uncertainty with a capital U out of respect—as it’s very fundamental, indeed!):

ΔpΔx ≥ ħ/2 and ΔEΔt ≥ ħ/2.

Note that the ħ/2 factor on the right-hand side quantifies the uncertainty, while the right-hand side of the two equations (ΔpΔx and ΔEΔt) are just an expression of that fundamental uncertainty. In other words, we have two equations (a pair), but there’s only one fundamental uncertainty, and it’s an uncertainty about a movement in spacetime. Hence, that uncertainty expresses itself in both time as well as in space.

Note the use of ħ rather than h, and the fact that the  1/2 factor makes it look like we’re splitting ħ over ΔpΔx and ΔEΔt respectively—which is actually a quite sensible explanation of what this pair of equations actually represent. Indeed, we can add both relations to get the following sum:

ΔpΔx + ΔEΔt ≥ ħ/2 + ħ/2 = ħ

Interesting, isn’t it? It explains that 1/2 factor which troubled us when playing with the de Broglie relations.

Let’s now think about our natural units again—about lP, and tP in particular. As mentioned above, we’ll want to think of them as the smallest distance and time units possible: so small, in fact, that both distance and time become countable variables, so we count x and t as 0, 1, 2, 3 etcetera. We may then imagine that the uncertainty in x and t is of the order of one unit only, so we write Δx = lP and Δt = tP. So we can now re-write the uncertainty relations as:

  • Δp·lP = ħ/2
  • ΔE·tP = ħ/2

Hey! Wait a minute! Do we have a solution for the value of lP and tP here? What if we equate the natural energy and momentum units to ΔE and Δp here? Well… Let’s try it. First note that we may think of the uncertainty in t, or in x, as being equal to plus or minus one unit, i.e. ±1. So the uncertainty is two units really. [Frankly, I just want to get rid of that 1/2 factor here.] Hence, we can re-write the ΔpΔx = ΔEΔt = ħ/2 equations as:

  • ΔpΔx = pPlP = FP∙tPlP = ħ
  • ΔEΔt = EP∙tP = FPlP∙tP = ħ

Hmm… What can we do with this? Nothing much, unfortunately. We’ve got the same problem: we need a value for FP (or for pP, or for EP) to get some specific value for lP and tP, so we’re stuck once again. We have three variables and two equations only, so we have no specific value for either of them. 😦

What to do? Well… I will give you the answer now—the answer you’ve been waiting for, really—but not the technicalities of it. There’s a thing called the Schwarzschild radius, aka as the gravitational radius. Let’s analyze it.

The Schwarzschild radius and the Planck length

The Schwarzschild radius is just the radius of a black hole. Its formal definition is the following: it is the radius of a sphere such that, if all the mass of an object were to be compressed within that sphere, the escape velocity from the surface of the sphere would equal the speed of light (c). The formula for the Schwartzschild radius is the following:

RS = 2m·G/c2

G is the gravitational constant here: G ≈ 6.674×10−11 N⋅m2/kg2. [Note that Newton’s F = m·Law tells us that 1 kg = 1 N·s2/m, as we’ll need to substitute units later.]

But what is the mass (m) in that RS = 2m·G/c2 equation? Using equivalent time and distance units (so = 1), we wrote the following for a zero-mass particle and for a photon respectively:

  • E = m = p = ħ/2 (zero-mass particle)
  • E = m = p = ħ (photon)

How can a zero-mass particle, or a photon, have some mass? Well… Because it moves at the speed of light. I’ve talked about that before, so I’ll just refer you to my post on that. Of course, the dimension of the right-hand side of these equations (i.e. ħ/2 or ħ) symbol has to be the same as the dimension on the left-hand side, so the ‘ħ’ in the E = ħ equation (or E = ħ/2 equation) is a different ‘ħ’ in the p = ħ equation (or p = ħ/2 equation). So we must be careful here. Let’s write it all out, so as to remind ourselves of the dimensions involved:

  • E [N·m] = ħ [N·m·s/s] = EP = FPlP∙tP/tP
  • p [N·s] = ħ [N·m·s/m] = pP = FPlP∙tP/lP

Now, let’s check this by cheating. I’ll just give you the numerical values—even if we’re not supposed to know them at this point—so you can see I am not writing nonsense here:

  • EP = 1.0545718×10−34 N·m·s/(5.391×10−44 s) = (1.21×1044 N)·(1.6162×10−35 m) = 1.9561×10N·m
  • pP =1.0545718×10−34 N·m·s/(1.6162×10−35 m) = (1.21×1044 N)·(5.391×10−44 s) = 6.52485 N·s

You can google the Planck units, and you’ll see I am not taking you for a ride here. 🙂

The associated Planck mass is mP = EP/c2 = 1.9561×10N·m/(2.998×10m/s)2 = 2.17651×108 N·s2/m = 2.17651×108 kg. So let’s plug that value into RS = 2m·G/cequation. We get:

RS = 2m·G/c= [(2.17651×108 kg)·(6.674×10−11 N⋅m2/kg)/(8.988×1016 m2·s−2)

= 1.6162×1035 kg·N⋅m2·kg−2·m2·s−2 = 1.6162×1035 kg·N⋅m2·kg−2·m2·s−2 = 1.6162×1035 m = lP

Bingo! You can look it up: 1.6162×1035 m is the Planck length indeed, so the Schwarzschild radius is the Planck length. We can now easily calculate the other Planck units:

  • t= lP/c = 1.6162×1035 m/(2.998×10m/s) = 5.391×10−44 s
  • F= ħ/(tPlP)= (1.0545718×10−34 N·m·s)/[(1.6162×1035 m)·(5.391×10−44 s) = 1.21×10−44 N

Bingo again! 🙂

[…] But… Well… Look at this: we’ve been cheating all the way. First, we just gave you that formula for the Schwarzschild radius. It looks like an easy formula but its derivation involves a profound knowledge of general relativity theory. So we’d need to learn about tensors and what have you. The formula is, in effect, a solution to what is known as Einstein’s field equations, and that’s pretty complicated stuff.

However, my crime is much worse than that: I also gave you those numerical values for the Planck energy and momentum, rather than calculating them. I just couldn’t calculate them with the knowledge we have so far. When everything is said and done, we have more than three unknowns. We’ve got five in total, including the Planck charge (qP) and, hence, we need five equations. Again, I’ll just copy them from Wikipedia, because… Well… What we’re discussing here is way beyond the undergraduate physics stuff that we’ve been presenting so far. The equations are the following. Just have a look at them and move on. 🙂

Planck units

Finally, I should note one more thing: I did not use 2m but m in Schwarzschild’s formula. Why? Well… I have no good answer to that. I did it to ensure I got the result we wanted to get. It’s that 1/2 factor again. In fact, the E = m = p = ħ/2 is the correct formula to use, and all would come out alright if we did that and defined the magnitude of the uncertainty as one unit only, but so we used the E = m = p = ħ formula instead, i.e. the equation that’s associated with a photon. You can re-do the calculations as an exercise: you’ll see it comes out alright.

Just to make things somewhat more real, let me note that the Planck energy is very substantial: 1.9561×10N·m ≈ 2×10J is equivalent to the energy that you’d get out of burning 60 liters of gasoline—or the mileage you’d get out of 16 gallons of fuel! In short, it’s huge,  and so we’re packing that into a unimaginably small space. To understand how that works, you can think of the E = h∙f ⇔ h = E/f relation once more. The h = E/f ratio implies that energy and frequency are directly proportional to each other, with h the coefficient of proportionality. Shortening the wavelength, amounts to increasing the frequency and, hence, the energy. So, as you think of our cycle becoming smaller and smaller, until it becomes the smallest cycle possible, you should think of the frequency becoming unimaginably large. Indeed, as I explained in one of my other posts on physical constants, we’re talking the the 1043 Hz scale here. However, we can always choose our time unit such that we measure the frequency as one cycle per time unit. Because the energy per cycle remains the same, it means the quantum of action (ħ = FPlP∙tP) expresses itself over extremely short time spans, which means the EP = FPlP product becomes huge, as we’ve shown above. The rest of the story is the same: gravity comes into play, and so our little blob in spacetime becomes a tiny black hole. Again, we should think of both space and time: they are joined in ‘some kind of union’ here, indeed, as they’re intimately connected through the wavefunction, which travels at the speed of light.

The wavefunction as an oscillation in and of spacetime

OK. Now I am going to present the big idea I started with. Let me first ask you a question: when thinking about the Planck-Einstein relation (I am talking about the E = ħ∙ω relation for a photon here, rather than the equivalent de Broglie equation for a matter-particle), aren’t you struck by the fact that the energy of a photon depends on the frequency of the electromagnetic wave only? I mean… It does not depend on its amplitude. The amplitude is mentioned nowhere. The amplitude is fixed, somehow—or considered to be fixed.

Isn’t that strange? I mean… For any other physical wave, the energy would not only depend on the frequency but also on the amplitude of the wave. For a photon, however, it’s just the frequency that counts. Light of the same frequency but higher intensity (read: more energy) is not a photon with higher amplitude, but just more photons. So it’s the photons that add up somehow, and so that explains the amplitude of the electric and magnetic field vectors (i.e. E and B) and, hence, the intensity of the light. However, every photon considered separately has the same amplitude apparently. We can only increase its energy by increasing the frequency. In short, ω is the only variable here.

Let’s look at that angular frequency once more. As you know, it’s expressed in radians per second but, if you multiply ω by 2π, you get the frequency you’re probably more acquainted with: f = 2πω = f cycles per second. The Planck-Einstein relation is then written as E = h∙f. That’s easy enough. But what if we’d change the time unit here? For example, what if our time unit becomes the time that’s needed for a photon to travel one meter? Let’s examine it.

Let’s denote that time unit by tm, so we write: 1 tm = 1/c s ⇔ tm1 = c s1, with c ≈ 3×108. The frequency, as measured using our new time unit, changes, obviously: we have to divide its former value by c now. So, using our little subscript once more, we could write: fm = f/c. [Why? Just add the dimension to make things more explicit: f s1 = f/c tm1 = f/c tm1.] But the energy of the photon should not depend on our time unit, should it?

Don’t worry. It doesn’t: the numerical value of Planck’s constant (h) would also change, as we’d replace the second in its dimension (N∙m∙s) by c times our new time unit tm. However, Planck’s constant remains what it is: some physical constant. It does not depend on our measurement units: we can use the SI units, or the Planck units (FP, lP, and tP), or whatever unit you can think of. It doesn’t matter: h (or ħ = h/2π) is what is—it’s the quantum of action, and so that’s a physical constant (as opposed to a mathematical constant) that’s associated with one cycle.

Now, I said we do not associate the wavefunction of a photon with an amplitude, but we do associate it with a wavelength. We do so using the standard formula for the velocity of a wave: c = f∙λ ⇔ λ = c/f. We can also write this using the angular frequency and the wavenumber: c = ω/k, with k = 2π/λ. We can double-check this, because we know that, for a photon, the following relation holds: E/p = c. Hence, using the E = ħ∙ω and p = ħ∙k relations, we get: (ħ∙ω)/(ħ∙k) = ω/k = c. So we have options here: h can express itself over a really long wavelength, or it can do so over an extremely short wavelength. We re-write p = ħ∙k as p = E/c = ħ∙2π/λ = h/λ ⇔ E = h∙c/λ ⇔ h∙c = E∙λ. We know this relationship: the energy and the wavelength of a photon (or an electromagnetic wave) are inversely proportional to each other.

Once again, we may want to think of the shortest wavelength possible. As λ gets a zillion times smaller, E gets a zillion times bigger. Is there a limit? There is. As I mentioned above, the gravitational force comes into play here: there’s a limit to the amount of energy we can pack into a tiny space. If we pack an awful lot of energy into a really tiny speck of space, then we get a black hole. In practical terms, that implies our photon can’t travel, as it can’t escape from the black hole it creates. That’s what that calculation of the Schwarzschild radius was all about.

We can—in fact, we should—now apply the same reasoning to the matter-wave. Instead of a photon, we should try to think of a zero-mass matter-particle. You’ll say: that’s a contradiction. Matter-particles – as opposed to force-carrying particles, like photons (or bosons in general) – must have some rest mass, so they can’t be massless. Well… Yes. You’re right. But we can throw the rest mass in later. I first want to focus on the abstract principles, i.e. the propagation mechanism of the matter-wave.

Using natural units, we know our particle will move in spacetime with velocity Δx/Δt = 1/1 = 1. Of course, it has to have some energy to move, or some momentum. We also showed that, if it’s massless, and the elementary wavefunction is ei[(p/ħ)x – (E/ħ)t), then we know the energy, and the momentum, has to be equal to ħ/2. Where does it get that energy, or momentum? Not sure. I like to think it borrows it from spacetime, as it breaks some potential barrier between those two points, and then it gives it back. Or, if it’s at point x = t = 0, then perhaps it gets it from some other massless particle moving from x = t = −1. In both cases, we’d like to think our particle keeps moving. So if the first description (borrowing) is correct, it needs to keep borrowing and returning energy in some kind of interaction with spacetime itself. If it’s the second description, it’s more like spacetime bumping itself forward.

In both cases, however, we’re actually trying to visualize (or should I say: imagine?) some oscillation of spacetime itself, as opposed to an oscillation in spacetime.

Huh? Yes. The idea is the following here: we like to think of the wavefunction as the dependent variable: both its real as well as its imaginary part are a function of x and t, indeed. But what if we’d think of x and t as dependent variables? In that case, the real and imaginary part of the wavefunction would be the independent variables. It’s just a matter of perspective. We sort of mirror our function: we switch its domain for its range, and its range for its domain, as shown below. It all makes sense, doesn’t it? Space and time appear as separate dimensions to us, but they’re intimately connected through c, ħ and the other fundamental physical constants. Likewise, the real and imaginary part of the wavefunction appear as separate dimensions, but they’re intimately connected through π and Euler’s number, i.e. through mathematical constants. That cannot be a coincidence: the mathematical and physical ‘space’ reflect each other through the wavefunction, just like the domain and range of a function reflect each other through that function. So physics and math must meet in some kind of union—at least in our mind, they do!

dependent independent

So, yes, we can—and probably should—be looking at the wavefunction as an oscillation of spacetime, rather than as an oscillation in spacetime only. As mentioned in my introduction, I’ll need to study general relativity theory—and very much in depth—to convincingly prove that point, but I am sure it can be done.

You’ll probably think I am arrogant when saying that—and I probably am—but then I am very much emboldened by the fact some nuclear scientist told me a photon doesn’t have any wavefunction: it’s just those E and B vectors, he told me—and then I found out he was dead wrong, as I showed in my previous post! So I’d rather think more independently now. I’ll keep you guys posted on progress—but it will probably take a while to figure it all out. In the meanwhile, please do let me know your ideas on this. 🙂

Let me wrap up this little excursion with two small notes:

  • We have this E/c = p relation. The mass-energy equivalence relation implies momentum must also have an equivalent mass. If E = m∙c2, then p = m∙c ⇔ m = p/c. It’s obvious, but I just thought it would be useful to highlight this.
  • When we studied the ammonia molecule as a two-state system, our space was not a continuum: we allowed just two positions—two points in space, which we defined with respect to the system. So x was a discrete variable. We assumed time to be continuous, however, and so we got those nice sinusoids as a solution to our set of Hamiltonian equations. However, if we look at space as being discrete, or countable, we should probably think of time as being countable as well. So we should, perhaps, think of a particle being at point x = t = 0 first, and, then, being at point x = t = 1. Instead of the nice sinusoids, we get some boxcar function, as illustrated below, but probably varying between 0 and 1—or whatever other normalized values. You get the idea, I hope. 🙂

boxcar1

Post Scriptum on the Principle of Least Action: As noted above, the Principle of Least Action is not very intuitive, even if Feynman’s exposé of it is not as impregnable as it may look at first. To put it simply, the Principle of Least Action says that the average kinetic energy less the average potential energy is as little as possible for the path of an object going from one point to another. So we have a path or line integral here. In a gravitation field, that integral is the following:

Least action

The integral is not all that important. Just note its dimension is the dimension of action indeed, as we multiply energy (the integrand) with time (dt). We can use the Principle of Least Action to re-state Newton’s Law, or whatever other classical law. Among other things, we’ll find that, in the absence of any potential, the trajectory of a particle will just be some straight line.

In quantum mechanics, however, we have uncertainty, as expressed in the ΔpΔx ≥ ħ/2 and ΔEΔt ≥ ħ/2 relations. Now, that uncertainty may express itself in time, or in distance, or in both. That’s where things become tricky. 🙂 I’ve written on this before, but let me copy Feynman himself here, with a more exact explanation of what’s happening (just click on the text to enlarge):

Feynman

The ‘student’ he speaks of above, is himself, of course. 🙂

Too complicated? Well… Never mind. I’ll come back to it later. 🙂

Re-visiting the speed of light, Planck’s constant, and the fine-structure constant

Note: I have published a paper that is very coherent and fully explains what the fine-structure constant actually is. There is nothing magical about it. It’s not some God-given number. It’s a scaling constant – and then some more. But not God-given. Check it out: The Meaning of the Fine-Structure Constant. No ambiguity. No hocus-pocus.

Jean Louis Van Belle, 23 December 2018

Original post:

A brother of mine sent me a link to an article he liked. Now, because we share some interest in physics and math and other stuff, I looked at it and…

Well… I was disappointed. Despite the impressive credentials of its author – a retired physics professor – it was very poorly written. It made me realize how much badly written stuff is around, and I am glad I am no longer wasting my time on it. However, I do owe my brother some explanation of (a) why I think it was bad, and of (b) what, in my humble opinion, he should be wasting his time on. 🙂 So what it is all about?

The article talks about physicists deriving the speed of light from “the electromagnetic properties of the quantum vacuum.” Now, it’s the term ‘quantum‘, in ‘quantum vacuum’, that made me read the article.

Indeed, deriving the theoretical speed of light in empty space from the properties of the classical vacuum – aka empty space – is a piece of cake: it was done by Maxwell himself as he was figuring out his equations back in the 1850s (see my post on Maxwell’s equations and the speed of light). And then he compared it to the measured value, and he saw it was right on the mark. Therefore, saying that the speed of light is a property of the vacuum, or of empty space, is like a tautology: we may just as well put it the other way around, and say that it’s the speed of light that defines the (properties of the) vacuum!

Indeed, as I’ll explain in a moment: the speed of light determines both the electric as well as the magnetic constants μand ε0, which are the (magnetic) permeability and the (electric) permittivity of the vacuum respectively. Both constants depend on the units we are working with (i.e. the units for electric charge, for distance, for time and for force – or for inertia, if you want, because force is defined in terms of overcoming inertia), but so they are just proportionality coefficients in Maxwell’s equations. So once we decide what units to use in Maxwell’s equations, then μand ε0 are just proportionality coefficients which we get from c. So they are not separate constants really – I mean, they are not separate from c – and all of the ‘properties’ of the vacuum, including these constants, are in Maxwell’s equations.

In fact, when Maxwell compared the theoretical value of c with its presumed actual value, he didn’t compare c‘s theoretical value with the speed of light as measured by astronomers (like that 17th century Ole Roemer, to which our professor refers: he had a first go at it by suggesting some specific value for it based on his observations of the timing of the eclipses of one of Jupiter’s moons), but with c‘s value as calculated from the experimental values of μand ε0! So he knew very well what he was looking at. In fact, to drive home the point, it may also be useful to note that the Michelson-Morley experiment – which accurately measured the speed of light – was done some thirty years later. So Maxwell had already left this world by then—very much in peace, because he had solved the mystery all 19th century physicists wanted to solve through his great unification: his set of equations covers it all, indeed: electricity, magnetism, light, and even relativity!

I think the article my brother liked so much does a very lousy job in pointing all of that out, but that’s not why I wouldn’t recommend it. It got my attention because I wondered why one would try to derive the speed of light from the properties of the quantum vacuum. In fact, to be precise, I hoped the article would tell me what the quantum vacuum actually is. Indeed, as far as I know, there’s only one vacuum—one ’empty space’: empty is empty, isn’t it? 🙂 So I wondered: do we have a ‘quantum’ vacuum? And, if so, what is it, really?

Now, that is where the article is really disappointing, I think. The professor drops a few names (like the Max Planck Institute, the University of Paris-Sud, etcetera), and then, promisingly, mentions ‘fleeting excitations of the quantum vacuum’ and ‘virtual pairs of particles’, but then he basically stops talking about quantum physics. Instead, he wanders off to share some philosophical thoughts on the fundamental physical constants. What makes it all worse is that even those thoughts on the ‘essential’ constants are quite off the mark.

So… This post is just a ‘quick and dirty’ thing for my brother which, I hope, will be somewhat more thought-provoking than that article. More importantly, I hope that my thoughts will encourage him to try to grind through better stuff.

On Maxwell’s equations and the properties of empty space

Let me first say something about the speed of light indeed. Maxwell’s four equations may look fairly simple, but that’s only until one starts unpacking all those differential vector equations, and it’s only when going through all of their consequences that one starts appreciating their deep mathematical structure. Let me quickly copy how another blogger jotted them down: 🙂

god-said-maxwell-equation

As I showed in my above-mentioned post, the speed of light (i.e. the speed with which an electromagnetic pulse or wave travels through space) is just one of the many consequences of the mathematical structure of Maxwell’s set of equations. As such, the speed of light is a direct consequence of the ‘condition’, or the properties, of the vacuum indeed, as Maxwell suggested when he wrote that “we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena”.

Of course, while Maxwell still suggests light needs some ‘medium’ here – so that’s a reference to the infamous aether theory – we now know that’s because he was a 19th century scientist, and so we’ve done away with the aether concept (because it’s a redundant hypothesis), and so now we also know there’s absolutely no reason whatsoever to try to “avoid the inference.” 🙂 It’s all OK, indeed: light is some kind of “transverse undulation” of… Well… Of what?

We analyze light as traveling fields, represented by two vectors, E and B, whose direction and magnitude varies both in space as well as in time. E and B are field vectors, and represent the electric and magnetic field respectively. An equivalent formulation – more or less, that is (see my post on the Liénard-Wiechert potentials) – for Maxwell’s equations when only one (moving) charge is involved is:

E

B

This re-formulation, which is Feynman’s preferred formula for electromagnetic radiation, is interesting in a number of ways. It clearly shows that, while we analyze the electric and magnetic field as separate mathematical entities, they’re one and the same phenomenon really, as evidenced by the B = –er×E/c equation, which tells us the magnetic field from a single moving charge is always normal (i.e. perpendicular) to the electric field vector, and also that B‘s magnitude is 1/times the magnitude of E, so |B| = B = |E|/c = E/c. In short, B is fully determined by E, or vice versa: if we have one of the two fields, we have the other, so they’re ‘one and the same thing’ really—not in a mathematical sense, but in a real sense.

Also note that E and B‘s magnitude is just the same if we’re using natural units, so if we equate c with 1. Finally, as I pointed out in my post on the relativity of electromagnetic fields, if we would switch from one reference frame to another, we’ll have a different mix of E and B, but that different mix obviously describes the same physical reality. More in particular, if we’d be moving with the charges, the magnetic field sort of disappears to re-appear as an electric field. So the Lorentz force F = Felectric + Fmagnetic = qE + qv×B is one force really, and its ‘electric’ and ‘magnetic’ component appear the way they appear in our reference frame only. In some other reference frame, we’d have the same force, but its components would look different, even if they, obviously, would and should add up to the same. [Well… Yes and no… You know there’s relativistic corrections to be made to the forces to, but that’s a minor point, really. The force surely doesn’t disappear!]

All of this reinforces what you know already: electricity and magnetism are part and parcel of one and the same phenomenon, the electromagnetic force field, and Maxwell’s equations are the most elegant way of ‘cutting it up’. Why elegant? Well… Click the Occam tab. 🙂

Now, after having praised Maxwell once more, I must say that Feynman’s equations above have another advantage. In Maxwell’s equations, we see two constants, the electric and magnetic constant (denoted by μand ε0 respectively), and Maxwell’s equations imply that the product of the electric and magnetic constant is the reciprocal of c2: μ0·ε= 1/c2. So here we see εand only, so no μ0, so that makes it even more obvious that the magnetic and electric constant are related one to another through c.

[…] Let me digress briefly: why do we have c2 in μ0·ε= 1/c2, instead of just c? That’s related to the relativistic nature of the magnetic force: think about that B = E/c relation. Or, better still, think about the Lorentz equation F = Felectric + Fmagnetic = qE + qv×B = q[E + (v/c)×(E×er)]: the 1/c factor is there because the magnetic force involves some velocity, and any velocity is always relative—and here I don’t mean relative to the frame of reference but relative to the (absolute) speed of light! Indeed, it’s the v/c ratio (usually denoted by β = v/c) that enters all relativistic formulas. So the left-hand side of the μ0·ε= 1/c2 equation is best written as (1/c)·(1/c), with one of the two 1/c factors accounting for the fact that the ‘magnetic’ force is a relativistic effect of the ‘electric’ force, really, and the other 1/c factor giving us the proper relationship between the magnetic and the electric constant. To drive home the point, I invite you to think about the following:

  • μ0 is expressed in (V·s)/(A·m), while εis expressed in (A·s)/(V·m), so the dimension in which the μ0·εproduct is expressed is [(V·s)/(A·m)]·[(A·s)/(V·m)] = s2/m2, so that’s the dimension of 1/c2.
  • Now, this dimensional analysis makes it clear that we can sort of distribute 1/c2 over the two constants. All it takes is re-defining the fundamental units we use to calculate stuff, i.e. the units for electric charge, for distance, for time and for force – or for inertia, as explained above. But so we could, if we wanted, equate both μ0 as well as εwith 1/c.
  • Now, if we would then equate c with 1, we’d have μ0 = ε= c = 1. We’d have to define our units for electric charge, for distance, for time and for force accordingly, but it could be done, and then we could re-write Maxwell’s set of equations using these ‘natural’ units.

In any case, the nitty-gritty here is less important: the point is that μand εare also related through the speed of light and, hence, they are ‘properties’ of the vacuum as well. [I may add that this is quite obvious if you look at their definition, but we’re approaching the matter from another angle here.]

In any case, we’re done with this. On to the next!

On quantum oscillations, Planck’s constant, and Planck units 

The second thought I want to develop is about the mentioned quantum oscillation. What is it? Or what could it be? An electromagnetic wave is caused by a moving electric charge. What kind of movement? Whatever: the charge could move up or down, or it could just spin around some axis—whatever, really. For example, if it spins around some axis, it will have a magnetic moment and, hence, the field is essentially magnetic, but then, again, E and B are related and so it doesn’t really matter if the first cause is magnetic or electric: that’s just our way of looking at the world: in another reference frame, one that’s moving with the charges, the field would essential be electric. So the motion can be anything: linear, rotational, or non-linear in some irregular way. It doesn’t matter: any motion can always be analyzed as the sum of a number of ‘ideal’ motions. So let’s assume we have some elementary charge in space, and it moves and so it emits some electromagnetic radiation.

So now we need to think about that oscillation. The key question is: how small can it be? Indeed, in one of my previous posts, I tried to explain some of the thinking behind the idea of the ‘Great Desert’, as physicists call it. The whole idea is based on our thinking about the limit: what is the smallest wavelength that still makes sense? So let’s pick up that conversation once again.

The Great Desert lies between the 1032 and 1043 Hz scale. 1032 Hz corresponds to a photon energy of Eγ = h·f = (4×10−15 eV·s)·(1032 Hz) = 4×1017 eV = 400,000 tera-electronvolt (1 TeV = 1012 eV). I use the γ (gamma) subscript in my Eγ symbol for two reasons: (1) to make it clear that I am not talking the electric field E here but energy, and (2) to make it clear we are talking ultra-high-energy gamma-rays here.

In fact, γ-rays of this frequency and energy are theoretical only. Ultra-high-energy gamma-rays are defined as rays with photon energies higher than 100 TeV, which is the upper limit for very-high-energy gamma-rays, which have been observed as part of the radiation emitted by so-called gamma-ray bursts (GRBs): flashes associated with extremely energetic explosions in distant galaxies. Wikipedia refers to them as the ‘brightest’ electromagnetic events know to occur in the Universe. These rays are not to be confused with cosmic rays, which consist of high-energy protons and atomic nuclei stripped of their electron shells. Cosmic rays aren’t rays really and, because they consist of particles with a considerable rest mass, their energy is even higher. The so-called Oh-My-God particle, for example, which is the most energetic particle ever detected, had an energy of 3×1020 eV, i.e. 300 million TeV. But it’s not a photon: its energy is largely kinetic energy, with the rest mass m0 counting for a lot in the m in the E = m·c2 formula. To be precise: the mentioned particle was thought to be an iron nucleus, and it packed the equivalent energy of a baseball traveling at 100 km/h! 

But let me refer you to another source for a good discussion on these high-energy particles, so I can get get back to the energy of electromagnetic radiation. When I talked about the Great Desert in that post, I did so using the Planck-Einstein relation (E = h·f), which embodies the idea of the photon being valid always and everywhere and, importantly, at every scale. I also discussed the Great Desert using real-life light being emitted by real-life atomic oscillators. Hence, I may have given the (wrong) impression that the idea of a photon as a ‘wave train’ is inextricably linked with these real-life atomic oscillators, i.e. to electrons going from one energy level to the next in some atom. Let’s explore these assumptions somewhat more.

Let’s start with the second point. Electromagnetic radiation is emitted by any accelerating electric charge, so the atomic oscillator model is an assumption that should not be essential. And it isn’t. For example, whatever is left of the nucleus after alpha or beta decay (i.e. a nuclear decay process resulting in the emission of an α- or β-particle) it likely to be in an excited state, and likely to emit a gamma-ray for about 10−12 seconds, so that’s a burst that’s about 10,000 times shorter than the 10–8 seconds it takes for the energy of a radiating atom to die out. [As for the calculation of that 10–8 sec decay time – so that’s like 10 nanoseconds – I’ve talked about this before but it’s probably better to refer you to the source, i.e. one of Feynman’s Lectures.]

However, what we’re interested in is not the energy of the photon, but the energy of one cycle. In other words, we’re not thinking of the photon as some wave train here, but what we’re thinking about is the energy that’s packed into a space corresponding to one wavelength. What can we say about that?

As you know, that energy will depend both on the amplitude of the electromagnetic wave as well as its frequency. To be precise, the energy is (1) proportional to the square of the amplitude, and (2) proportional to the frequency. Let’s look at the first proportionality relation. It can be written in a number of ways, but one way of doing it is stating the following: if we know the electric field, then the amount of energy that passes per square meter per second through a surface that is normal to the direction in which the radiation is going (which we’ll denote by S – the s from surface – in the formula below), must be proportional to the average of the square of the field. So we write S ∝ 〈E2〉, and so we should think about the constant of proportionality now. Now, let’s not get into the nitty-gritty, and so I’ll just refer to Feynman for the derivation of the formula below:

S = ε0c·〈E2

So the constant of proportionality is ε0c. [Note that, in light of what we wrote above, we can also write this as S = (1/μ0·c)·〈(c·B)2〉 = (c0)·〈B2〉, so that underlines once again that we’re talking one electromagnetic phenomenon only really.] So that’s a nice and rather intuitive result in light of all of the other formulas we’ve been jotting down. However, it is a ‘wave’ perspective. The ‘photon’ perspective assumes that, somehow, the amplitude is given and, therefore, the Planck-Einstein relation only captures the frequency variable: Eγ = h·f.

Indeed, ‘more energy’ in the ‘wave’ perspective basically means ‘more photons’, but photons are photons: they have a definite frequency and a definite energy, and both are given by that Planck-Einstein relation. So let’s look at that relation by doing a bit of dimensional analysis:

  • Energy is measured in electronvolt or, using SI units, joule: 1 eV ≈ 1.6×10−19 J. Energy is force times distance: 1 joule = 1 newton·meter, which means that a larger force over a shorter distance yields the same energy as a smaller force over a longer distance. The oscillations we’re talking about here involve very tiny distances obviously. But the principle is the same: we’re talking some moving charge q, and the power – which is the time rate of change of the energy – that goes in or out at any point of time is equal to dW/dt = F·v, with W the work that’s being done by the charge as it emits radiation.
  • I would also like to add that, as you know, forces are related to the inertia of things. Newton’s Law basically defines a force as that what causes a mass to accelerate: F = m·a = m·(dv/dt) = d(m·v)/dt = dp/dt, with p the momentum of the object that’s involved. When charges are involved, we’ve got the same thing: a potential difference will cause some current to change, and one of the equivalents of Newton’s Law F = m·a = m·(dv/dt) in electromagnetism is V = L·(dI/dt). [I am just saying this so you get a better ‘feel’ for what’s going on.]
  • Planck’s constant is measured in electronvolt·seconds (eV·s) or in, using SI units, in joule·seconds (J·s), so its dimension is that of (physical) action, which is energy times time: [energy]·[time]. Again, a lot of energy during a short time yields the same energy as less energy over a longer time. [Again, I am just saying this so you get a better ‘feel’ for these dimensions.]
  • The frequency f is the number of cycles per time unit, so that’s expressed per second, i.e. in herz (Hz) = 1/second = s−1.

So… Well… It all makes sense: [x joule] = [6.626×10−34 joule]·[1 second]×[f cycles]/[1 second]. But let’s try to deepen our understanding even more: what’s the Planck-Einstein relation really about?

To answer that question, let’s think some more about the wave function. As you know, it’s customary to express the frequency as an angular frequency ω, as used in the wave function A(x, t) = A0·sin(kx − ωt). The angular frequency is the frequency expressed in radians per second. That’s because we need an angle in our wave function, and so we need to relate x and t to some angle. The way to think about this is as follows: one cycle takes a time T (i.e. the period of the wave) which is equal to T = 1/f. Yes: one second divided by the number of cycles per second gives you the time that’s needed for one cycle. One cycle is also equivalent to our argument ωt going around the full circle (i.e. 2π), so we write:  ω·T = 2π and, therefore:

ω = 2π/T = 2π·f

Now we’re ready to play with the Planck-Einstein relation. We know it gives us the energy of one photon really, but what if we re-write our equation Eγ = h·f as Eγ/f = h? The dimensions in this equation are:

[x joule]·[1 second]/[cyles] = [6.626×10−34 joule]·[1 second]

⇔ = 6.626×10−34 joule per cycle

So that means that the energy per cycle is equal to 6.626×10−34 joule, i.e. the value of Planck’s constant.

Let me rephrase truly amazing result, so you appreciate it—perhaps: regardless of the frequency of the light (or our electromagnetic wave, in general) involved, the energy per cycle, i.e. per wavelength or per period, is always equal to 6.626×10−34 joule or, using the electronvolt as the unit, 4.135667662×10−15 eV. So, in case you wondered, that is the true meaning of Planck’s constant!

Now, if we have the frequency f, we also have the wavelength λ, because the velocity of the wave is the frequency times the wavelength: = λ·f and, therefore, λ = c/f. So if we increase the frequency, the wavelength becomes smaller and smaller, and so we’re packing the same amount of energy – admittedly, 4.135667662×10−15 eV is a very tiny amount of energy – into a space that becomes smaller and smaller. Well… What’s tiny, and what’s small? All is relative, of course. 🙂 So that’s where the Planck scale comes in. If we pack that amount of energy into some tiny little space of the Planck dimension, i.e. a ‘length’ of 1.6162×10−35 m, then it becomes a tiny black hole, and it’s hard to think about how that would work.

[…] Let me make a small digression here. I said it’s hard to think about black holes but, of course, it’s not because it’s ‘hard’ that we shouldn’t try it. So let me just mention a few basic facts. For starters, black holes do emit radiation! So they swallow stuff, but they also spit stuff out. More in particular, there is the so-called Hawking radiation, as Roger Penrose and Stephen Hawking discovered.

Let me quickly make a few remarks on that: Hawking radiation is basically a form of blackbody radiation, so all frequencies are there, as shown below: the distribution of the various frequencies depends on the temperature of the black body, i.e. the black hole in this case. [The black curve is the curve that Lord Rayleigh and Sir James Jeans derived in the late 19th century, using classical theory only, so that’s the one that does not correspond to experimental fact, and which led Max Planck to become the ‘reluctant’ father of quantum mechanics. In any case, that’s history and so I shouldn’t dwell on this.]

600px-Black_body

The interesting thing about blackbody radiation, including Hawking radiation, is that it reduces energy and, hence, the equivalent mass of our blackbody. So Hawking radiation reduces the mass and energy of black holes and is therefore also known as black hole evaporation. So black holes that lose more mass than they gain through other means are expected to shrink and ultimately vanish. Therefore, there’s all kind of theories that say why micro black holes, like that Planck scale black hole we’re thinking of right now, should be much larger net emitters of radiation than large black holes and, hence, whey they should shrink and dissipate faster.

Hmm… Interesting… What do we do with all of this information? Well… Let’s think about it as we continue our trek on this long journey to reality over the next year or, more probably, years (plural). 🙂

The key lesson here is that space and time are intimately related because of the idea of movement, i.e. the idea of something having some velocity, and that it’s not so easy to separate the dimensions of time and distance in any hard and fast way. As energy scales become larger and, therefore, our natural time and distance units become smaller and smaller, it’s the energy concept that comes to the fore. It sort of ‘swallows’ all other dimensions, and it does lead to limiting situations which are hard to imagine. Of course, that just underscores the underlying unity of Nature, and the mysteries involved.

So… To relate all of this back to the story that our professor is trying to tell, it’s a simple story really. He’s talking about two fundamental constants basically, c and h, pointing out that c is a property of empty space, and h is related to something doing something. Well… OK. That’s really nothing new, and surely not ground-breaking research. 🙂

Now, let me finish my thoughts on all of the above by making one more remark. If you’ve read a thing or two about this – which you surely have – you’ll probably say: this is not how people usually explain it. That’s true, they don’t. Anything I’ve seen about this just associates the 1043 Hz scale with the 1028 eV energy scale, using the same Planck-Einstein relation. For example, the Wikipedia article on micro black holes writes that “the minimum energy of a microscopic black hole is 1019 GeV [i.e. 1028 eV], which would have to be condensed into a region on the order of the Planck length.” So that’s wrong. I want to emphasize this point because I’ve been led astray by it for years. It’s not the total photon energy, but the energy per cycle that counts. Having said that, it is correct, however, and easy to verify, that the 1043 Hz scale corresponds to a wavelength of the Planck scale: λ = c/= (3×10m/s)/(1043 s−1) = 3×10−35 m. The confusion between the photon energy and the energy per wavelength arises because of the idea of a photon: it travels at the speed of light and, hence, because of the relativistic length contraction effect, it is said to be point-like, to have no dimension whatsoever. So that’s why we think of packing all of its energy in some infinitesimally small place. But you shouldn’t think like that. The photon is dimensionless in our reference frame: in its own ‘world’, it is spread out, so it is a wave train. And it’s in its ‘own world’ that the contradictions start… 🙂

OK. Done!

My third and final point is about what our professor writes on the fundamental physical constants, and more in particular on what he writes on the fine-structure constant. In fact, I could just refer you to my own post on it, but that’s probably a bit too easy for me and a bit difficult for you 🙂 so let me summarize that post and tell you what you need to know about it.

The fine-structure constant

The fine-structure constant α is a dimensionless constant which also illustrates the underlying unity of Nature, but in a way that’s much more fascinating than the two or three things the professor mentions. Indeed, it’s quite incredible how this number (α = 0.00729735…, but you’ll usually see it written as its reciprocal, which is a number that’s close to 137.036…) links charge with the relative speeds, radii, and the mass of fundamental particles and, therefore, how this number also these concepts with each other. And, yes, the fact that it is, effectively, dimensionless, unlike h or c, makes it even more special. Let me quickly sum up what the very same number α all stands for:

(1) α is the square of the electron charge expressed in Planck units: α = eP2.

(2) α is the square root of the ratio of (a) the classical electron radius and (b) the Bohr radius: α = √(re /r). You’ll see this more often written as re = α2r. Also note that this is an equation that does not depend on the units, in contrast to equation 1 (above), and 4 and 5 (below), which require you to switch to Planck units. It’s the square of a ratio and, hence, the units don’t matter. They fall away.

(3) α is the (relative) speed of an electron: α = v/c. [The relative speed is the speed as measured against the speed of light. Note that the ‘natural’ unit of speed in the Planck system of units is equal to c. Indeed, if you divide one Planck length by one Planck time unit, you get (1.616×10−35 m)/(5.391×10−44 s) = m/s. However, this is another equation, just like (2), that does not depend on the units: we can express v and c in whatever unit we want, as long we’re consistent and express both in the same units.]

(4) α is also equal to the product of (a) the electron mass (which I’ll simply write as me here) and (b) the classical electron radius re (if both are expressed in Planck units): α = me·re. Now think that’s, perhaps, the most amazing of all of the expressions for α. [If you don’t think that’s amazing, I’d really suggest you stop trying to study physics. :-)]

Also note that, from (2) and (4), we find that:

(5) The electron mass (in Planck units) is equal me = α/r= α/α2r = 1/αr. So that gives us an expression, using α once again, for the electron mass as a function of the Bohr radius r expressed in Planck units.

Finally, we can also substitute (1) in (5) to get:

(6) The electron mass (in Planck units) is equal to me = α/r = eP2/re. Using the Bohr radius, we get me = 1/αr = 1/eP2r.

So… As you can see, this fine-structure constant really links all of the fundamental properties of the electron: its charge, its radius, its distance to the nucleus (i.e. the Bohr radius), its velocity, its mass (and, hence, its energy),…

So… Why is what it is?

Well… We all marvel at this, but what can we say about it, really? I struggle how to interpret this, just as much – or probably much more 🙂 – as the professor who wrote the article I don’t like (because it’s so imprecise, and that’s what made me write all what I am writing here).

Having said that, it’s obvious that it points to a unity beyond these numbers and constants that I am only beginning to appreciate for what it is: deep, mysterious, and very beautiful. But so I don’t think that professor does a good job at showing how deep, mysterious and beautiful it all is. But then that’s up to you, my brother and you, my imaginary reader, to judge, of course. 🙂

[…] I forgot to mention what I mean with ‘Planck units’. Well… Once again, I should refer you to one of my other posts. But, yes, that’s too easy for me and a bit difficult for you. 🙂 So let me just note we get those Planck units by equating not less than five fundamental physical constants to 1, notably (1) the speed of light, (2) Planck’s (reduced) constant, (3) Boltzmann’s constant, (4) Coulomb’s constant and (5) Newton’s constant (i.e. the gravitational constant). Hence, we have a set of five equations here (ħ = kB = ke = G = 1), and so we can solve that to get the five Planck units, i.e. the Planck length unit, the Planck time unit, the Planck mass unit, the Planck energy unit, the Planck charge unit and, finally (oft forgotten), the Planck temperature unit. Of course, you should note that all mass and energy units are directly related because of the mass-energy equivalence relation E = mc2, which simplifies to E = m if c is equated to 1. [I could also say something about the relation between temperature and (kinetic) energy, but I won’t, as it would only further confuse you.]

OK. Done! 🙂

Addendum: How to think about space and time?

If you read the argument on the Planck scale and constant carefully, then you’ll note that it does not depend on the idea of an indivisible photon. However, it does depend on that Planck-Einstein relation being valid always and everywhere. Now, the Planck-Einstein relation is, in its essence, a fairly basic result from classical electromagnetic theory: it incorporates quantum theory – remember: it’s the equation that allowed Planck to solve the black-body radiation problem, and so it’s why they call Planck the (reluctant) ‘Father of Quantum Theory’ – but it’s not quantum theory.

So the obvious question is: can we make this reflection somewhat more general, so we can think of the electromagnetic force as an example only. In other words: can we apply the thoughts above to any force and any movement really?

The truth is: I haven’t advanced enough in my little study to give the equations for the other forces. Of course, we could think of gravity, and I developed some thoughts on how gravity waves might look like, but nothing specific really. And then we have the shorter-range nuclear forces, of course: the strong force, and the weak force. The laws involved are very different. The strong force involves color charges, and the way distances work is entirely different. So it would surely be some different analysis. However, the results should be the same. Let me offer some thoughts though:

  • We know that the relative strength of the nuclear force is much larger, because it pulls like charges (protons) together, despite the strong electromagnetic force that wants to push them apart! So the mentioned problem of trying to ‘pack’ some oscillation in some tiny little space should be worse with the strong force. And the strong force is there, obviously, at tiny little distances!
  • Even gravity should become important, because if we’ve got a lot of energy packed into some tiny space, its equivalent mass will ensure the gravitational forces also become important. In fact, that’s what the whole argument was all about!
  • There’s also all this talk about the fundamental forces becoming one at the Planck scale. I must, again, admit my knowledge is not advanced enough to explain how that would be possible, but I must assume that, if physicists are making such statements, the argument must be fairly robust.

So… Whatever charge or whatever force we are talking about, we’ll be thinking of waves or oscillations—or simply movement, but it’s always a movement in a force field, and so there’s power and energy involved (energy is force times distance, and power is the time rate of change of energy). So, yes, we should expect the same issues in regard to scale. And so that’s what’s captured by h.

As we’re talking the smallest things possible, I should also mention that there are also other inconsistencies in the electromagnetic theory, which should (also) have their parallel for other forces. For example, the idea of a point charge is mathematically inconsistent, as I show in my post on fields and charges. Charge, any charge really, must occupy some space. It cannot all be squeezed into one dimensionless point. So the reasoning behind the Planck time and distance scale is surely valid.

In short, the whole argument about the Planck scale and those limits is very valid. However, does it imply our thinking about the Planck scale is actually relevant? I mean: it’s not because we can imagine how things might look like  – they may look like those tiny little black holes, for example – that these things actually exist. GUT or string theorists obviously think they are thinking about something real. But, frankly, Feynman had a point when he said what he said about string theory, shortly before his untimely death in 1988: “I don’t like that they’re not calculating anything. I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation—a fix-up to say, ‘Well, it still might be true.'”

It’s true that the so-called Standard Model does not look very nice. It’s not like Maxwell’s equations. It’s complicated. It’s got various ‘sectors’: the electroweak sector, the QCD sector, the Higgs sector,… So ‘it looks like it’s got too much going on’, as a friend of mine said when he looked at a new design for mountainbike suspension. 🙂 But, unlike mountainbike designs, there’s no real alternative for the Standard Model. So perhaps we should just accept it is what it is and, hence, in a way, accept Nature as we can see it. So perhaps we should just continue to focus on what’s here, before we reach the Great Desert, rather than wasting time on trying to figure out how things might look like on the other side, especially because we’ll never be able to test our theories about ‘the other side.’

On the other hand, we can see where the Great Desert sort of starts (somewhere near the 1032 Hz scale), and so it’s only natural to think it should also stop somewhere. In fact, we know where it stops: it stops at the 1043 Hz scale, because everything beyond that doesn’t make sense. The question is: is there actually there? Like fundamental strings or whatever you want to call it. Perhaps we should just stop where the Great Desert begins. And what’s the Great Desert anyway? Perhaps it’s a desert indeed, and so then there is absolutely nothing there. 🙂

Hmm… There’s not all that much one can say about it. However, when looking at the history of physics, there’s one thing that’s really striking. Most of what physicists can think of, in the sense that it made physical sense, turned out to exist. Think of anti-matter, for instance. Paul Dirac thought it might exist, that it made sense to exist, and so everyone started looking for it, and Carl Anderson found in a few years later (in 1932). In fact, it had been observed before, but people just didn’t pay attention, so they didn’t want to see it, in a way. […] OK. I am exaggerating a bit, but you know what I mean. The 1930s are full of examples like that. There was a burst of scientific creativity, as the formalism of quantum physics was being developed, and the experimental confirmations of the theory just followed suit.

In the field of astronomy, or astrophysics I should say, it was the same with black holes. No one could really imagine the existence of black holes until the 1960s or so: they were thought of a mathematical curiosity only, a logical possibility. However, the circumstantial evidence now is quite large and so… Well… It seems a lot of what we can think of actually has some existence somewhere. 🙂

So… Who knows? […] I surely don’t. And so I need to get back to the grind and work my way through the rest of Feynman’s Lectures and the related math. However, this was a nice digression, and so I am grateful to my brother he initiated it. 🙂

Bad thinking: photons versus the matter wave

In my previous post, I wrote that I was puzzled by that relation between the energy and the size of a particle: higher-energy photons are supposed to be smaller and, pushing that logic to the limit, we get photons becoming black holes at the Planck scale. Now, understanding what the Planck scale is all about, is important to understand why we’d need a GUT, and so I do want to explore that relation between size and energy somewhat further.

I found the answer by a coincidence. We’ll call it serendipity. 🙂 Indeed, an acquaintance of mine who is very well versed in physics pointed out a terrible mistake in (some of) my reasoning in the previous posts: photons do not have a de Broglie wavelength. They just have a wavelength. Full stop. It immediately reduced my bemusement about that energy-size relation and, in the end, eliminated it completely. So let’s analyze that mistake – which seems to be a fairly common freshman mistake judging from what’s being written about it in some of the online discussions on physics.

If photons are not to be associated with a de Broglie wave, it basically means that the Planck relation has nothing to do with the de Broglie relation, even if these two relations are identical from a pure mathematical point of view:

  1. The Planck relation E = hν states that electromagnetic waves with frequency ν are a bunch of discrete packets of energy referred to as photons, and that the energy of these photons is proportional to the frequency of the electromagnetic wave, with the Planck constant h as the factor of proportionality. In other words, the natural unit to measure their energy is h, which is why h is referred to as the quantum of action.
  2. The de Broglie relation E = hf assigns de Broglie wave with frequency f to a matter particle with energy E = mc2 = γm0c2. [The factor γ in this formula is the Lorentz factor: γ = (1 – v2/c2)–1/2. It just corrects for the relativistic effect on mass as the velocity of the particle (v) gets closer to the speed of light (c).]

These are two very different things: photons do not have rest mass (which is why they can travel at light speed) and, hence, they are not to be considered as matter particles. Therefore, one should not assign a de Broglie wave to them. So what are they then? A photon is a wave packet but it’s an electromagnetic wave packet. Hence, its wave function is not some complex-valued psi function Ψ(x, t). What is oscillating in the illustration below (let’s say this is a procession of photons) is the electric field vector E. [To get the full picture of the electromagnetic wave, you should also imagine a (tiny) magnetic field vector B, which oscillates perpendicular to E), but that does not make much of a difference. Finally, in case you wonder about these dots: the red and green dot just make it clear that phase and group velocity of the wave are the same: vg = vp = v = c.] Wave - same group and phase velocityThe point to note is that we have a real wave here: it is not de Broglie wave. A de Broglie wave is a complex-valued function Ψ(x, t) with two oscillating parts: (i) the so-called real part of the complex value Ψ, and (ii) the so-called imaginary part (and, despite its name, that counts as much as the real part when working with Ψ !). That’s what’s shown in the examples of complex (standing) waves below: the blue part is one part (let’s say the real part), and then the salmon color is the other part. We need to square the modulus of that complex value to find the probability P of detecting that particle in space at point x at time t: P(x, t) = |Ψ(x, t)|2. Now, if we would write Ψ(x, t) as Ψ = u(x, t) + iv(x, t), then u(x, t) is the real part, and v(x, t) is the imaginary part. |Ψ(x, t)|2 is then equal to u2 + u2 so that shows that both the blue as well as the salmon amplitude matter when doing the math.  

StationaryStatesAnimation

So, while I may have given the impression that the Planck relation was like a limit of the de Broglie relation for particles with zero rest mass traveling at speed c, that’s just plain wrong ! The description of a particle with zero rest mass fits a photon but the Planck relation is not the limit of the de Broglie relation: photons are photons, and electrons are electrons, and an electron wave has nothing to do with a photon. Electrons are matter particles (fermions as physicists would say), and photons are bosons, i.e. force carriers.

Let’s now re-examine the relationship between the size and the energy of a photon. If the wave packet below would represent an (ideal) photon, what is its energy E as a function of the electric and magnetic field vectors E and B[Note that the (non-boldface) E stands for energy (i.e. a scalar quantity, so it’s just a number) indeed, while the (italic and bold) E stands for the (electric) field vector (so that’s something with a magnitude (E – with the symbol in italics once again to distinguish it from energy E) and a direction).] Indeed, if a photon is nothing but a disturbance of the electromagnetic field, then the energy E of this disturbance – which obviously depends on E and B – must also be equal to E = hν according to the Planck relation. Can we show that?

Well… Let’s take a snapshot of a plane-wave photon, i.e. a photon oscillating in a two-dimensional plane only. That plane is perpendicular to our line of sight here:

photon

Because it’s a snapshot (time is not a variable), we may look at this as an electrostatic field: all points in the interval Δx are associated with some magnitude (i.e. the magnitude of our electric field E), and points outside of that interval have zero amplitude. It can then be shown (just browse through any course on electromagnetism) that the energy density (i.e. the energy per unit volume) is equal to (1/2)ε0Eis the electric constant which we encountered in previous posts already). To calculate the total energy of this photon, we should integrate over the whole distance Δx, from left to right. However, rather than bothering you with integrals, I think that (i) the ε0E2/2 formula and (ii) the illustration above should be sufficient to convince you that:

  1. The energy of a photon is proportional to the square of the amplitude of the electric field. Such E ∝ Arelation is typical of any real wave, be they water waves or electromagnetic waves. So if we would double, triple, or quadruple its amplitude (i.e. the magnitude E of the electric field E), then the energy of this photon with be multiplied with four, nine times and sixteen respectively.
  2. If we would not change the amplitude of the wave above but double, triple or quadruple its frequency, then we would only double, triple or quadruple its energy: there’s no exponential relation here. In other words, the Planck relation E = hν makes perfect sense, because it reflects that simple proportionality: there is nothing to be squared.
  3. If we double the frequency but leave the amplitude unchanged, then we can imagine a photon with the same energy occupying only half of the Δx space. In fact, because we also have that universal relationship between frequency and wavelength (the propagation speed of a wave equals the product of its wavelength and its frequency: v = λf), we would have to halve the wavelength (and, hence, that would amount to dividing the Δx by two) to make sure our photon is still traveling at the speed of light.

Now, the Planck relation only says that higher energy is associated with higher frequencies: it does not say anything about amplitudes. As mentioned above, if we leave amplitudes unchanged, then the same Δx space will accommodate a photon with twice the frequency and twice the energy. However, if we would double both frequency and amplitude, then the photon would occupy only half of the Δx space, and still have twice as much energy. So the only thing I now need to prove is that higher-frequency electromagnetic waves are associated with larger-amplitude E‘s. Now, while that is something that we get straight out of the the laws of electromagnetic radiation: electromagnetic radiation is caused by oscillating electric charges, and it’s the magnitude of the acceleration (written as a in the formula below) of the oscillating charge that determines the amplitude. Indeed, for a full write-up of these ‘laws’, I’ll refer to a textbook (or just download Feynman’s 28th Lecture on Physics), but let me just give the formula for the (vertical) component of E: EMR law

You will recognize all of the variables and constants in this one: the electric constant ε0, the distance r, the speed of light (and our wave) c, etcetera. The ‘a’ is the acceleration: note that it’s a function not of t but of (t – r/c), and so we’re talking the so-called retarded acceleration here, but don’t worry about that.

Now, higher frequencies effectively imply a higher magnitude of the acceleration vector, and so that’s what’s I had to prove and so we’re done: higher-energy photons not only have higher frequency but also larger amplitude, and so they take less space.

It would be nice if I could derive some kind of equation to specify the relation between energy and size, but I am not that advanced in math (yet). 🙂 I am sure it will come.

Post scriptum 1: The ‘mistake’ I made obviously fully explains why Feynman is only interested in the amplitude of a photon to go from point A to B, and not in the amplitude of a photon to be at point x at time t. The question of the ‘size of the arrows’ then becomes a question related to the so-called propagator function, which gives the probability amplitude for a particle (a photon in this case) to travel from one place to another in a given time. The answer seems to involve another important buzzword when studying quantum mechanics: the gauge parameter. However, that’s also advanced math which I don’t master (as yet). I’ll come back on it… Hopefully… 🙂

Post scriptum 2: As I am re-reading some of my post now (i.e. on 12 January 2015), I noted how immature this post is. I wanted to delete it, but finally I didn’t, as it does illustrate my (limited) progress. I am still struggling with the question of a de Broglie wave for a photon, but I dare to think that my analysis of the question at least is a bit more mature now: please see one of my other posts on it.

The Uncertainty Principle re-visited: Fourier transforms and conjugate variables

In previous posts, I presented a time-independent wave function for a particle (or wavicle as we should call it – but so that’s not the convention in physics) – let’s say an electron – traveling through space without any external forces (or force fields) acting upon it. So it’s just going in some random direction with some random velocity v and, hence, its momentum is p = mv. Let me be specific – so I’ll work with some numbers here – because I want to introduce some issues related to units for measurement.

So the momentum of this electron is the product of its mass m (about 9.1×10−28 grams) with its velocity v (typically something in the range around 2,200 km/s, which is fast but not even close to the speed of light – and, hence, we don’t need to worry about relativistic effects on its mass here). Hence, the momentum p of this electron would be some 20×10−25 kg·m/s. Huh? Kg·m/s?Well… Yes, kg·m/s or N·s are the usual measures of momentum in classical mechanics: its dimension is [mass][length]/[time] indeed. However, you know that, in atomic physics, we don’t want to work with these enormous units (because we then always have to add these ×10−28 and ×10−25 factors and so that’s a bit of a nuisance indeed). So the momentum p will usually be measured in eV/c, with c representing what it usually represents, i.e. the speed of light. Huh? What’s this strange unit? Electronvolts divided by c? Well… We know that eV is an appropriate unit for measuring energy in atomic physics: we can express eV in Joule and vice versa: 1 eV = 1.6×10−19 Joule, so that’s OK – except for the fact that this Joule is a monstrously large unit at the atomic scale indeed, and so that’s why we prefer electronvolt. But the Joule is a shorthand unit for kg·m2/s2, which is the measure for energy expressed in SI units, so there we are: while the SI dimension for energy is actually [mass][length]2/[time]2, using electronvolts (eV) is fine. Now, just divide the SI dimension for energy, i.e. [mass][length]2/[time]2, by the SI dimension for velocity, i.e. [length]/[time]: we get something expressed in [mass][length]/[time]. So that’s the SI dimension for momentum indeed! In other words, dividing some quantity expressed in some measure for energy (be it Joules or electronvolts or erg or calories or coulomb-volts or BTUs or whatever – there’s quite a lot of ways to measure energy indeed!) by the speed of light (c) will result in some quantity with the right dimensions indeed. So don’t worry about it. Now, 1 eV/c is equivalent to 5.344×10−28 kg·m/s, so the momentum of this electron will be 3.75 eV/c.

Let’s go back to the main story now. Just note that the momentum of this electron that we are looking at is a very tiny amount – as we would expect of course.

Time-independent means that we keep the time variable (t) in the wave function Ψ(x, t) fixed and so we only look at how Ψ(x, t) varies in space, with x as the (real) space variable representing position. So we have a simplified wave function Ψ(x) here: we can always put the time variable back in when we’re finished with the analysis. By now, it should also be clear that we should distinguish between real-valued wave functions and complex-valued wave functions. Real-valued wave functions represent what Feynman calls “real waves”, like a sound wave, or an oscillating electromagnetic field. Complex-valued wave functions describe probability amplitudes. They are… Well… Feynman actually stops short of saying that they are not real. So what are they?

They are, first and foremost complex numbers, so they have a real and a so-called imaginary part (z = a + ib or, if we use polar coordinates, reθ = cosθ + isinθ). Now, you may think – and you’re probably right to some extent – that the distinction between ‘real’ waves and ‘complex’ waves is, perhaps, less of a dichotomy than popular writers – like me 🙂 – suggest. When describing electromagnetic waves, for example, we need to keep track of both the electric field vector E as well as the magnetic field vector B (both are obviously related through Maxwell’s equations). So we have two components as well, so to say, and each of these components has three dimensions in space, and we’ll use the same mathematical tools to describe them (so we will also represent them using complex numbers). That being said, these probability amplitudes usually denoted by Ψ(x), describe something very different. What exactly? Well… By now, it should be clear that that is actually hard to explain: the best thing we can do is to work with them, so they start feeling familiar. The main thing to remember is that we need to square their modulus (or magnitude or absolute value if you find these terms more comprehensible) to get a probability (P). For example, the expression below gives the probability of finding a particle – our electron for example – in in the (space) interval [a, b]:

probability versus amplitude

Of course, we should not be talking intervals but three-dimensional regions in space. However, we’ll keep it simple: just remember that the analysis should be extended to three (space) dimensions (and, of course, include the time dimension as well) when we’re finished (to do that, we’d use so-called four-vectors – another wonderful mathematical invention).

Now, we also used a simple functional form for this wave function, as an example: Ψ(x) could be proportional, we said, to some idealized function eikx. So we can write: Ψ(x) ∝ eikx (∝ is the standard symbol expressing proportionality). In this function, we have a wave number k, which is like the frequency in space of the wave (but then measured in radians because the phase of the wave function has to be expressed in radians). In fact, we actually wrote Ψ(x, t) = (1/x)ei(kx – ωt) (so the magnitude of this amplitude decreases with distance) but, again, let’s keep it simple for the moment: even with this very simple function eikx , things will become complex enough.

We also introduced the de Broglie relation, which gives this wave number k as a function of the momentum p of the particle: k = p/ħ, with ħ the (reduced) Planck constant, i.e. a very tiny number in the neighborhood of 6.582 ×10−16 eV·s. So, using the numbers above, we’d have a value for k equal to 3.75 eV/c divided by 6.582 ×10−16 eV·s. So that’s 0.57×1016 (radians) per… Hey, how do we do it with the units here? We get an incredibly huge number here (57 with 14 zeroes after it) per second? We should get some number per meter because k is expressed in radians per unit distance, right? Right. We forgot c. We are actually measuring distance here, but in light-seconds instead of meter: k is 0.57×1016/s. Indeed, a light-second is the distance traveled by light in one second, so that’s s, and if we want k expressed in radians per meter, then we need to divide this huge number 0.57×1016 (in rad) by 2.998×108 ( in (m/s)·s) and so then we get a much more reasonable value for k, and with the right dimension too: to be precise, k is about 19×106 rad/m in this case. That’s still huge: it corresponds with a wavelength of 0.33 nanometer (1 nm = 10-6 m) but that’s the correct order of magnitude indeed.

[In case you wonder what formula I am using to calculate the wavelength: it’s λ = 2π/k. Note that our electron’s wavelength is more than a thousand times shorter than the wavelength of (visible) light (we humans can see light with wavelengths ranging from 380 to 750 nm) but so that’s what gives the electron its particle-like character! If we would increase their velocity (e.g. by accelerating them in an accelerator, using electromagnetic fields to propel them to speeds closer to and also to contain them in a beam), then we get hard beta rays. Hard beta rays are surely not as harmful as high-energy electromagnetic rays. X-rays and gamma rays consist of photons with wavelengths ranging from 1 to 100 picometer (1 pm = 10–12 m) – so that’s another factor of a thousand down – and thick lead shields are needed to stop them: they are the cause of cancer (Marie Curie’s cause of death), and the hard radiation of a nuclear blast will always end up killing more people than the immediate blast effect. In contrast, hard beta rays will cause skin damage (radiation burns) but they won’t go deeper than that.]

Let’s get back to our wave function Ψ(x) ∝ eikx. When we introduced it in our previous posts, we said it could not accurately describe a particle because this wave function (Ψ(x) = Aeikx) is associated with probabilities |Ψ(x)|2 that are the same everywhere. Indeed,  |Ψ(x)|2 = |Aeikx|2 = A2. Apart from the fact that these probabilities would add up to infinity (so this mathematical shape is unacceptable anyway), it also implies that we cannot locate our electron somewhere in space. It’s everywhere and that’s the same as saying it’s actually nowhere. So, while we can use this wave function to explain and illustrate a lot of stuff (first and foremost the de Broglie relations), we actually need something different if we would want to describe anything real (which, in the end, is what physicists want to do, right?). We already said in our previous posts: real particles will actually be represented by a wave packet, or a wave train. A wave train can be analyzed as a composite wave consisting of a (potentially infinite) number of component waves. So we write:

Composite wave

Note that we do not have one unique wave number k or – what amounts to saying the same – one unique value p for the momentum: we have n values. So we’re introducing a spread in the wavelength here, as illustrated below:

Explanation of uncertainty principle

In fact, the illustration above talks of a continuous distribution of wavelengths and so let’s take the continuum limit of the function above indeed and write what we should be writing:

Composite wave - integral

Now that is an interesting formula. [Note that I didn’t care about normalization issues here, so it’s not quite what you’d see in a more rigorous treatment of the matter. I’ll correct that in the Post Scriptum.] Indeed, it shows how we can get the wave function Ψ(x) from some other function Φ(p). We actually encountered that function already, and we referred to it as the wave function in the momentum space. Indeed, Nature does not care much what we measure: whether it’s position (x) or momentum (p), Nature will not share her secrets with us and, hence, the best we can do – according to quantum mechanics – is to find some wave function associating some (complex) probability amplitude with each and every possible (real) value of x or p. What the equation above shows, then, is these wave functions come as a pair: if we have Φ(p), then we can calculate Ψ(x) – and vice versa. Indeed, the particular relation between Ψ(x) and Φ(p) as established above, makes Ψ(x) and Φ(p) a so-called Fourier transform pair, as we can transform Φ(p) into Ψ(x) using the above Fourier transform (that’s how that  integral is called), and vice versa. More in general, a Fourier transform pair can be written as:

Fourier transform pair

Instead of x and p, and Ψ(x) and Φ(p), we have x and y, and f(x) and g(y), in the formulas above, but so that does not make much of a difference when it comes to the interpretation: x and p (or x and y in the formulas above) are said to be conjugate variables. What it means really is that they are not independent. There are quite a few of such conjugate variables in quantum mechanics such as, for example: (1) time and energy (and time and frequency, of course, in light of the de Broglie relation between both), and (2) angular momentum and angular position (or orientation). There are other pairs too but these involve quantum-mechanical variables which I do not understand as yet and, hence, I won’t mention them here. [To be complete, I should also say something about that 1/2π factor, but so that’s just something that pops up when deriving the Fourier transform from the (discrete) Fourier series on which it is based. We can put it in front of either integral, or split that factor across both. Also note the minus sign in the exponent of the inverse transform.]

When you look at the equations above, you may think that f(x) and g(y) must be real-valued functions. Well… No. The Fourier transform can be used for both real-valued as well as complex-valued functions. However, at this point I’ll have to refer those who want to know each and every detail about these Fourier transforms to a course in complex analysis (such as Brown and Churchill’s Complex Variables and Applications (2004) for instance) or, else, to a proper course on real and complex Fourier transforms (they are used in signal processing – a very popular topic in engineering – and so there’s quite a few of those courses around).

The point to note in this post is that we can derive the Uncertainty Principle from the equations above. Indeed, the (complex-valued) functions Ψ(x) and Φ(p) describe (probability) amplitudes, but the (real-valued) functions |Ψ(x)|2 and |Φ(p)|2 describe probabilities or – to be fully correct – they are probability (density) functions. So it is pretty obvious that, if the functions Ψ(x) and Φ(p) are a Fourier transform pair, then |Ψ(x)|2 and |Φ(p)|2 must be related to. They are. The derivation is a bit lengthy (and, hence, I will not copy it from the Wikipedia article on the Uncertainty Principle) but one can indeed derive the so-called Kennard formulation of the Uncertainty Principle from the above Fourier transforms. This Kennard formulation does not use this rather vague Δx and Δp symbols but clearly states that the product of the standard deviation from the mean of these two probability density functions can never be smaller than ħ/2:

σxσ≥ ħ/2

To be sure: ħ/2 is a rather tiny value, as you should know by now, 🙂 but, so, well… There it is.

As said, it’s a bit lengthy but not that difficult to do that derivation. However, just for once, I think I should try to keep my post somewhat shorter than usual so, to conclude, I’ll just insert one more illustration here (yes, you’ve seen that one before), which should now be very easy to understand: if the wave function Ψ(x) is such that there’s relatively little uncertainty about the position x of our electron, then the uncertainty about its momentum will be huge (see the top graphs). Vice versa (see the bottom graphs), precise information (or a narrow range) on its momentum, implies that its position cannot be known.

2000px-Quantum_mechanics_travelling_wavefunctions_wavelength

Does all this math make it any easier to understand what’s going on? Well… Yes and no, I guess. But then, if even Feynman admits that he himself “does not understand it the way he would like to” (Feynman Lectures, Vol. III, 1-1), who am I? In fact, I should probably not even try to explain it, should I? 🙂

So the best we can do is try to familiarize ourselves with the language used, and so that’s math for all practical purposes. And, then, when everything is said and done, we should probably just contemplate Mario Livio’s question: Is God a mathematician? 🙂

Post scriptum:

I obviously cut corners above, and so you may wonder how that ħ factor can be related to σand σ if it doesn’t appear in the wave functions. Truth be told, it does. Because of (i) the presence of ħ in the exponent in our ei(p/ħ)x function, (ii) normalization issues (remember that probabilities (i.e. Ψ|(x)|2 and |Φ(p)|2) have to add up to 1) and, last but not least, (iii) the 1/2π factor involved in Fourier transforms , Ψ(x) and Φ(p) have to be written as follows:

Position and momentum wave functionNote that we’ve also re-inserted the time variable here, so it’s pretty complete now. One more thing we could do is to substitute x for a proper three-dimensional space vector or, better still, introduce four-vectors, which would allow us to also integrate relativistic effects (most notably the slowing of time with motion – as observed from the stationary reference frame) – which become important when, for instance, we’re looking at electrons being accelerated, which is the rule, rather than the exception, in experiments.

Remember (from a previous post) that we calculated that an electron traveling at its usual speed in orbit (2200 km/s, i.e. less than 1% of the speed of light) had an energy of about 70 eV? Well, the Large Electron-Positron Collider (LEP) did accelerate them to speeds close to light, thereby giving them energy levels topping 104.5 billion eV (or 104.5 GeV as it’s written) so they could hit each other with collision energies topping 209 GeV (they come from opposite directions so it’s two times 104.5 GeV). Now, 209 GeV is tiny when converted to everyday energy units: 209 GeV is 33×10–9 Joule only indeed – and so note the minus sign in the exponent here: we’re talking billionths of a Joule here. Just to put things into perspective: 1 Watt is the energy consumption of an LED (and 1 Watt is 1 Joule per second), so you’d need to combine the energy of billions of these fast-traveling electrons to power just one little LED lamp. But, of course, that’s not the right comparison: 104.5 GeV is more than 200,000 times the electron’s rest mass (0.511 MeV), so that means that – in practical terms – their mass (remember that mass is a measure for inertia) increased by the same factor (204,500 times to be precise). Just to give an idea of the effort that was needed to do this: CERN’s LEP collider was housed in a tunnel with a circumference of 27 km. Was? Yes. The tunnel is still there but it now houses the Large Hadron Collider (LHC) which, as you surely know, is the world’s largest and most powerful particle accelerator: its experiments confirmed the existence of the Higgs particle in 2013, thereby confirming the so-called Standard Model of particle physics. [But I’ll see a few things about that in my next post.]

Oh… And, finally, in case you’d wonder where we get the inequality sign in σxσ≥ ħ/2, that’s because – at some point in the derivation – one has to use the Cauchy-Schwarz inequality (aka as the triangle inequality): |z1+ z1| ≤ |z1|+| z1|. In fact, to be fully complete, the derivation uses the more general formulation of the Cauchy-Schwarz inequality, which also applies to functions as we interpret them as vectors in a function space. But I would end up copying the whole derivation here if I add any more to this – and I said I wouldn’t do that. 🙂 […]