Some thoughts on the nature of reality

Some other comment on an article on my other blog, inspired me to structure some thoughts that are spread over various blog posts. What follows below, is probably the first draft of an article or a paper I plan to write. Or, who knows, I might re-write my two introductory books on quantum physics and publish a new edition soon. 馃檪

Physical dimensions and Uncertainty

The physical dimension of the quantum of action (h or聽魔 = h/2蟺) is force (expressed in newton)聽times distance (expressed in meter)聽times time (expressed in seconds): N路m路s. Now, you may think this N路m路s dimension is kinda hard to imagine. We can imagine its individual components, right? Force, distance and time. We know what they are. But the product of all three? What is it, really?

It shouldn’t be all that hard to imagine what it might be, right? The N路m路s unit is also the unit in which angular momentum is expressed – and you can sort of imagine what that is, right? Think of a spinning top, or a gyroscope. We may also think of the following:

  1. [h] = N路m路s = (N路m)路s = [E]路[t]
  2. [h] = N路m路s = (N路s)路m = [p]路[x]

Hence, the physical dimension of action is that of energy (E) multiplied by time (t) or, alternatively, that of momentum (p) times distance (x). To be precise, the second dimensional equation should be written as [h] = [p]路[x], because both the momentum and the distance traveled will be associated with some direction. It’s a moot point for the discussion at the moment, though. Let’s think about the first equation first:聽[h] = [E]路[t]. What does it mean?

Energy… Hmm… In聽real life, we are usually not interested in the energy of a system as such, but by the energy it can deliver, or absorb, per second. This is referred to as the聽power聽of a system, and it’s expressed in J/s, or watt. Power is also defined as the (time) rate at which work is done. Hmm… But so here we’re multiplying energy and time. So what’s that? After Hiroshima and Nagasaki, we can sort of imagine the energy of an atomic bomb. We can also sort of imagine the power that’s being released by the Sun in light and other forms of radiation, which is about 385脳1024 joule per second. But energy times time? What’s that?

I am not sure. If we think of the Sun as a huge reservoir of energy, then the physical dimension of action is just like having that reservoir of energy guaranteed for some time, regardless of how fast or how slow we use it. So, in short, it’s just like the Sun – or the Earth, or the Moon, or whatever object – just being there, for some聽definite聽amount of time. So, yes: some聽definite amount of mass or energy (E) for some聽definite聽amount of time (t).

Let’s bring the mass-energy equivalence formula in here: E = mc2. Hence, the physical dimension of action can also be written as [h] = [E]路[t] = [mc]2路[t] = (kg路m2/s2)路s =聽kg路m2/s.聽What does that say? Not all that much – for the time being, at least. We can get this聽[h] = kg路m2/s through some other substitution as well. A force of one newton will give a mass of 1 kg an acceleration of 1 m/s per second. Therefore, 1 N = 1 kg路m/s2聽and, hence, the physical dimension of h, or the unit of angular momentum, may also be written as 1 N路m路s = 1 (kg路m/s2)路m路s = 1 kg路m2/s, i.e. the product of mass, velocity and distance.

Hmm… What can we do with that? Nothing much for the moment: our first reading of it is just that it reminds us of the definition of angular momentum – some mass with some velocity rotating around an axis. What about the distance? Oh… The distance here is just the distance from the axis, right? Right. But… Well… It’s like having some amount of linear momentum available over some distance – or in some space, right? That’s sufficiently significant as an interpretation for the moment, I’d think…

Fundamental units

This makes one think about what units would be fundamental – and what units we’d consider as being derived. Formally, the聽newton is a聽derived聽unit in the metric system, as opposed to the units of mass, length and time (kg, m, s). Nevertheless, I personally like to think of force as being fundamental:聽 a force is what causes an object to deviate from its straight trajectory in spacetime. Hence, we may want to think of the聽quantum of action as representing three fundamental physical dimensions: (1)聽force, (2)聽time and (3) distance – or space. We may then look at energy and (linear) momentum as physical quantities combining (1) force and distance and (2) force and time respectively.

Let me write this out:

  1. Force times length (think of a force that is聽acting on some object over some distance) is energy: 1 joule聽(J) =聽1 newtonmeter (N). Hence, we may think of the concept of energy as a projection聽of action in space only: we make abstraction of time. The physical dimension of the quantum of action should then be written as [h] = [E]路[t]. [Note the square brackets tell us we are looking at a聽dimensional聽equation only, so [t] is just the physical dimension of the time variable. It’s a bit confusing because I also use square brackets as parentheses.]
  2. Conversely, the magnitude of linear momentum (p = m路v) is expressed in newtonseconds: 1 kg路m/s = 1 (kg路m/s2)路s = 1 N路s. Hence, we may think of (linear) momentum as a projection of action in time only: we make abstraction of its spatial dimension. Think of a force that is acting on some object聽during some time.聽The physical dimension of the quantum of action should then be written as [h] = [p]路[x]

Of course, a force that is acting on some object during some time, will usually also act on the same object over some distance but… Well… Just try, for once, to make abstraction of one of the two dimensions here: time聽or聽distance.

It is a difficult thing to do because, when everything is said and done, we don’t live in space or in time alone, but in spacetime and, hence, such abstractions are not easy. [Of course, now you’ll say that it’s easy to think of something that moves in time only: an object that is standing still does just that – but then we know movement is relative, so there is no such thing as an object that is standing still in space聽in an absolute sense: Hence, objects never stand still in spacetime.] In any case, we should try such abstractions, if only because of the principle of least action聽is so essential and deep in physics:

  1. In classical physics, the path of some object in a force field will minimize聽the total action (which is usually written as S) along that path.
  2. In quantum mechanics, the same action integral will give us various values S – each corresponding to a particular path – and each path (and, therefore, each value of S, really) will be associated with a probability amplitude that will be proportional to some constant times e鈭抜路胃聽=聽ei路(S/魔). Because is so tiny, even a small change in S will give a completely different phase angle 胃. Therefore, most amplitudes will cancel each other out as we take the sum of the amplitudes over all possible paths: only the paths that nearly聽give the same phase matter. In practice, these are the paths that are associated with a variation in S of an order of magnitude that is equal to .

The paragraph above summarizes, in essence, Feynman’s path integral formulation of quantum mechanics. We may, therefore, think of the quantum of action聽expressing聽itself (1) in time only, (2) in space only, or – much more likely – (3) expressing itself in both dimensions at the same time. Hence, if the quantum of action gives us the order of magnitude聽of the uncertainty – think of writing something like S 卤 , we may re-write our dimensional [] = [E]路[t] and [] = [p]路[x] equations as the uncertainty equations:

  • 螖E路螖t =
  • 螖p路螖x =

You should note here that it is best to think of the uncertainty relations as a聽pair聽of equations, if only because you should also think of the concept of energy and momentum as representing different aspects聽of the same reality, as evidenced by the (relativistic) energy-momentum relation (E2聽= p2c2聽鈥 m02c4). Also, as illustrated below, the actual path – or, to be more precise, what we might associate with the concept of the actual path – is likely to be some mix of 螖x and 螖t. If 螖t is very small, then 螖x will be very large. In order to move over such distance, our particle will require a larger energy, so 螖E will be large. Likewise, if 螖t is very large, then 螖x will be very small and, therefore, 螖E will be very small. You can also reason in terms of 螖x, and talk about momentum rather than energy. You will arrive at the same conclusions: the 螖E路螖t = h and 螖p路螖x = h聽relations represent two aspects of the same reality – or, at the very least, what we might聽think聽of as reality.


Also think of the following: if聽螖E路螖t =聽h聽and 螖p路螖x =聽h, then聽螖E路螖t =聽螖p路螖x and, therefore,聽螖E/螖p must be equal to 螖x/螖t. Hence, the聽ratio聽of the uncertainty about x (the distance) and the uncertainty about t (the time) equals the聽ratio聽of the uncertainty about E (the energy) and the uncertainty about p (the momentum).

Of course, you will note that the actual uncertainty relations have a factor 1/2 in them. This may be explained by thinking of both negative as well as positive variations in space and in time.

We will obviously want to do some more thinking about those physical dimensions. The idea of a force implies the idea of some object – of some mass on which the force is acting. Hence, let’s think about the concept of mass now. But… Well… Mass and energy are supposed to be equivalent, right? So let’s look at the concept of energy聽too.

Action, energy and mass

What is聽energy, really? In聽real life, we are usually not interested in the energy of a system as such, but by the energy it can deliver, or absorb, per second. This is referred to as the聽power聽of a system, and it’s expressed in J/s. However, in physics, we always talk energy – not power – so… Well… What is the energy of a system?

According to the de Broglie聽and Einstein – and so many other eminent physicists, of course – we should not only think of the kinetic energy of its parts, but also of their potential energy, and their rest聽energy, and – for an atomic system – we may add some internal energy, which may be binding energy, or excitation energy (think of a hydrogen atom in an excited state, for example). A lot of stuff. 馃檪 But, obviously, Einstein’s mass-equivalence formula comes to mind here, and summarizes it all:

E = m路c2

The m in this formula refers to mass – not to meter, obviously. Stupid remark, of course… But… Well… What is energy, really? What is mass,聽really? What’s that聽equivalence聽between mass and energy,聽really?

I don’t have the definite answer to that question (otherwise I’d be famous), but… Well… I do think physicists and mathematicians should invest more in exploring some basic intuitions here. As I explained in several posts, it is very tempting to think of energy as some kind of two-dimensional oscillation of mass. A force over some distance will cause a mass to accelerate. This is reflected in the聽dimensional analysis:

[E] = [m]路[c2] = 1 kg路m2/s2聽= 1 kg路m/s2路m = 1 N路m

The kg and m/s2聽factors make this abundantly clear: m/s2聽is the physical dimension of acceleration: (the change in) velocity per time unit.

Other formulas now come to mind, such as the Planck-Einstein relation: E = h路f = 蠅路魔. We could also write: E = h/T. Needless to say, T = 1/f聽is the聽period聽of the oscillation. So we could say, for example, that the energy of some particle times the period of the oscillation gives us Planck’s constant again. What does that mean? Perhaps it’s easier to think of it the other way around: E/f = h = 6.626070040(81)脳10鈭34聽J路s. Now, f聽is the number of oscillations聽per second. Let’s write it as聽f聽= n/s, so we get:

E/f聽= E/(n/s) = E路s/n聽= 6.626070040(81)脳10鈭34聽J路s 鈬 E/n聽= 6.626070040(81)脳10鈭34聽J

What an amazing result! Our wavicle – be it a photon or a matter-particle – will always聽pack聽6.626070040(81)脳10鈭34joule聽in聽one聽oscillation, so that’s the numerical聽value of Planck’s constant which, of course, depends on our fundamental聽units (i.e. kg, meter, second, etcetera in the SI system).

Of course, the obvious question is: what’s one聽oscillation? If it’s a wave packet, the oscillations may not have the same amplitude, and we may also not be able to define an exact period. In fact, we should expect the amplitude and duration of each oscillation to be slightly different, shouldn’t we? And then…

Well… What’s an oscillation? We’re used to聽counting聽them:聽n聽oscillations per second, so that’s聽per time unit. How many do we have in total? We wrote about that in our posts on the shape and size of a photon. We know photons are emitted by atomic oscillators – or, to put it simply, just atoms going from one energy level to another. Feynman calculated the Q of these atomic oscillators: it鈥檚 of the order of 108聽(see his聽Lectures,聽I-33-3: it鈥檚 a wonderfully simple exercise, and one that really shows his greatness as a physics teacher), so… Well… This wave train will last about 10鈥8聽seconds (that鈥檚 the time it takes for the radiation to die out by a factor 1/e). To give a somewhat more precise example,聽for sodium light, which has a frequency of 500 THz (500脳1012聽oscillations per second) and a wavelength of 600 nm (600脳10鈥9聽meter), the radiation will lasts about 3.2脳10鈥8聽seconds. [In fact, that鈥檚 the time it takes for the radiation鈥檚 energy to die out by a factor 1/e, so(i.e. the so-called decay time 蟿), so the wavetrain will actually last聽longer, but so the amplitude becomes quite small after that time.]聽So… Well… That鈥檚 a very short time but… Still, taking into account the rather spectacular frequency (500 THz) of sodium light, that makes for some 16 million oscillations and, taking into the account the rather spectacular speed of light (3脳108聽m/s), that makes for a wave train with a length of, roughly,聽9.6 meter. Huh? 9.6 meter!? But a photon is supposed to be pointlike, isn’it it? It has no length, does it?

That’s where relativity helps us out: as I wrote in one of my posts, relativistic length contraction may explain the apparent paradox. Using the reference frame of the photon聽– so if we’d be traveling at speed c,鈥 riding鈥 with the photon, so to say, as it鈥檚 being emitted – then we’d 鈥榮ee鈥 the electromagnetic transient as it鈥檚 being radiated into space.

However, while we can associate some mass聽with the energy of the photon, none of what I wrote above explains what the (rest) mass of a matter-particle could possibly be.There is no real answer to that, I guess. You’ll think of the Higgs field now but… Then… Well. The Higgs field is a scalar field. Very simple: some number that’s associated with some position in spacetime. That doesn’t explain very much, does it? 馃槮 When everything is said and done, the scientists who, in 2013 only, got the Nobel Price for their theory on the Higgs mechanism, simply tell us mass is some number. That’s something we knew already, right? 馃檪

The reality of the wavefunction

The wavefunction is, obviously, a mathematical construct: a聽description聽of reality using a very specific language. What language? Mathematics, of course! Math may not be universal (aliens might not be able to decipher our mathematical models) but it’s pretty good as a global聽tool of communication, at least.

The real聽question is: is the description聽accurate? Does it match reality and, if it does, how聽good聽is the match? For example, the wavefunction for an electron in a hydrogen atom looks as follows:

蠄(r, t) = ei路(E/魔)路tf(r)

As I explained in previous posts (see, for example, my recent post聽on reality and perception), the聽f(r) function basically provides some envelope for the two-dimensional ei路胃聽=聽ei路(E/魔)路t聽= cos胃 + isin胃聽oscillation, with r= (x, y, z),聽胃 = (E/魔)路t聽= 蠅路t聽and 蠅 = E/魔. So it presumes the聽duration of each oscillation is some constant. Why? Well… Look at the formula: this thing has a constant frequency in time. It’s only the amplitude that is varying as a function of the r= (x, y, z) coordinates. 馃檪 So… Well… If each oscillation is to always聽pack聽6.626070040(81)脳10鈭34joule, but the amplitude of the oscillation varies from point to point, then… Well… We’ve got a problem. The wavefunction above is likely to be an approximation of reality only. 馃檪 The associated energy is the same, but… Well… Reality is probably聽not聽the nice geometrical shape we associate with those wavefunctions.

In addition, we should think of the聽Uncertainty Principle: there聽must聽be some uncertainty in the energy of the photons when our hydrogen atom makes a transition from one energy level to another. But then… Well… If our photon packs something like 16 million oscillations, and the order of magnitude of the uncertainty is only of the order of聽h聽(or 魔 = h/2蟺) which, as mentioned above, is the (average) energy of one聽oscillation only, then we don’t have much of a problem here, do we? 馃檪

Post scriptum: In previous posts, we offered some analogies – or metaphors – to a two-dimensional oscillation (remember the V-2 engine?). Perhaps it’s all relatively simple. If we have some tiny little ball of mass – and its center of mass has to stay where it is – then any rotation – around any axis – will be some combination of a rotation around our聽x- and z-axis – as shown below. Two axes only. So we may want to think of a two-dimensional聽oscillation as an oscillation of the polar and azimuthal angle. 馃檪

oscillation of a ball

Occam’s Razor

The analysis of a two-state system (i.e.聽the rather famous example of an ammonia molecule ‘flipping’ its spin direction from ‘up’ to ‘down’, or vice versa) in my previous post is a good opportunity to think about Occam’s Razor once more. What are we doing? What does the math聽tell us?

In the example we chose, we didn’t need to worry about space. It was all about聽time: an evolving聽state聽over time.聽We also knew the answers we wanted to get: if there is聽some聽probability for the system to ‘flip’ from one state to another, we know it will, at聽some聽point in time. We also want probabilities to add up to one, so we knew the graph below had to be the result we would find: if our molecule can be in two states only, and it starts of in one, then the probability that it will聽remain in that state will gradually decline, while the probability that it flips into the other state will gradually increase, which is what is depicted below.


However, the graph above is only a Platonic idea: we don’t bother to actually verify聽what state the molecule is in. If we did, we’d have to ‘re-set’ our t = 0 point, and start all over again. The wavefunction would collapse, as they say, because we’ve made a measurement. However, having said that, yes, in the physicist’s Platonic world of ideas, the probability functions above make perfect sense. They are beautiful. You should note, for example, that P1聽(i.e. the probability to be in state 1) and P2聽(i.e. the probability to be in state 2) add up to 1 all of the time, so we don’t need to integrate over a cycle or something: so it’s all聽perfect!

These probability functions are聽based on ideas that are even more Platonic: interfering amplitudes. Let me explain.

Quantum physics is based on the idea that these probabilities聽are determined by some wavefunction, a complex-valued聽amplitude聽that varies in time and space. It’s a two-dimensional thing, and then it’s not. It’s two-dimensional because it combines a sine and cosine, i.e. a real and an imaginary part, but the argument of the sine and the cosine is the same, and the sine and cosine are the same function, except for a phase shift equal to 蟺. We write:

a路ei胃聽= a路cos(胃) 鈥撀a路sin(鈭捨) =聽a路cos胃 鈥撀a路sin胃

The minus sign is there because it turns out that Nature measures angles, i.e. our phase,聽clockwise, rather than counterclockwise, so that’s not聽as per our聽mathematical convention. But that’s a minor detail, really. [It should give you some food for thought, though.] For the rest, the聽related graph is as simple as the formula:

graph sin and cos

Now, the聽phase聽of this wavefunction is written as聽胃 =聽(蠅路t 鈭 k聽鈭x). Hence, 蠅 determines how this wavefunction varies in time, and the wavevector聽k tellsus how this wave varies in space. The young Frenchman Comte Louis de Broglie noted the mathematical similarity between the聽蠅路t 鈭 k聽鈭x聽expression and Einstein’s four-vector product px渭聽=聽E路t 鈭 px, which remains invariant under a Lorentz transformation. He also understood that the Planck-Einstein relation E =聽魔路蠅 actually defines聽the energy unit and, therefore, that any frequency,聽any oscillation really, in space or in time, is to be expressed in terms of 魔.

[To be precise, the fundamental quantum of energy is h = 魔路2蟺, because that’s the energy of one cycle. To illustrate the point, think of the Planck-Einstein relation. It gives us the energy of a聽photon with frequency f: E聽= h路f. If we re-write this equation as E/f聽= h, and we do a dimensional analysis, we get:聽h = E/f聽鈬斅6.626脳10鈭34聽joule路second =聽[x joule]/[f聽cycles per second]聽鈬斅爃=聽6.626脳10鈭34聽joule聽per cycle. It’s only because we are expressing 蠅 and k as angular聽frequencies (i.e.聽in radians聽per second or per meter, rather than in cycles聽per second or per meter) that we have to think of 魔 = h/2蟺 rather than h.]

Louis de Broglie connected the dots between some other equations too. He was fully familiar with the equations determining the phase and group velocity of composite聽waves, or a聽wavetrain聽that actually might represent a聽wavicle聽traveling through spacetime. In short, he boldly equated聽蠅 with 蠅 = E/魔 and k with k聽= p/魔, and all came out alright. It made perfect sense!

I’ve written enough about this. What I want to write about here is how this also makes for the situation on hand: a simple two-state system that depends on time only. So its phase is聽胃 = 蠅路t =聽E0/魔. What’s E0? It is the聽total聽energy of the system, including the equivalent energy of the particle’s rest mass and any potential energy that may be there聽because of the presence of one or the other force field. What about kinetic energy? Well… We said it: in this case, there is no translational or linear momentum, so p = 0. So our Platonic wavefunction reduces to:

a路ei胃聽= ae(i/魔)路(E0路t)

Great! […] But… Well… No! The problem with this wavefunction is that it yields a constant聽probability. To be precise, when we take the absolute聽square of this wavefunction聽鈥 which is what we do when calculating a probability from a wavefunction聽鈭 we get P = a2, always. The ‘normalization’ condition (so that’s the condition that probabilities have to add up to one) implies that P1聽= P2聽= a2= 1/2. Makes sense, you’ll say, but the problem is that this doesn’t reflect reality: these probabilities do not evolve over time and, hence, our聽ammonia molecule never ‘flips’ its spin direction from ‘up’ to ‘down’, or vice versa. In short, our wavefunction does聽not聽explain reality.

The problem is not unlike the problem we’d had with a similar function relating the momentum聽and the聽position聽of a particle. You’ll remember it: we wrote it as聽a路ei胃聽= ae(i/魔)路(px). [Note that we can write聽a路ei胃聽= a路e鈭(i/魔)路(E0路t 鈭捖px)= ae鈭(i/魔)路(E0路t)e(i/魔)路(px), so we can always split our wavefunction in a ‘time’ and a ‘space’ part.] But then we found that this wavefunction also yielded a constant and equal probability all over space, which implies our particle is everywhere (and, therefore, nowhere, really).

In quantum physics, this problem is solved by introducing uncertainty.聽Introducing some uncertainty about the energy, or about the momentum, is mathematically equivalent to saying that we’re actually looking at a composite聽wave, i.e. the sum of a finite or infinite set of component waves. So we have the same聽蠅 = E/魔 and k聽= p/魔 relations, but we apply them to n energy levels, or to some continuous聽range聽of energy levels聽螖E. It amounts to saying that our wave function doesn鈥檛 have a specific frequency: it now has n frequencies, or a range of frequencies聽螖蠅 =聽螖E/魔.

We know what that does: it ensures our wavefunction is being ‘contained’ in some ‘envelope’. It becomes a wavetrain, or a kind of beat note, as illustrated below:


[The animation also shows the difference between the聽group聽and聽phase聽velocity: the green dot shows the group velocity, while the red dot travels at the phase velocity.]

This begs the following question: what鈥檚 the uncertainty really? Is it an uncertainty in the energy, or is it an uncertainty in the wavefunction? I mean: we have a function relating the energy to a frequency. Introducing some uncertainty about the energy is mathematically equivalent to introducing uncertainty about the frequency.聽Of course, the answer is: the uncertainty is in both, so it鈥檚 in thefrequency and in the energy and both are related through the wavefunction. So鈥 Well… Yes. In some way, we鈥檙e chasing our own tail. 馃檪

However, the trick does the job, and perfectly so. Let me summarize what we did in the previous post: we had the ammonia molecule, i.e. an聽NH3聽molecule, with the nitrogen 鈥榝lipping鈥 across the hydrogens from time to time, as illustrated below:


This聽鈥榝lip鈥 requires energy, which is why we associate two聽energy levels with the molecule, rather than just one. We wrote these two energy levels as E0聽+ A and E0聽鈭 A. That聽assumption solved all of our problems. [Note that we don’t specify what the energy barrier really consists of: moving the center of mass obviously requires some energy, but it is likely that a ‘flip’ also involves overcoming some electrostatic forces, as shown by the reversal of the electric聽dipole moment in the illustration above.]聽To be specific, it gave us the following wavefunctions for the amplitude聽to be in the ‘up’ or ‘1’ state versus the ‘down’ or ‘2’ state respectivelly:

  • C1聽= (1/2)路e(i/魔)路(E0聽鈭 A)路t聽+ (1/2)路e(i/魔)路(E0聽+ A)路t
  • C2聽= (1/2)路e(i/魔)路(E0聽鈭 A)路t聽鈥 (1/2)路e(i/魔)路(E0聽+ A)路t

Both are聽composite聽waves. To be precise, they are the sum of two component waves with a聽temporal聽frequency equal to 蠅1聽=聽(E0聽鈭 A)/魔 and 蠅1聽=聽(E0聽+ A)/魔 respectively. [As for the minus sign in front of the second term in the wave equation for聽C2, 鈭1 = ei, so + (1/2)路e(i/魔)路(E0聽+ A)路t聽and 鈥 (1/2)路e(i/魔)路(E0聽+ A)路t聽are the same wavefunction: they only differ because their聽relative聽phase is shifted by 卤蟺.] So the聽so-called聽base states of the molecule themselves聽are associated with two different energy levels: it’s not聽like one state has more energy than the other.

You’ll say: so what?

Well… Nothing. That’s it really. That’s all I wanted to say here. The absolute square of those two wavefunctions gives us those time-dependent probabilities above, i.e. the graph we started this post with. So… Well… Done!

You’ll say: where’s the ‘envelope’? Oh! Yes! Let me tell you. The聽C1(t) and C2(t) equations can be re-written as:


Now, remembering our rules for adding and subtracting complex conjugates (ei聽+聽e鈥搃聽=聽2cos胃 and聽ei聽鈭捖e鈥搃聽=聽2sin胃), we can re-write this as:


So there we are! We’ve got wave equations whose聽temporal variation is basically defined by E0聽but, on top of that, we have an envelope here: the cos(A路t/魔) and sin(A路t/魔) factor respectively. So their magnitude聽is no longer time-independent: both the phase as well as the amplitude now vary with time. The associated probabilities are the ones we plotted:

  • |C1(t)|2聽= cos2[(A/魔)路t], and
  • |C2(t)|2聽= sin2[(A/魔)路t].

So, to summarize it all once more, allowing the nitrogen atom to push its way through the three hydrogens, so as to flip to the other side, thereby breaking the energy barrier, is equivalent to associating two聽energy levels to the ammonia molecule as a whole, thereby introducing some聽uncertainty, or聽indefiniteness聽as to its energy, and that, in turn, gives us the amplitudes and probabilities that we鈥檝e just calculated. [And you may want to note here that the probabilities 鈥渟loshing back and forth鈥, or 鈥渄umping into each other鈥澛犫 as Feynman puts it聽鈥 is the result of the varying聽magnitudes聽of our amplitudes, so that’s the ‘envelope’ effect. It’s only because the magnitudes vary in time that their absolute square, i.e. the associated聽probability,聽varies too.

So鈥 Well鈥 That鈥檚 it. I think this and all of the previous posts served as a nice introduction to quantum physics. More in particular, I hope this post made you appreciate the mathematical framework is not as horrendous as it often seems to be.

When thinking about it, it’s actually all quite straightforward, and it surely respects聽Occam’s principle of parsimony in philosophical and scientific thought, also know as Occam鈥檚 Razor: 鈥淲hen trying to explain something, it is vain to do with more what can be done with less.鈥 So the math we need is the math we need, really: nothing more, nothing less. As I’ve said a couple of times already, Occam would have loved the math behind QM:聽the physics call for the math, and the math becomes the physics.

That’s what makes it beautiful. 馃檪

Post scriptum:

One might think that the addition of a term in the argument in itself would lead to a beat note and, hence, a varying probability but, no! We may look at聽e(i/魔)路(E0聽+ A)路t聽as a聽product聽of two amplitudes:

e(i/魔)路(E0聽+ A)路t聽=聽e(i/魔)路E0路te(i/魔)路A路t

But, when writing this all out, one just gets a cos(伪路t+尾路t)鈥搒in(伪路t+尾路t), whose absolute square |cos(伪路t+尾路t)鈥搒in(伪路t+尾路t)|2聽= 1. However, writing e(i/魔)路(E0聽+ A)路t聽as a product聽of two amplitudes in itself is interesting. We聽multiply聽amplitudes when an event consists of two sub-events. For example, the amplitude for some particle to go from s to聽x via some point a is written as:

x聽| s聽鈱via a聽=聽鈱 x聽| a 鈱尒 a聽| s聽鈱

Having said that, the graph of the product is uninteresting: the real and imaginary part of the wavefunction are a simple sine and cosine function, and their absolute square is constant, as shown below.聽graph

Adding聽two waves with very different frequencies聽鈥撀燗 is a聽fraction聽of聽E0聽鈥 gives a much more interesting pattern, like the one below, which shows an聽ei伪t+ei尾t聽= cos(伪t)鈭i路sin(伪t)+cos(尾t)鈭i路sin(尾t) =聽cos(伪t)+cos(尾t)鈭i路[sin(伪t)+sin(尾t)] pattern for聽伪 = 1 and 尾 = 0.1.

graph 2

That doesn’t look a beat note, does it? The graphs below, which use 0.5 and 0.01 for 尾 respectively, are not typical beat notes either.

graph 3graph 4

We get our typical ‘beat note’ only when we’re looking at a wave聽traveling in space, so then we involve the space variable聽x聽again, and the relations that come with in, i.e. a聽phase聽velocity vp聽= 蠅/k 聽= (E/魔)/(p/魔) = E/p = c2/v (read: all component聽waves travel at the same speed), and a group velocity vg聽= d蠅/dk = v (read: the composite聽wave or wavetrain聽travels at the classical speed of our particle, so it travels with聽the particle, so to speak). That’s what’s I’ve shown numerous times already, but I’ll insert one more animation here, just to make sure you see what we’re talking about. [Credit for the animation goes to another site, one on acoustics, actually!]


So what’s left? Nothing much. The only thing you may want to do is to continue thinking about that wavefunction. It’s tempting to think it actually聽is聽the particle, somehow. But it isn’t. So what is it then? Well… Nobody knows, really, but I like to think it does聽travel聽with the particle. So it’s like a fundamental聽property聽of the particle. We need it every time when we try to measure something: its position, its momentum, its spin (i.e. angular momentum) or, in the example of our ammonia molecule, its orientation in space. So the funny thing is that, in quantum mechanics,

  1. We can measure probabilities聽only, so there’s always some randomness. That’s how Nature works: we don’t really know what’s happening. We don’t know the internal wheels and gears, so to speak, or the ‘hidden variables’, as one interpretation of quantum mechanics would say. In fact, the most commonly accepted interpretation of quantum mechanics says there are no ‘hidden variables’.
  2. But then, as Polonius famously put, there is a method in this madness, and the pioneers 鈥 I mean Werner Heisenberg, Louis de Broglie, Niels Bohr, Paul Dirac, etcetera 鈥 discovered. All probabilities can be found by taking the square of the absolute value of a complex-valued wavefunction聽(often denoted by 唯), whose argument, or phase聽(胃),聽is given by the de Broglie relations聽蠅 = E/魔 and k聽=聽p/魔:

胃 =聽(蠅路t 鈭 k聽鈭x) = (E/魔)路t聽鈭 (p/魔)路x

That should be obvious by now, as I’ve written dozens of posts on this by now. 馃檪 I still have trouble interpreting this, however鈥攁nd I am not ashamed, because the Great Ones I just mentioned have trouble with that too. But let’s try to go as far as we can by making a few remarks:

  • 聽Adding two terms in math implies the two terms should have the same聽dimension: we can only add apples to apples, and oranges to oranges. We shouldn’t mix them. Now, the聽(E/魔)路t and (p/魔)路x termsare actually dimensionless: they are pure numbers. So that’s even better. Just check it: energy is expressed in newton路meter聽(force over distance, remember?) or electronvolts聽(1 eV聽=聽1.6脳10鈭19 J = 1.6脳10鈭19 N路m); Planck’s constant, as the quantum of action,聽is expressed in J路s or eV路s; and the unit of聽(linear) momentum is 1聽N路s = 1聽kg路m/s = 1聽N路s. E/魔 gives a number expressed per second, and p/魔 a number expressed per meter. Therefore, multiplying it by t and x respectively gives us a dimensionless number indeed.
  • It’s also an invariant number, which means we’ll always get the same value for it. As mentioned above, that’s because the聽four-vector product px渭聽=聽E路t 鈭 px聽is invariant: it doesn’t change when analyzing a phenomenon in one聽reference frame (e.g. our inertial reference frame)聽or another (i.e. in a moving frame).
  • Now, Planck’s quantum of action聽h or 魔 (they only differ in their dimension: h is measured in cycles聽per second and聽魔 is measured in聽radians聽per second) is the quantum of energy really. Indeed, if “energy is the currency of the Universe”, and it’s real and/or virtual photons who are exchanging it, then it’s good to know the currency unit is h, i.e. the energy that’s associated with one cycle聽of a photon.
  • It’s not only time and space that are related, as evidenced by the fact that t 鈭 x itself is an invariant four-vector, E and p are related too, of course! They are related through the classical velocity of the particle that we’re looking at: E/p = c2/v and, therefore, we can write:聽E路尾 = p路c, with 尾 = v/c, i.e. the relative聽velocity of our particle, as measured as a聽ratio聽of the speed of light.聽Now, I should add that the聽t 鈭 x聽four-vector is invariant only if we measure time and space in equivalent units. Otherwise, we have to write c路t 鈭 x. If we do that, so our unit of distance becomes c聽meter, rather than one meter, or our unit of time becomes the time that is needed for light to travel one meter, then聽c聽= 1, and the E路尾 = p路c聽becomes E路尾 = p, which we also write as 尾 = p/E: the ratio of the energy聽and the聽momentum聽of our particle is its (relative) velocity.

Combining all of the above, we may want to assume that we are measuring energy聽and聽momentum in terms of the Planck constant, i.e. the聽‘natural’聽unit for both. In addition, we may also want to assume that we’re measuring time and distance in equivalent units. Then the equation for the phase of our wavefunctions reduces to:

胃 =聽(蠅路t 鈭 k聽鈭檟) = E路t聽鈭 p路x

Now,聽胃 is the argument of a wavefunction, and we can always聽re-scale聽such argument by multiplying or dividing it by some聽constant. It’s just like writing the argument of a wavefunction as聽v路t鈥搙 or (v路t鈥搙)/v聽= t 鈥搙/v聽 with聽v聽the velocity of the waveform that we happen to be looking at. [In case you have trouble following this argument, please check the post I did for my kids on waves and wavefunctions.] Now, the energy conservation principle tells us the energy of a free particle won’t change. [Just to remind you, a ‘free particle’ means it is present in a ‘field-free’ space, so our particle is in a region of聽uniform potential.] You see what I am going to do now: we can, in this case, treat E as a constant, and divide聽E路t聽鈭 p路x by E, so we get a re-scaled phase for our wavefunction, which I’ll write as:

蠁 =聽(E路t聽鈭 p路x)/E = t 鈭 (p/E)路x = t 鈭 尾路x

Now that’s the argument of a wavefunction with the argument expressed in distance units. Alternatively, we could also look at p as some constant, as there is no variation in potential energy that will cause a change in momentum, i.e. in kinetic energy. We’d then divide by p and we’d get聽(E路t聽鈭 p路x)/p = (E/p)路t 鈭 x) = t/尾 鈭 x, which amounts to the same, as we can always re-scale by multiplying it with 尾, which would then yield the same t 鈭 尾路x argument.

The point is, if we measure energy and momentum in terms of the Planck unit (I mean:聽in terms of the Planck constant, i.e. the聽quantum of energy), and if we measure time and distance in ‘natural’ units too, i.e. we take the speed of light to be unity, then our Platonic wavefunction becomes as simple as:

桅(蠁) =聽a路ei蠁聽= a路ei(t 鈭 尾路x)

This is a wonderful formula, but let me first answer your most likely question: why would we use a relative聽velocity?Well… Just think of it: when everything is said and done, the whole theory of relativity and, hence, the whole of physics, is based on聽one fundamental and experimentally verified fact: the speed of light is聽absolute. In whatever reference frame, we will聽always聽measure it as聽299,792,458 m/s. That’s obvious, you’ll say, but it’s actually the weirdest thing ever if you start thinking about it, and it explains why those Lorentz transformations look so damn complicated. In any case, this聽fact聽legitimately establishes c聽as some kind of聽absolute聽measure against which all speeds can be measured. Therefore, it is only聽natural聽indeed to express a velocity as some number between 0 and 1. Now that amounts to expressing it as the聽尾 = v/c ratio.

Let’s now go back to that 桅(蠁) =聽a路ei蠁聽= a路ei(t 鈭 尾路x)聽wavefunction. Its temporal frequency 蠅 is equal to one, and its spatial frequency k is equal to 尾 = v/c. It couldn’t be simpler but, of course, we’ve got this remarkably simple result because we re-scaled the argument of our wavefunction using the聽energy聽and聽momentum聽itself as the scale factor. So, yes, we can re-write the wavefunction of our particle in a particular elegant and simple form using the only information that we have when looking at quantum-mechanical stuff: energy and momentum, because that’s what everything reduces to at that level.

Of course, the analysis above does not聽include uncertainty. Our information on the energy and the momentum of our particle will be incomplete: we’ll write E = E0聽卤 蟽E, and p = p0聽卤 蟽p. [I am a bit tired of using the 螖 symbol, so I am using the 蟽 symbol here, which denotes a standard deviation聽of some density function. It underlines the probabilistic, or statistical, nature of our approach.] But, including that, we’ve pretty much explained what quantum physics is about here.

You just need to get used to that complex exponential: ei聽= cos(鈭捪) + i路sin(鈭捪) =聽cos(蠁) 鈭 i路sin(蠁). Of course, it would have been nice if Nature would have given us a simple sine or cosine function. [Remember the sine and cosine function are actually the same, except for a phase difference of 90 degrees: sin(蠁) = cos(蟺/2鈭捪) = cos(蠁+蟺/2). So we can go always from one to the other by shifting the origin of our axis.] But… Well… As we’ve shown so many times already, a real-valued wavefunction doesn’t explain the interference we observe, be it interference of electrons or whatever other particles or, for that matter, the interference of electromagnetic waves itself, which, as you know, we also need to look at as a stream of聽photons聽, i.e. light quanta, rather than as some kind of infinitely flexible聽aether聽that’s undulating, like water or air.

So… Well… Just accept that聽ei聽is a very simple periodic function, consisting of two sine waves rather than just one, as illustrated below.


And then you need to think of stuff like this (the animation is taken from Wikipedia), but then with a projection of the sine聽of those聽phasors聽too. It’s all great fun, so I’ll let you play with it now. 馃檪