Playing with amplitudes

Pre-script (dated 26 June 2020): This post got mutilated by the removal of some material by the dark force. You should be able to follow the main story line, however. If anything, the lack of illustrations might actually help you to think things through for yourself. In any case, we now have different views on these concepts as part of our realist interpretation of quantum mechanics, so we recommend you read our recent papers instead of these old blog posts.

Original post:

Let’s play a bit with the stuff we found in our previous post. This is going to be unconventional, or experimental, if you want. The idea is to give you… Well… Some ideas. So you can play yourself. 🙂 Let’s go.

Let’s first look at Feynman’s (simplified) formula for the amplitude of a photon to go from point a to point b. If we identify point a by the position vector r₁ and point b by the position vector r₂, and using Dirac’s fancy bra-ket notation, then it’s written as:

propagator

So we have a vector dot product here: p∙r₁₂ = |p|∙|r₁₂|· cosθ = p∙r₁₂·cosα. The angle here (α) is the angle between the p and r₁₂ vector. All good. Well… No. We’ve got a problem. When it comes to calculating probabilities, the α angle doesn’t matter: |e^i·θ/r|² = 1/r². Hence, for the probability, we get: P = | 〈r₂|r₁〉 |² = 1/r₁₂². Always ! Now that’s strange. The θ = p∙r₁₂/ħ argument gives us a different phase depending on the angle (α) between p and r₁₂. But… Well… Think of it: cosα goes from 1 to 0 when α goes from 0 to ±90° and, of course, is negative when p and r₁₂ have opposite directions but… Well… According to this formula, the probabilities do not depend on the direction of the momentum. That’s just weird, I think. Did Feynman, in his iconic Lectures, give us a meaningless formula?

Maybe. We may also note this function looks like the elementary wavefunction for any particle, which we wrote as:

ψ(x, t) = a·e⁻ⁱ^∙θ = a·e⁻ⁱ^∙^{(E∙t − p∙x)/ħ}= a·e⁻ⁱ^∙^(E∙t)/ħ·eⁱ^∙^(p∙x)/ħ

The only difference is that the 〈r₂|r₁〉 sort of abstracts away from time, so… Well… Let’s get a feel for the quantities. Let’s think of a photon carrying some typical amount of energy. Hence, let’s talk visible light and, therefore, photons of a few eV only – say 5.625 eV = 5.625×1.6×10⁻¹⁹ J = 9×10⁻¹⁹ J. Hence, their momentum is equal to p = E/c = (9×10⁻¹⁹ N·m)/(3×10⁵ m/s) = 3×10⁻²⁴ N·s. That’s tiny but that’s only because newtons and seconds are enormous units at the (sub-)atomic scale. As for the distance, we may want to use the thickness of a playing card as a starter, as that’s what Young used when establishing the experimental fact of light interfering with itself. Now, playing cards in Young’s time were obviously rougher than those today, but let’s take the smaller distance: modern cards are as thin as 0.3 mm. Still, that distance is associated with a value of θ that is equal to 13.6 million. Hence, the density of our wavefunction is enormous at this scale, and it’s a bit of a miracle that Young could see any interference at all ! As shown in the table below, we only get meaningful values (remember: θ is a phase angle) when we go down to the nanometer scale (10⁻⁹ m) or, even better, the angstroms scale ((10⁻⁹ m).

So… Well… Again: what can we do with Feynman’s formula? Perhaps he didn’t give us a propagator function but something that is more general (read: more meaningful) at our (limited) level of knowledge. As I’ve been reading Feynman for quite a while now – like three or four years 🙂 – I think… Well… Yes. That’s it. Feynman wants us to think about it. 🙂 Are you joking again, Mr. Feynman? 🙂 So let’s assume the reasonable thing: let’s assume it gives us the amplitude to go from point a to point b by the position vector along some path r. So, then, in line with what we wrote in our previous post, let’s say p·r (momentum over a distance) is the action (S) we’d associate with this particular path (r) and then see where we get. So let’s write the formula like this:

ψ = a·e^i·θ = (1/r)·e^i·S/ħ = e^i·p∙r/ħ/r

We’ll use an index to denote the various paths: r₀ is the straight-line path and r_i is any (other) path. Now, quantum mechanics tells us we should calculate this amplitude for every possible path. The illustration below shows the straight-line path and two nearby paths. So each of these paths is associated with some amount of action, which we measure in Planck units: θ = S/ħ. alternative paths

The time interval is given by t = t₀= r₀/c, for all paths. Why is the time interval the same for all paths? Because we think of a photon going from some specific point in space and in time to some other specific point in space and in time. Indeed, when everything is said and done, we do think of light as traveling from point a to point b at the speed of light (c). In fact, all of the weird stuff here is all about trying to explain how it does that. 🙂

Now, if we would think of the photon actually traveling along this or that path, then this implies its velocity along any of the nonlinear paths will be larger than c, which is OK. That’s just the weirdness of quantum mechanics, and you should actually not think of the photon actually traveling along one of these paths anyway although we’ll often put it that way. Think of something fuzzier, whatever that may be. 🙂

So the action is energy times time, or momentum times distance. Hence, the difference in action between two paths i and j is given by:

δS = p·r_j − p·r_i = p·(r_j − r_i) = p·Δr

I’ll explain the δS < 2πħ/3 thing in a moment. Let’s first pause and think about the uncertainty and how we’re modeling it. We can effectively think of the variation in S as some uncertainty in the action: δS = ΔS = p·Δr. However, if S is also equal to energy times time (S = E·t), and we insist t is the same for all paths, then we must have some uncertainty in the energy, right? Hence, we can write δS as ΔS = ΔE·t. But, of course, E = E = m·c² = p·c, so we will have an uncertainty in the momentum as well. Hence, the variation in S should be written as:

δS = ΔS = Δp·Δr

That’s just logical thinking: if we, somehow, entertain the idea of a photon going from some specific point in spacetime to some other specific point in spacetime along various paths, then the variation, or uncertainty, in the action will effectively combine some uncertainty in the momentum and the distance. We can calculate Δp as ΔE/c, so we get the following:

δS = ΔS = Δp·Δr = ΔE·Δr/c = ΔE·Δt with Δt = Δr/c

So we have the two expressions for the Uncertainty Principle here: ΔS = Δp·Δr = ΔE·Δt. Just be careful with the interpretation of Δt: it’s just the equivalent of Δr. We just express the uncertainty in distance in seconds using the (absolute) speed of light. We are not changing our spacetime interval: we’re still looking at a photon going from a to b in t seconds, exactly. Let’s now look at the δS < 2πħ/3 thing. If we’re adding two amplitudes (two arrows or vectors, so to speak) and we want the magnitude of the result to be larger than the magnitude of the two contributions, then the angle between them should be smaller than 120 degrees, so that’s 2π/3 rad. The illustration below shows how you can figure that out geometrically.Hence, if S₀ is the action for r₀, then S₁ = S₀ + ħ and S₂ = S₀ + 2·ħ are still good, but S₃ = S₀ + 3·ħ is not good. Why? Because the difference in the phase angles is Δθ = S₁/ħ − S₀/ħ = (S₀ + ħ)/ħ − S₀/ħ = 1 and Δθ = S₂/ħ − S₀/ħ = (S₀ + 2·ħ)/ħ − S₀/ħ = 2 respectively, so that’s 57.3° and 114.6° respectively and that’s, effectively, less than 120°. In contrast, for the next path, we find that Δθ = S₃/ħ − S₀/ħ = (S₀ + 3·ħ)/ħ − S₀/ħ = 3, so that’s 171.9°. So that amplitude gives us a negative contribution.

Let’s do some calculations using a spreadsheet. To simplify things, we will assume we measure everything (time, distance, force, mass, energy, action,…) in Planck units. Hence, we can simply write: S_n = S₀ + n. Of course, n = 1, 2,… etcetera, right? Well… Maybe not. We are measuring action in units of ħ, but do we actually think action comes in units of ħ? I am not sure. It would make sense, intuitively, but… Well… There’s uncertainty on the energy (E) and the momentum (p) of our photon, right? And how accurately can we measure the distance? So there’s some randomness everywhere. 😦 So let’s leave that question open as for now.

We will also assume that the phase angle for S₀ is equal to 0 (or some multiple of 2π, if you want). That’s just a matter of choosing the origin of time. This makes it really easy: ΔS_n = S_n − S₀ = n, and the associated phase angle θ_n = Δθ_n is the same. In short, the amplitude for each path reduces to ψ_n = e^i·n/r₀. So we need to add these first and then calculate the magnitude, which we can then square to get a probability. Of course, there is also the issue of normalization (probabilities have to add up to one) but let’s tackle that later. For the calculations, we use Euler’s r·e^i·θ = r·(cosθ + i·sinθ) = r·cosθ + i·r·sinθ formula. Needless to say, |r·e^i·θ|² = |r|²·|e^i·θ|² = |r|²·(cos²θ + sin²θ) = r. Finally, when adding complex numbers, we add the real and imaginary parts respectively, and we’ll denote the ψ₀ + ψ₁ +ψ₂ + … sum as Ψ.

Now, we also need to see how our ΔS = Δp·Δr works out. We may want to assume that the uncertainty in p and in r will both be proportional to the overall uncertainty in the action. For example, we could try writing the following: ΔS_n = Δp_n·Δr_n = n·Δp₁·Δr₁. It also makes sense that you may want Δp_n and Δr_n to be proportional to Δp₁ and Δr₁ respectively. Combining both, the assumption would be this:

Δp_n = √n·Δp₁and Δr_n = √n·Δr₁

So now we just need to decide how we will distribute ΔS₁ = ħ = 1 over Δp₁ and Δr₁ respectively. For example, if we’d assume Δp₁ = 1, then Δr₁ = ħ/Δp₁ = 1/1 = 1. These are the calculations. I will let you analyze them. 🙂 newnew Well… We get a weird result. It reminds me of Feynman’s explanation of the partial reflection of light, shown below, but… Well… That doesn’t make much sense, does it?

partial reflection

Hmm… Maybe it does. 🙂 Look at the graph more carefully. The peaks sort of oscillate out so… Well… That might make sense… 🙂

Does it? Are we doing something wrong here? These amplitudes should reflect the ones that are reflected in those nice animations (like this one, for example, which is part of that’s part of the Wikipedia article on Feynman’s path integral formulation of quantum mechanics). So what’s wrong, if anything? Well… Our paths differ by some fixed amount of action, which doesn’t quite reflect the geometric approach that’s used in those animations. The graph below shows how the distance r varies as a function of n. geometry

If we’d use a model in which the distance would increase linearly or, preferably, exponentially, then we’d get the result we want to get, right?

Well… Maybe. Let’s try it. Hmm… We need to think about the geometry here. Look at the triangle below. triangle side If b is the straight-line path (r₀), then ac could be one of the crooked paths (r_n). To simplify, we’ll assume isosceles triangles, so a equals c and, hence, r_n = 2·a = 2·c. We will also assume the successive paths are separated by the same vertical distance (h = h₁) right in the middle, so h_b = h_n = n·h₁. It is then easy to show the following:This gives the following graph for r_n = 10 and h₁= 0.01. r graph

Is this the right step increase? Not sure. We can vary the values in our spreadsheet. Let’s first build it. The photon will have to travel faster in order to cover the extra distance in the same time, so its momentum will be higher. Let’s think about the velocity. Let’s start with the first path (n = 1). In order to cover the extra distance Δr₁, the velocity c₁ must be equal to (r₀ + Δr₁)/t = r₀/t + Δr₁/t = c + Δr₁/t = c₀ + Δr₁/t. We can write c₁ as c₁ = c₀ + Δc₁, so Δc₁ = Δr₁/t. Now, the ratio of p₁ and p₀ will be equal to the ratio of c₁ and c₀ because p₁/p₀= (mc₁)/mc₀) = c₁/c₀. Hence, we have the following formula for p₁:

p₁ = p₀·c₁/c₀ = p₀·(c₀ + Δc₁)/c₀ = p₀·[1 + Δr₁/(c₀·t) = p₀·(1 + Δr₁/r₀)

For p_n, the logic is the same, so we write:

p_n = p₀·c_n/c₀ = p₀·(c₀ + Δc_n)/c₀ = p₀·[1 + Δr_n/(c₀·t) = p₀·(1 + Δr_n/r₀)

Let’s do the calculations, and let’s use meaningful values, so the nanometer scale and actual values for Planck’s constant and the photon momentum. The results are shown below. original

Pretty interesting. In fact, this looks really good. The probability first swings around wildly, because of these zones of constructive and destructive interference, but then stabilizes. [Of course, I would need to normalize the probabilities, but you get the idea, right?] So… Well… I think we get a very meaningful result with this model. Sweet ! 🙂 I’m lovin’ it ! 🙂 And, here you go, this is (part of) the calculation table, so you can see what I am doing. 🙂 newnew

The graphs below look even better: I just changed the h₁/r₀ ratio from 1/100 to 1/10. The probability stabilizes almost immediately. 🙂 So… Well… It’s not as fancy as the referenced animation, but I think the educational value of this thing here is at least as good ! 🙂 great

🙂 This is good stuff… 🙂

Post scriptum (19 September 2017): There is an obvious inconsistency in the model above, and in the calculations. We assume there is a path r₁ = , r₂, r₂,etcetera, and then we calculate the action for it, and the amplitude, and then we add the amplitude to the sum. But, surely, we should count these paths twice, in two-dimensional space, that is. Think of the graph: we have positive and negative interference zones that are sort of layered around the straight-line path, as shown below. zones

In three-dimensional space, these lines become surfaces. Hence, rather than adding one arrow for every δ having one contribution only, we may want to add… Well… In three-dimensional space, the formula for the surface around the straight-line path would probably look like π·h_n·r₁, right? Hmm… Interesting idea. I changed my spreadsheet to incorporate that idea, and I got the graph below. It’s a nonsensical result, because the probability does swing around, but it gradually spins out of control: it never stabilizes. revised That’s because we increase the weight of the paths that are further removed from the center. So… Well… We shouldn’t be doing that, I guess. 🙂 I’ll you look for the right formula, OK? Let me know when you found it. 🙂

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The Poynting vector for the matter-wave

Original post:

In my various posts on the wavefunction – which I summarized in my e-book – I wrote at the length on the structural similarities between the matter-wave and the electromagnetic wave. Look at the following images once more:

Animation 5d_euler_f

Both are the same, and then they are not. The illustration on the right-hand side is a regular quantum-mechanical wavefunction, i.e. an amplitude wavefunction: the x-axis represents time, so we are looking at the wavefunction at some particular point in space. [Of course, we could just switch the dimensions and it would all look the same.] The illustration on the left-hand side looks similar, but it is not an amplitude wavefunction. The animation shows how the electric field vector (E) of an electromagnetic wave travels through space. Its shape is the same. So it is the same function. Is it also the same reality?

Yes and no. The two energy propagation mechanisms are structurally similar. The key difference is that, in electromagnetics, we get two waves for the price of one. Indeed, the animation above does not show the accompanying magnetic field vector (B), which is equally essential. But, for the rest, Schrödinger’s equation and Maxwell’s equation model a similar energy propagation mechanism, as shown below.

They have to, as the force laws are similar too:

Coulomb Law

gravitation law

The only difference is that mass comes in one color only, so to speak: it’s always positive. In contrast, electric charge comes in two colors: positive and negative. You can now guess what comes next: quantum chromodynamics, but I won’t write about that here, because I haven’t studied that yet. I won’t repeat what I wrote elsewhere, but I want to make good on one promise, and that is to develop the idea of the Poynting vector for the matter-wave. So let’s do that now. Let me first remind you of the basic ideas, however.

Basics

The animation below shows the two components of the archetypal wavefunction, i.e. the sine and cosine:

circle_cos_sin

Think of the two oscillations as (each) packing half of the total energy of a particle (like an electron or a photon, for example). Look at how the sine and cosine mutually feed into each other: the sine reaches zero as the cosine reaches plus or minus one, and vice versa. Look at how the moving dot accelerates as it goes to the center point of the axis, and how it decelerates when reaching the end points, so as to switch direction. The two functions are exactly the same function, but for a phase difference of 90 degrees, i.e. a right angle. Now, I love engines, and so it makes me think of a V-2 engine with the pistons at a 90-degree angle. Look at the illustration below. If there is no friction, we have a perpetual motion machine: it would store energy in its moving parts, while not requiring any external energy to keep it going.

two-timer-576-px-photo-369911-s-original

If it is easier for you, you can replace each piston by a physical spring, as I did below. However, I should learn how to make animations myself, because the image below does not capture the phase difference. Hence, it does not show how the real and imaginary part of the wavefunction mutually feed into each other, which is (one of the reasons) why I like the V-2 image much better. 🙂

summary 2

The point to note is: all of the illustrations above are true representations – whatever that means – of (idealized) stationary particles, and both for matter (fermions) as well as for force-carrying particles (bosons). Let me give you an example. The (rest) energy of an electron is tiny: about 8.2×10⁻¹⁴ joule. Note the minus 14 exponent: that’s an unimaginably small amount. It sounds better when using the more commonly used electronvolt scale for the energy of elementary particles: 0.511 MeV. Despite its tiny mass (or energy, I should say, but then mass and energy are directly proportional to each other: the proportionality coefficient is given by the E = m·c² formula), the frequency of the matter-wave of the electron is of the order of 1×10²⁰ = 100,000,000,000,000,000,000 cycles per second. That’s an unimaginably large number and – as I will show when we get there – that’s not because the second is a huge unit at the atomic or sub-atomic scale.

We may refer to this as the natural frequency of the electron. Higher rest masses increase the frequency and, hence, give the wavefunction an even higher density in spacetime. Let me summarize things in a very simple way:

The (total) energy that is stored in an oscillating spring is the sum of the kinetic and potential energy (T and U) and is given by the following formula: E = T + U = a₀²·m·ω₀²/2. The a₀factor is the maximum amplitude – which depends on the initial conditions, i.e. the initial pull or push. The ω₀in the formula is the natural frequency of our spring, which is a function of the stiffness of the spring (k) and the mass on the spring (m): ω₀² = k/m.
Hence, the total energy that’s stored in two springs is equal to a₀²·m·ω₀².
The similarity between the E = a₀²·m·ω₀² and the E = m·c² formula is much more than just striking. It is fundamental: the two oscillating components of the wavefunction each store half of the total energy of our particle.
To emphasize the point: ω₀ = √(k/m) is, obviously, a characteristic of the system. Likewise, c = √(E/m) is just the same: a property of spacetime.

Of course, the key question is: what is that is oscillating here? In our V-2 engine, we have the moving parts. Now what exactly is moving when it comes to the wavefunction? The easy answer is: it’s the same thing. The V-2 engine, or our springs, store energy because of the moving parts. Hence, energy is equivalent only to mass that moves, and the frequency of the oscillation obviously matters, as evidenced by the E = a₀²·m·ω₀²/2 formula for the energy in a oscillating spring. Mass. Energy is moving mass. To be precise, it’s oscillating mass. Think of it: mass and energy are equivalent, but they are not the same. That’s why the dimension of the c² factor in Einstein’s famous E = m·c² formula matters. The equivalent energy of a 1 kg object is approximately 9×10¹⁶ joule. To be precise, it is the following monstrous number:

89,875,517,873,681,764 kg·m²/s²

Note its dimension: the joule is the product of the mass unit and the square of the velocity unit. So that, then, is, perhaps, the true meaning of Einstein’s famous formula: energy is not just equivalent to mass. It’s equivalent to mass that’s moving. In this case, an oscillating mass. But we should explore the question much more rigorously, which is what I do in the next section. Let me warn you: it is not an easy matter and, even if you are able to work your way through all of the other material below in order to understand the answer, I cannot promise you that the answer will satisfy you entirely. However, it will surely help you to phrase the question.

The Poynting vector for the matter-wave

For the photon, we have the electric and magnetic field vectors E and B. The boldface highlights the fact that these are vectors indeed: they have a direction as well as a magnitude. Their magnitude has a physical dimension. The dimension of E is straightforward: the electric field strength (E) is a quantity expressed in newton per coulomb (N/C), i.e. force per unit charge. This follows straight from the F = q·E force relation.

The dimension of B is much less obvious: the magnetic field strength (B) is measured in (N/C)/(m/s) = (N/C)·(s/m). That’s what comes out of the F = q·v×B force relation. Just to make sure you understand: v×B is a vector cross product, and yields another vector, which is given by the following formula:

a×b = |a×b|·n = |a|·|b|·sinφ·n

The φ in this formula is the angle between a and b (in the plane containing them) and, hence, is always some angle between 0 and π. The n is the unit vector that is perpendicular to the plane containing a and b in the direction given by the right-hand rule. The animation below shows it works for some rather special angles:

We may also need the vector dot product, so let me quickly give you that formula too. The vector dot product yields a scalar given by the following formula:

a•b = |a|·|b|·cosφ

Let’s get back to the F = q·v×B relation. A dimensional analysis shows that the dimension of B must involve the reciprocal of the velocity dimension in order to ensure the dimensions come out alright:

[F]= [q·v×B] = [q]·[v]·[B] = C·(m/s)·(N/C)·(s/m) = N

We can derive the same result in a different way. First, note that the magnitude of B will always be equal to E/c (except when none of the charges is moving, so B is zero), which implies the same:

[B] = [E/c] = [E]/[c] = (N/C)/(m/s) = (N/C)·(s/m)

Finally, the Maxwell equation we used to derive the wavefunction of the photon was ∂E/∂t = c²∇×B, which also tells us the physical dimension of B must involve that s/m factor. Otherwise, the dimensional analysis would not work out:

[∂E/∂t] = (N/C)/s = N/(C·s)
[c²∇×B] = [c²]·[∇×B] = (m²/s²)·[(N/C)·(s/m)]/m= N/(C·s)

This analysis involves the curl operator ∇×, which is a rather special vector operator. It gives us the (infinitesimal) rotation of a three-dimensional vector field. You should look it up so you understand what we’re doing here.

Now, when deriving the wavefunction for the photon, we gave you a purely geometric formula for B:

B = e_x×E = i·E

Now I am going to ask you to be extremely flexible: wouldn’t you agree that the B = E/c and the B = e_x×E = i·E formulas, jointly, only make sense if we’d assign the s/m dimension to e_x and/or to i? I know you’ll think that’s nonsense because you’ve learned to think of the e_x× and/or i· operation as a rotation only. What I am saying here is that it also transforms the physical dimension of the vector on which we do the operation: it multiplies it with the reciprocal of the velocity dimension. Don’t think too much about it, because I’ll do yet another hat trick. We can think of the real and imaginary part of the wavefunction as being geometrically equivalent to the E and B vector. Just compare the illustrations below:

e-and-b Rising_circular

Of course, you are smart, and you’ll note the phase difference between the sine and the cosine (illustrated below). So what should we do with that? Not sure. Let’s hold our breath for the moment.

Let’s first think about what dimension we could possible assign to the real part of the wavefunction. We said this oscillation stores half of the energy of the elementary particle that is being described by the wavefunction. How does that storage work for the E vector? As I explained in my post on the topic, the Poynting vector describes the energy flow in a varying electromagnetic field. It’s a bit of a convoluted story (which I won’t repeat here), but the upshot is that the energy density is given by the following formula:

Its shape should not surprise you. The formula is quite intuitive really, even if its derivation is not. The formula represents the one thing that everyone knows about a wave, electromagnetic or not: the energy in it is proportional to the square of its amplitude, and so that’s E•E = E² and B•B = B². You should also note he c²factor that comes with the B•B product. It does two things here:

As a physical constant, with some dimension of its own, it ensures that the dimensions on both sides of the equation come out alright.
The magnitude of B is 1/c of that of E, so cB = E, and so that explains the extra c² factor in the second term: we do get two waves for the price of one here and, therefore, twice the energy.

Speaking of dimensions, let’s quickly do the dimensional analysis:

E is measured in newton per coulomb, so [E•E] = [E²] = N²/C².
B is measured in (N/C)/(m/s), so we get [B•B] = [B²] = (N²/C²)·(s²/m²). However, the dimension of our c² factor is (m²/s²) and so we’re left with N²/C². That’s nice, because we need to add stuff that’s expressed in the same units.
The ε₀ is that ubiquitous physical constant in electromagnetic theory: the electric constant, aka as the vacuum permittivity. Besides ensuring proportionality, it also ‘fixes’ our units, and so we should trust it to do the same thing here, and it does: [ε₀] = C²/(N·m²), so if we multiply that with N²/C², we find that u is expressed in N/m².

Why is N/m² an energy density? The correct answer to that question involves a rather complicated analysis, but there is an easier way to think about it: just multiply N/m²with m/m, and then its dimension becomes N·m/m³= J/m³, so that’s joule per cubic meter. That looks more like an energy density dimension, doesn’t it? But it’s actually the same thing. In any case, I need to move on.

We talked about the Poynting vector, and said it represents an energy flow. So how does that work? It is also quite intuitive, as its formula really speaks for itself. Let me write it down:

Just look at it: u is the energy density, so that’s the amount of energy per unit volume at a given point, and so whatever flows out of that point must represent its time rate of change. As for the –∇•S expression… Well… The ∇• operator is the divergence, and so it give us the magnitude of a (vector) field’s source or sink at a given point. If C is a vector field (any vector field, really), then ∇•C is a scalar, and if it’s positive in a region, then that region is a source. Conversely, if it’s negative, then it’s a sink. To be precise, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. So, in this case, it gives us the volume density of the flux of S. If you’re somewhat familiar with electromagnetic theory, then you will immediately note that the formula has exactly the same shape as the ∇•j = −∂ρ/∂t formula, which represents a flow of electric charge.

But I need to get on with my own story here. In order to not create confusion, I will denote the total energy by U, rather than E, because we will continue to use E for the magnitude of the electric field. We said the real and the imaginary component of the wavefunction were like the E and B vector, but what’s their dimension? It must involve force, but it should obviously not involve any electric charge. So what are our options here? You know the electric force law (i.e. Coulomb’s Law) and the gravitational force law are structurally similar:

Coulomb Law

gravitation law

So what if we would just guess that the dimension of the real and imaginary component of our wavefunction should involve a newton per kg factor (N/kg), so that’s force per mass unit rather than force per unit charge? But… Hey! Wait a minute! Newton’s force law defines the newton in terms of mass and acceleration, so we can do a substitution here: 1 N = 1 kg·m/s² ⇔ 1 kg = 1 N·s²/m. Hence, our N/kg dimension becomes:

N/kg = N/(N·s²/m)= m/s²

What is this: m/s²? Is that the dimension of the a·cosθ term in the a·e^−i·θ= a·cosθ − i·a·sinθ wavefunction? I hear you. This is getting quite crazy, but let’s see where it leads us. To calculate the equivalent energy density, we’d then need an equivalent for the ε₀ factor, which – replacing the C by kg in the [ε₀] = C²/(N·m²) expression – would be equal to kg²/(N·m²). Because we know what we want (energy is defined using the force unit, not the mass unit), we’ll want to substitute the kg unit once again, so – temporarily using the μ₀ symbol for the equivalent of that ε₀ constant – we get:

[μ₀] = [N·s²/m]²/(N·m²) = N·s⁴/m⁴

Hence, the dimension of the equivalent of that ε₀·E² term becomes:

[(μ₀/2)]·[cosθ]² = (N·s⁴/m⁴)·m²/s⁴= N/m²

Bingo! How does it work for the other component? The other component has the imaginary unit (i) in front. If we continue to pursue our comparison with the E and B vectors, we should assign an extra s/m dimension because of the e_x and/or i factor, so the physical dimension of the i·sinθ term would be (m/s²)·(s/m) = s. What? Just the second? Relax. That second term in the energy density formula has the c² factor, so it all works out:

[(μ₀/2)]·[c²]·[i·sinθ]² = [(μ₀/2)]·[c²]·[i]²·[sinθ]² (N·s⁴/m⁴)·(m²/s²)·(s²/m²)·m²/s⁴= N/m²

As weird as it is, it all works out. We can calculate u and, hence, we can now also calculate the equivalent Poynting vector (S). However, I will let you think about that as an exercise. 🙂 Just note the grand conclusions:

The physical dimension of the argument of the wavefunction is physical action (newton·meter·second) and Planck’s quantum of action is the scaling factor.
The physical dimension of both the real and imaginary component of the elementary wavefunction is newton per kg (N/kg). This allows us to analyze the wavefunction as an energy propagation mechanism that is structurally similar to Maxwell’s equations, which represent the energy propagation mechanism when electromagnetic energy is involved.

As such, all we presented so far was a deep exploration of the mathematical equivalence between the gravitational and electromagnetic force laws:

Coulomb Law

gravitation law

Despite our grand conclusions, you should note we have not answered the most fundamental question of all. What is mass? What is electric charge? We have all these relations and equations, but are we any wiser, really? The answer to that question probably lies in general relativity: mass is that what curves spacetime. Likewise, we may look at electric charge as causing a very special type of spacetime curvature. However, even such answer – which would involve a much more complicated mathematical analysis – may not satisfy you. In any case, I will let you digest this post. I hope you enjoyed it as much as I enjoyed writing it. 🙂

Post scriptum: Of all of the weird stuff I presented here, I think the dimensional analyses were the most interesting. Think of the N/kg = N/(N·s²/m)= m/s²identity, for example. The m/s² dimension is the dimension of physical acceleration (or deceleration): the rate of change of the velocity of an object. The identity comes straight out of Newton’s force law:

F = m·a ⇔ F/m = a

Now look, once again, at the animation, and remember the formula for the argument of the wavefunction: θ = E₀∙t’. The energy of the particle that is being described is the (angular) frequency of the real and imaginary components of the wavefunction.

circle_cos_sin

The relation between (1) the (angular) frequency of a harmonic oscillator (which is what the sine and cosine represent here) and (2) the acceleration along the axis is given by the following equation:

a(x) = −ω₀²·x

I’ll let you think about what that means. I know you will struggle with it – because I did – and, hence, let me give you the following hint:

The energy of an ordinary string wave, like a guitar string oscillating in one dimension only, will be proportional to the square of the frequency.
However, for two-dimensional waves – such as an electromagnetic wave – we find that the energy is directly proportional to the frequency. Think of Einstein’s E = h·f = ħ·ω relation, for example. There is no squaring here!

It is a strange observation. Those two-dimensional waves – the matter-wave, or the electromagnetic wave – give us two waves for the price of one, each carrying half of the total energy but, as a result, we no longer have that square function. Think about it. Solving the mystery will make you feel like you’ve squared the circle, which – as you know – is impossible. 🙂

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Quantum Mechanics: The Other Introduction

About three weeks ago, I brought my most substantial posts together in one document: it’s the Deep Blue page of this site. I also published it on Amazon/Kindle. It’s nice. It crowns many years of self-study, and many nights of short and bad sleep – as I was mulling over yet another paradox haunting me in my dreams. It’s been an extraordinary climb but, frankly, the view from the top is magnificent. 🙂

The offer is there: anyone who is willing to go through it and offer constructive and/or substantial comments will be included in the book’s acknowledgements section when I go for a second edition (which it needs, I think). First person to be acknowledged here is my wife though, Maria Elena Barron, as she has given me the spacetime and, more importantly, the freedom to take this bull by its horns.Below I just copy the foreword, just to give you a taste of it. 🙂

Foreword

Another introduction to quantum mechanics? Yep. I am not hoping to sell many copies, but I do hope my unusual background—I graduated as an economist, not as a physicist—will encourage you to take on the challenge and grind through this.

I’ve always wanted to thoroughly understand, rather than just vaguely know, those quintessential equations: the Lorentz transformations, the wavefunction and, above all, Schrödinger’s wave equation. In my bookcase, I’ve always had what is probably the most famous physics course in the history of physics: Richard Feynman’s Lectures on Physics, which have been used for decades, not only at Caltech but at many of the best universities in the world. Plus a few dozen other books. Popular books—which I now regret I ever read, because they were an utter waste of time: the language of physics is math and, hence, one should read physics in math—not in any other language.

But Feynman’s Lectures on Physics—three volumes of about fifty chapters each—are not easy to read. However, the experimental verification of the existence of the Higgs particle in CERN’s LHC accelerator a couple of years ago, and the award of the Nobel prize to the scientists who had predicted its existence (including Peter Higgs and François Englert), convinced me it was about time I take the bull by its horns. While, I consider myself to be of average intelligence only, I do feel there’s value in the ideal of the ‘Renaissance man’ and, hence, I think stuff like this is something we all should try to understand—somehow. So I started to read, and I also started a blog (www.readingfeynman.org) to externalize my frustration as I tried to cope with the difficulties involved. The site attracted hundreds of visitors every week and, hence, it encouraged me to publish this booklet.

So what is it about? What makes it special? In essence, it is a common-sense introduction to the key concepts in quantum physics. However, while common-sense, it does not shy away from the math, which is complicated, but not impossible. So this little book is surely not a Guide to the Universe for Dummies. I do hope it will guide some Not-So-Dummies. It basically recycles what I consider to be my more interesting posts, but combines them in a comprehensive structure.

It is a bit of a philosophical analysis of quantum mechanics as well, as I will – hopefully – do a better job than others in distinguishing the mathematical concepts from what they are supposed to describe, i.e. physical reality.

Last but not least, it does offer some new didactic perspectives. For those who know the subject already, let me briefly point these out:

I. Few, if any, of the popular writers seems to have noted that the argument of the wavefunction (θ = E·t – p·t) – using natural units (hence, the numerical value of ħ and c is one), and for an object moving at constant velocity (hence, x = v·t) – can be written as the product of the proper time of the object and its rest mass:

θ = E·t – p·x = E·t − p·x = m_v·t − m_v·v·x = m_v·(t − v·x)

⇔ θ = m₀·(t − v·x)/√(1 – v²) = m₀·t’

Hence, the argument of the wavefunction is just the proper time of the object with the rest mass acting as a scaling factor for the time: the internal clock of the object ticks much faster if it’s heavier. This symmetry between the argument of the wavefunction of the object as measured in its own (inertial) reference frame, and its argument as measured by us, in our own reference frame, is remarkable, and allows to understand the nature of the wavefunction in a more intuitive way.

While this approach reflects Feynman’s idea of the photon stopwatch, the presentation in this booklet generalizes the concept for all wavefunctions, first and foremost the wavefunction of the matter-particles that we’re used to (e.g. electrons).

II. Few, if any, have thought of looking at Schrödinger’s wave equation as an energy propagation mechanism. In fact, when helping my daughter out as she was trying to understand non-linear regression (logit and Poisson regressions), it suddenly realized we can analyze the wavefunction as a link function that connects two physical spaces: the physical space of our moving object, and a physical energy space.

Re-inserting Planck’s quantum of action in the argument of the wavefunction – so we write θ as θ = (E/ħ)·t – (p/ħ)·x = [E·t – p·x]/ħ – we may assign a physical dimension to it: when interpreting ħ as a scaling factor only (and, hence, when we only consider its numerical value, not its physical dimension), θ becomes a quantity expressed in newton·meter·second, i.e. the (physical) dimension of action. It is only natural, then, that we would associate the real and imaginary part of the wavefunction with some physical dimension too, and a dimensional analysis of Schrödinger’s equation tells us this dimension must be energy.

This perspective allows us to look at the wavefunction as an energy propagation mechanism, with the real and imaginary part of the probability amplitude interacting in very much the same way as the electric and magnetic field vectors E and B. This leads me to the next point, which I make rather emphatically in this booklet: the propagation mechanism for electromagnetic energy – as described by Maxwell’s equations – is mathematically equivalent to the propagation mechanism that’s implicit in the Schrödinger equation.

I am, therefore, able to present the Schrödinger equation in a much more coherent way, describing not only how this famous equation works for electrons, or matter-particles in general (i.e. fermions or spin-1/2 particles), which is probably the only use of the Schrödinger equation you are familiar with, but also how it works for bosons, including the photon, of course, but also the theoretical zero-spin boson!

In fact, I am personally rather proud of this. Not because I am doing something that hasn’t been done before (I am sure many have come to the same conclusions before me), but because one always has to trust one’s intuition. So let me say something about that third innovation: the photon wavefunction.

III. Let me tell you the little story behind my photon wavefunction. One of my acquaintances is a retired nuclear scientist. While he knew I was delving into it all, I knew he had little time to answer any of my queries. However, when I asked him about the wavefunction for photons, he bluntly told me photons didn’t have a wavefunction. I should just study Maxwell’s equations and that’s it: there’s no wavefunction for photons: just this traveling electric and a magnetic field vector. Look at Feynman’s Lectures, or any textbook, he said. None of them talk about photon wavefunctions. That’s true, but I knew he had to be wrong. I mulled over it for several months, and then just sat down and started doing to fiddle with Maxwell’s equations, assuming the oscillations of the E and B vector could be described by regular sinusoids. And – Lo and behold! – I derived a wavefunction for the photon. It’s fully equivalent to the classical description, but the new expression solves the Schrödinger equation, if we modify it in a rather logical way: we have to double the diffusion constant, which makes sense, because E and B give you two waves for the price of one!

[…]

In any case, I am getting ahead of myself here, and so I should wrap up this rather long introduction. Let me just say that, through my rather long journey in search of understanding – rather than knowledge alone – I have learned there are so many wrong answers out there: wrong answers that hamper rather than promote a better understanding. Moreover, I was most shocked to find out that such wrong answers are not the preserve of amateurs alone! This emboldened me to write what I write here, and to publish it. Quantum mechanics is a logical and coherent framework, and it is not all that difficult to understand. One just needs good pointers, and that’s what I want to provide here.

As of now, it focuses on the mechanics in particular, i.e. the concept of the wavefunction and wave equation (better known as Schrödinger’s equation). The other aspect of quantum mechanics – i.e. the idea of uncertainty as implied by the quantum idea – will receive more attention in a later version of this document. I should also say I will limit myself to quantum electrodynamics (QED) only, so I won’t discuss quarks (i.e. quantum chromodynamics, which is an entirely different realm), nor will I delve into any of the other more recent advances of physics.

In the end, you’ll still be left with lots of unanswered questions. However, that’s quite OK, as Richard Feynman himself was of the opinion that he himself did not understand the topic the way he would like to understand it. But then that’s exactly what draws all of us to quantum physics: a common search for a deep and full understanding of reality, rather than just some superficial description of it, i.e. knowledge alone.

So let’s get on with it. I am not saying this is going to be easy reading. In fact, I blogged about much easier stuff than this in my blog—treating only aspects of the whole theory. This is the whole thing, and it’s not easy to swallow. In fact, it may well too big to swallow as a whole. But please do give it a try. I wanted this to be an intuitive but formally correct introduction to quantum math. However, when everything is said and done, you are the only who can judge if I reached that goal.

Of course, I should not forget the acknowledgements but… Well… It was a rather lonely venture, so I am only going to acknowledge my wife here, Maria, who gave me all of the spacetime and all of the freedom I needed, as I would get up early, or work late after coming home from my regular job. I sacrificed weekends, which we could have spent together, and – when mulling over yet another paradox – the nights were often short and bad. Frankly, it’s been an extraordinary climb, but the view from the top is magnificent.

I just need to insert one caution, my site (www.readingfeynman.org) includes animations, which make it much easier to grasp some of the mathematical concepts that I will be explaining. Hence, I warmly recommend you also have a look at that site, and its Deep Blue page in particular – as that page has the same contents, more or less, but the animations make it a much easier read.

Have fun with it!

Jean Louis Van Belle, BA, MA, BPhil, Drs.

All what you ever wanted to know about the photon wavefunction…

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 read the more recent exposé on the matter that is being presented here, which you can find by clicking on the provided link.

Original post:

This post is, essentially, a continuation of my previous post, in which I juxtaposed the following images:

Both are the same, and then they’re not. The illustration on the right-hand side is a regular quantum-mechanical wavefunction, i.e. an amplitude wavefunction. You’ve seen that one before. In this case, the x-axis represents time, so we’re looking at the wavefunction at some particular point in space. ]You know we can just switch the dimensions and it would all look the same.] The illustration on the left-hand side looks similar, but it’s not an amplitude wavefunction. The animation shows how the electric field vector (E) of an electromagnetic wave travels through space. Its shape is the same. So it’s the same function. Is it also the same reality?

Yes and no. And I would say: more no than yes—in this case, at least. Note that the animation does not show the accompanying magnetic field vector (B). That vector is equally essential in the electromagnetic propagation mechanism according to Maxwell’s equations, which—let me remind you—are equal to:

∂B/∂t = –∇×E
∂E/∂t = ∇×B

In fact, I should write the second equation as ∂E/∂t = c²∇×B, but then I assume we measure time and distance in equivalent units, so c = 1.

You know that E and B are two aspects of one and the same thing: if we have one, then we have the other. To be precise, B is always orthogonal to E in the direction that’s given by the right-hand rule for the following vector cross-product: B = e_x×E, with e_x the unit vector pointing in the x-direction (i.e. the direction of propagation). The reality behind is illustrated below for a linearly polarized electromagnetic wave.

E and b

The B = e_x×E equation is equivalent to writing B= i·E, which is equivalent to:

B = i·E = eⁱ^(π/2)·eⁱ^{(kx − ωt)} = cos(kx − ωt + π/2) + i·sin(kx − ωt + π/2)

= −sin((kx − ωt) + i·cos(kx − ωt)

Now, E and B have only two components: E_yand E_z, and B_yand B_z. That’s only because we’re looking at some ideal or elementary electromagnetic wave here but… Well… Let’s just go along with it. 🙂 It is then easy to prove that the equation above amounts to writing:

B_y= cos(kx − ωt + π/2) = −sin(kx − ωt) = −E_z
B_z= sin(kx − ωt + π/2) = cos(kx − ωt) = E_y

We should now think of E_y and E_zas the real and imaginary part of some wavefunction, which we’ll denote as ψ_E = eⁱ^{(kx − ωt)}. So we write:

E = (E_y, E_z) = E_y + i·E_z= cos(kx − ωt) + i∙sin(kx − ωt) = Re(ψ_E) + i·Im(ψ_E) = ψ_E = eⁱ^{(kx − ωt)}

What about B? We just do the same, so we write:

B = (B_y, B_z) = B_y + i·B_z= ψ_B = i·E = i·ψ_E = −sin(kx − ωt) + i∙sin(kx − ωt) = − Im(ψ_E) + i·Re(ψ_E)

Now we need to prove that ψ_E and ψ_B are regular wavefunctions, which amounts to proving Schrödinger’s equation, i.e. ∂ψ/∂t = i·(ħ/m)·∇²ψ, for both ψ_E and ψ_B. [Note I use the Schrödinger’s equation for a zero-mass spin-zero particle here, which uses the ħ/m factor rather than the ħ/(2m) factor.] To prove that ψ_E and ψ_B are regular wavefunctions, we should prove that:

Re(∂ψ_E/∂t) = −(ħ/m)·Im(∇²ψ_E) and Im(∂ψ_E/∂t) = (ħ/m)·Re(∇²ψ_E), and
Re(∂ψ_B/∂t) = −(ħ/m)·Im(∇²ψ_B) and Im(∂ψ_B/∂t) = (ħ/m)·Re(∇²ψ_B).

Let’s do the calculations for the second pair of equations. The time derivative on the left-hand side is equal to:

∂ψ_B/∂t = −iω·ieⁱ^{(kx − ωt)}= ω·[cos(kx − ωt) + i·sin(kx − ωt)] = ω·cos(kx − ωt) + iω·sin(kx − ωt)

The second-order derivative on the right-hand side is equal to:

∇²ψ_B= ∂²ψ_B/∂x²= i·k²·eⁱ^{(kx − ωt)}= k²·cos(kx − ωt) + i·k²·sin(kx − ωt)

So the two equations for ψ_Bare equivalent to writing:

Re(∂ψ_B/∂t) = −(ħ/m)·Im(∇²ψ_B) ⇔ ω·cos(kx − ωt) = k²·(ħ/m)·cos(kx − ωt)
Im(∂ψ_B/∂t) = (ħ/m)·Re(∇²ψ_B) ⇔ ω·sin(kx − ωt) = k²·(ħ/m)·sin(kx − ωt)

So we see that both conditions are fulfilled if, and only if, ω = k²·(ħ/m).

Now, we also demonstrated in that post of mine that Maxwell’s equations imply the following:

∂B_y/∂t = –(∇×E)_y = ∂E_z/∂x = ∂[sin(kx − ωt)]/∂x = k·cos(kx − ωt) = k·E_y
∂B_z/∂t = –(∇×E)_z = – ∂E_y/∂x = – ∂[cos(kx − ωt)]/∂x = k·sin(kx − ωt) = k·E_z

Hence, using those B_y= −E_zand B_z= E_yequations above, we can also calculate these derivatives as:

∂B_y/∂t = −∂E_z/∂t = −∂sin(kx − ωt)/∂t = ω·cos(kx − ωt) = ω·E_y
∂B_z/∂t = ∂E_y/∂t = ∂cos(kx − ωt)/∂t = −ω·[−sin(kx − ωt)] = ω·E_z

In other words, Maxwell’s equations imply that ω = k, which is consistent with us measuring time and distance in equivalent units, so the phase velocity is c = 1 = ω/k.

So far, so good. We basically established that the propagation mechanism for an electromagnetic wave, as described by Maxwell’s equations, is fully coherent with the propagation mechanism—if we can call it like that—as described by Schrödinger’s equation. We also established the following equalities:

ω = k
ω = k²·(ħ/m)

The second of the two de Broglie equations tells us that k = p/ħ, so we can combine these two equations and re-write these two conditions as:

ω/k = 1 = k·(ħ/m) = (p/ħ)·(ħ/m) = p/m ⇔ p = m

What does this imply? The p here is the momentum: p = m·v, so this condition implies v must be equal to 1 too, so the wave velocity is equal to the speed of light. Makes sense, because we actually are talking light here. 🙂 In addition, because it’s light, we also know E/p = c = 1, so we have – once again – the general E = p = m equation, which we’ll need!

OK. Next. Let’s write the Schrödinger wave equation for both wavefunctions:

∂ψ_E/∂t = i·(ħ/m_E)·∇²ψ_E, and
∂ψ_B/∂t = i·(ħ/m_B)·∇²ψ_B.

Huh? What’s m_E and m_E? We should only associate one mass concept with our electromagnetic wave, shouldn’t we? Perhaps. I just want to be on the safe side now. Of course, if we distinguish m_E and m_B, we should probably also distinguish p_E and p_B, and E_E and E_B as well, right? Well… Yes. If we accept this line of reasoning, then the mass factor in Schrödinger’s equations is pretty much like the 1/c² = μ₀ε₀ factor in Maxwell’s (1/c²)·∂E/∂t = ∇×B equation: the mass factor appears as a property of the medium, i.e. the vacuum here! [Just check my post on physical constants in case you wonder what I am trying to say here, in which I explain why and how c defines the (properties of the) vacuum.]

To be consistent, we should also distinguish p_E and p_B, and E_E and E_B, and so we should write ψ_Eand ψ_B as:

ψ_E = eⁱ^{(k_Ex − ω_Et)}, and
ψ_B = eⁱ^{(k_Bx − ω_Bt)}.

Huh? Yes. I know what you think: we’re talking one photon—or one electromagnetic wave—so there can be only one energy, one momentum and, hence, only one k, and one ω. Well… Yes and no. Of course, the following identities should hold: k_E = k_B and, likewise, ω_E= ω_B. So… Yes. They’re the same: one k and one ω. But then… Well… Conceptually, the two k’s and ω’s are different. So we write:

p_E = E_E = m_E, and
p_B = E_B = m_B.

The obvious question is: can we just add them up to find the total energy and momentum of our photon? The answer is obviously positive: E = E_E + E_B, p = p_E + p_B and m = m_E + m_B.

Let’s check a few things now. How does it work for the phase and group velocity of ψ_Eand ψ_B? Simple:

v_g = ∂ω_E/∂k_E = ∂[E_E/ħ]/∂[p_E/ħ] = ∂E_E/∂p_E = ∂p_E/∂p_E = 1
v_p = ω_E/k_E = (E_E/ħ)/(p_E/ħ) = E_E/p_E = p_E/p_E = 1

So we’re fine, and you can check the result for ψ_Bby substituting the subscript E for B. To sum it all up, what we’ve got here is the following:

We can think of a photon having some energy that’s equal to E = p = m (assuming c = 1), but that energy would be split up in an electric and a magnetic wavefunction respectively: ψ_Eand ψ_B.
Schrödinger’s equation applies to both wavefunctions, but the E, p and m in those two wavefunctions are the same and not the same: their numerical value is the same (p_E =E_E = m_E = p_B =E_B = m_B), but they’re conceptually different. They must be: if not, we’d get a phase and group velocity for the wave that doesn’t make sense.

Of course, the phase and group velocity for the sum of the ψ_Eand ψ_Bwaves must also be equal to c. This is obviously the case, because we’re adding waves with the same phase and group velocity c, so there’s no issue with the dispersion relation.

So let’s insert those p_E =E_E = m_E = p_B =E_B = m_B values in the two wavefunctions. For ψ_E, we get:

ψ_E= eⁱ^{[k_Ex − ω_Et)}= eⁱ^{[(p_E/ħ)·x − (E_E/ħ)·t]}

You can do the calculation for ψ_Byourself. Let’s simplify our life a little bit and assume we’re using Planck units, so ħ = 1, and so the wavefunction simplifies to ψ_E= eⁱ^{·(p_E·x − E_E·t)}. We can now add the components of E and B using the summation formulas for sines and cosines:

1. B_y+ E_y = cos(p_B·x − E_B·t + π/2) + cos(p_E·x − E_E·t) = 2·cos[(p·x − E·t + π/2)/2]·cos(π/4) = √2·cos(p·x/2 − E·t/2 + π/4)

2. B_z+ E_z = sin(p_B·x − E_B·t+π/2) + sin(p_E·x − E_E·t) = 2·sin[(p·x − E·t + π/2)/2]·cos(π/4) = √2·sin(p·x/2 − E·t/2 + π/4)

Interesting! We find a composite wavefunction for our photon which we can write as:

E + B = ψ_E+ ψ_B= E + i·E = √2·eⁱ^{(p·x/2 − E·t/2 + π/4)}= √2·eⁱ^(π/4)·eⁱ^{(p·x/2 − E·t/2)}= √2·eⁱ^(π/4)·E

What a great result! It’s easy to double-check, because we can see the E + i·E = √2·eⁱ^(π/4)·E formula implies that 1 + i should equal √2·eⁱ^(π/4). Now that’s easy to prove, both geometrically (just do a drawing) or formally: √2·eⁱ^(π/4) = √2·cos(π/4) + i·sin(π/4eⁱ^(π/4) = (√2/√2) + i·(√2/√2) = 1 + i. We’re bang on! 🙂

We can double-check once more, because we should get the same from adding E and B = i·E, right? Let’s try:

E + B = E + i·E = cos(p_E·x − E_E·t) + i·sin(p_E·x − E_E·t) + i·cos(p_E·x − E_E·t) − sin(p_E·x − E_E·t)

= [cos(p_E·x − E_E·t) – sin(p_E·x − E_E·t)] + i·[sin(p_E·x − E_E·t) – cos(p_E·x − E_E·t)]

Indeed, we can see we’re going to obtain the same result, because the −sinθ in the real part of our composite wavefunction is equal to cos(θ+π/2), and the −cosθ in its imaginary part is equal to sin(θ+π/2). So the sum above is the same sum of cosines and sines that we did already.

So our electromagnetic wavefunction, i.e. the wavefunction for the photon, is equal to:

ψ = ψ_E+ ψ_B= √2·eⁱ^{(p·x/2 − E·t/2 + π/4)} = √2·eⁱ^(π/4)·eⁱ^{(p·x/2 − E·t/2)}

What about the √2 factor in front, and the π/4 term in the argument itself? No sure. It must have something to do with the way the magnetic force works, which is not like the electric force. Indeed, remember the Lorentz formula: the force on some unit charge (q = 1) will be equal to F = E + v×B. So… Well… We’ve got another cross-product here and so the geometry of the situation is quite complicated: it’s not like adding two forces F₁and F₂to get some combined force F = F₁and F₂.

In any case, we need the energy, and we know that its proportional to the square of the amplitude, so… Well… We’re spot on: the square of the √2 factor in the √2·cos product and √2·sin product is 2, so that’s twice… Well… What? Hold on a minute! We’re actually taking the absolute square of the E + B = ψ_E+ ψ_B= E + i·E = √2·eⁱ^{(p·x/2 − E·t/2 + π/4)}wavefunction here. Is that legal? I must assume it is—although… Well… Yes. You’re right. We should do some more explaining here.

We know that we usually measure the energy as some definite integral, from t = 0 to some other point in time, or over the cycle of the oscillation. So what’s the cycle here? Our combined wavefunction can be written as √2·eⁱ^{(p·x/2 − E·t/2 + π/4)}= √2·eⁱ^{(θ/2 + π/4)}, so a full cycle would correspond to θ going from 0 to 4π here, rather than from 0 to 2π. So that explains the √2 factor in front of our wave equation.

Bingo! If you were looking for an interpretation of the Planck energy and momentum, here it is. And, while everything that’s written above is not easy to understand, it’s close to the ‘intuitive’ understanding to quantum mechanics that we were looking for, isn’t it? The quantum-mechanical propagation model explains everything now. 🙂 I only need to show one more thing, and that’s the different behavior of bosons and fermions:

The amplitudes of identitical bosonic particles interfere with a positive sign, so we have Bose-Einstein statistics here. As Feynman writes it: (amplitude direct) + (amplitude exchanged).
The amplitudes of identical fermionic particles interfere with a negative sign, so we have Fermi-Dirac statistics here: (amplitude direct) − (amplitude exchanged).

I’ll think about it. I am sure it’s got something to do with that B= i·E formula or, to put it simply, with the fact that, when bosons are involved, we get two wavefunctions (ψ_Eand ψ_B) for the price of one. The reasoning should be something like this:

I. For a massless particle (i.e. a zero-mass fermion), our wavefunction is just ψ = eⁱ^{(p·x − E·t)}. So we have no √2 or √2·eⁱ^(π/4)factor in front here. So we can just add any number of them – ψ₁+ ψ₂+ ψ₃+ … – and then take the absolute square of the amplitude to find a probability density, and we’re done.

II. For a photon (i.e. a zero-mass boson), our wavefunction is √2·eⁱ^(π/4)·eⁱ^{(p·x − E·t)/2}, which – let’s introduce a new symbol – we’ll denote by φ, so φ = √2·eⁱ^(π/4)·eⁱ^{(p·x − E·t)/2}. Now, if we add any number of these, we get a similar sum but with that √2·eⁱ^(π/4)factor in front, so we write: φ₁+ φ₂+ φ₃+ … = √2·eⁱ^(π/4)·(ψ₁+ ψ₂+ ψ₃+ …). If we take the absolute square now, we’ll see the probability density will be equal to twice the density for the ψ₁+ ψ₂+ ψ₃+ … sum, because

|√2·eⁱ^(π/4)·(ψ₁+ ψ₂+ ψ₃+ …)|² = |√2·eⁱ^(π/4)|²·|ψ₁+ ψ₂+ ψ₃+ …)|² = 2·|ψ₁+ ψ₂+ ψ₃+ …)|²

So… Well… I still need to connect this to Feynman’s (amplitude direct) ± (amplitude exchanged) formula, but I am sure it can be done.

Now, we haven’t tested the complete √2·eⁱ^(π/4)·eⁱ^{(p·x − E·t)/2}wavefunction. Does it respect Schrödinger’s ∂ψ/∂t = i·(1/m)·∇²ψ or, including the 1/2 factor, the ∂ψ/∂t = i·[1/2m)]·∇²ψ equation? [Note we assume, once again, that ħ = 1, so we use Planck units once more.] Let’s see. We can calculate the derivatives as:

∂ψ/∂t = −√2·eⁱ^(π/4)·e^{−i∙[p·x − E·t]/2}·(i·E/2)
∇²ψ = ∂²[√2·eⁱ^(π/4)·e^{−i∙[p·x − E·t]/2}]/∂x²= √2·eⁱ^(π/4)·∂[√2·eⁱ^(π/4)·e^{−i∙[p·x − E·t]/2}·(i·p/2)]/∂x = −√2·eⁱ^(π/4)·e^{−i∙[p·x − E·t]/2}·(p²/4)

So Schrödinger’s equation becomes:

−i·√2·eⁱ^(π/4)·e^{−i∙[p·x − E·t]/2}·(i·E/2) = −i·(1/m)·√2·eⁱ^(π/4)·e^{−i∙[p·x − E·t]/2}·(p²/4) ⇔ 1/2 = 1/4!?

That’s funny ! It doesn’t work ! The E and m and p² are OK because we’ve got that E = m = p equation, but we’ve got problems with yet another factor 2. It only works when we use the 2/m coefficient in Schrödinger’s equation.

So… Well… There’s no choice. That’s what we’re going to do. The Schrödinger equation for the photon is ∂ψ/∂t = i·(2/m)·∇²ψ !

It’s a very subtle point. This is all great, and very fundamental stuff! Let’s now move on to Schrödinger’s actual equation, i.e. the ∂ψ/∂t = i·(ħ/2m)·∇²ψ equation.

Post scriptum on the Planck units:

If we measure time and distance in equivalent units, say seconds, we can re-write the quantum of action as:

1.0545718×10⁻³⁴N·m·s = (1.21×10⁴⁴N)·(1.6162×10⁻³⁵m)·(5.391×10⁻⁴⁴ s)

⇔ (1.0545718×10⁻³⁴/2.998×10⁸) N·s² = (1.21×10⁴⁴N)·(1.6162×10⁻³⁵/2.998×10⁸ s)(5.391×10⁻⁴⁴ s)

⇔ (1.21×10⁴⁴N) = [(1.0545718×10⁻³⁴/2.998×10⁸)]/[(1.6162×10⁻³⁵/2.998×10⁸ s)(5.391×10⁻⁴⁴ s)] N·s²/s²

You’ll say: what’s this? Well… Look at it. We’ve got a much easier formula for the Planck force—much easier than the standard formulas you’ll find on Wikipedia, for example. If we re-interpret the symbols ħ and c so they denote the numerical value of the quantum of action and the speed of light in standard SI units (i.e. newton, meter and second)—so ħ and c become dimensionless, or mathematical constants only, rather than physical constants—then the formula above can be written as:

F_P newton = (ħ/c)/[(l_P/c)·t_P] newton ⇔ F_P = ħ/(l_P·t_P)

Just double-check it: 1.0545718×10⁻³⁴/(1.6162×10⁻³⁵·5.391×10⁻⁴⁴) = 1.21×10⁴⁴. Bingo!

You’ll say: what’s the point? The point is: our model is complete. We don’t need the other physical constants – i.e. the Coulomb, Boltzmann and gravitational constant – to calculate the Planck units we need, i.e. the Planck force, distance and time units. It all comes out of our elementary wavefunction! All we need to explain the Universe – or, let’s be more modest, quantum mechanics – is two numerical constants (c and ħ) and Euler’s formula (which uses π and e, of course). That’s it.

If you don’t think that’s a great result, then… Well… Then you’re not reading this. 🙂

The photon wavefunction

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:

In my previous posts, I juxtaposed the following images:

Both are the same, and then they’re not. The illustration on the left-hand side shows how the electric field vector (E) of an electromagnetic wave travels through space, but it does not show the accompanying magnetic field vector (B), which is as essential in the electromagnetic propagation mechanism according to Maxwell’s equations:

∂B/∂t = –∇×E
∂E/∂t = c²∇×B = ∇×B for c = 1

The second illustration shows a wavefunction eⁱ^{(kx − ωt)}= cos(kx − ωt) + i∙sin(kx − ωt). Its propagation mechanism—if we can call it like that—is Schrödinger’s equation:

∂ψ/∂t = i·(ħ/2m)·∇²ψ

We already drew attention to the fact that an equation like this models some flow. To be precise, the Laplacian on the right-hand side is the second derivative with respect to x here, and, therefore, expresses a flux density: a flow per unit surface area, i.e. per square meter. To be precise: the Laplacian represents the flux density of the gradient flow of ψ.

On the left-hand side of Schrödinger’s equation, we have a time derivative, so that’s a flow per second. The ħ/2m factor is like a diffusion constant. In fact, strictly speaking, that ħ/2m factor is a diffusion constant, because it does exactly the same thing as the diffusion constant D in the diffusion equation ∂φ/∂t = D·∇²φ, i.e:

As a constant of proportionality, it quantifies the relationship between both derivatives.
As a physical constant, it ensures the dimensions on both sides of the equation are compatible.

So our diffusion constant here is ħ/2m. Because of the Uncertainty Principle, m is always going to be some integer multiple of ħ/2, so ħ/2m = 1, 1/2, 1/3, 1/4 etcetera. In other words, the ħ/2m term is the inverse of the mass measured in units of ħ/2. We get the terms of the harmonic series here. How convenient! 🙂

In our previous posts, we studied the wavefunction for a zero-mass particle. Such particle has zero rest mass but – because of its movement – does have some energy, and, therefore, some mass and momentum. In fact, measuring time and distance in equivalent units (so c = 1), we found that E = m = p = ħ/2 for the zero-mass particle. It had to be. If not, our equations gave us nonsense. So Schrödinger’s equation was reduced to:

∂ψ/∂t = i·∇²ψ

How elegant! We only need to explain that imaginary unit (i) in the equation. It does a lot of things. First, it gives us two equations for the price of one—thereby providing a propagation mechanism indeed. It’s just like the E and B vectors. Indeed, we can write that ∂ψ/∂t = i·∇²ψ equation as:

Re(∂ψ/∂t) = −Im(∇²ψ)
Im(∂ψ/∂t) = Re(∇²ψ)

You should be able to show that the two equations above are effectively equivalent to Schrödinger’s equation. If not… Well… Then you should not be reading this stuff.] The two equations above show that the real part of the wavefunction feeds into its imaginary part, and vice versa. Both are as essential. Let me say this one more time: the so-called real and imaginary part of a wavefunction are equally real—or essential, I should say!

Second, i gives us the circle. Huh? Yes. Writing the wavefunction as ψ = a + i·b is not just like writing a vector in terms of its Cartesian coordinates, even if it looks very much that way. Why not? Well… Never forget: i²= −1, and so—let me use mathematical lingo here—the introduction of i makes our metric space complete. To put it simply: we can now compute everything. In short, the introduction of the imaginary unit gives us that wonderful mathematical construct, eⁱ^{(kx − ωt)}, which allows us to model everything. In case you wonder, I mean: everything! Literally. 🙂

However, we’re not going to impose any pre-conditions here, and so we’re not going to make that E = m = p = ħ/2 assumption now. We’ll just re-write Schrödinger’s equation as we did last time—so we’re going to keep our ‘diffusion constant’ ħ/2m as for now:

Re(∂ψ/∂t) = −(ħ/2m)·Im(∇²ψ)
Im(∂ψ/∂t) = (ħ/2m)·Re(∇²ψ)

So we have two pairs of equations now. Can they be related? Well… They look the same, so they had better be related! 🙂 Let’s explore it. First note that, if we’d equate the direction of propagation with the x-axis, we can write the E vector as the sum of two y- and z-components: E = (E_y, E_z). Using complex number notation, we can write E as:

E = (E_y, E_z) = E_y + i·E_z

In case you’d doubt, just think of this simple drawing:

The next step is to imagine—funny word when talking complex numbers—that E_y and E_zare the real and imaginary part of some wavefunction, which we’ll denote as ψ_E = eⁱ^{(kx − ωt)}. So now we can write:

E = (E_y, E_z) = E_y + i·E_z= cos(kx − ωt) + i∙sin(kx − ωt) = Re(ψ_E) + i·Im(ψ_E)

What’s k and ω? Don’t worry about it—for the moment, that is. We’ve done nothing special here. In fact, we’re used to representing waves as some sine or cosine function, so that’s what we are doing here. Nothing more. Nothing less. We just need two sinusoids because of the circular polarization of our electromagnetic wave.

What’s next? Well… If ψ_E is a regular wavefunction, then we should be able to check if it’s a solution to Schrödinger’s equation. So we should be able to write:

Re(∂ψ_E/∂t) = −(ħ/2m)·Im(∇²ψ_E)
Im(∂ψ_E/∂t) = (ħ/2m)·Re(∇²ψ_E)

Are we? How does that work? The time derivative on the left-hand side is equal to:

∂ψ_E/∂t = −iω·eⁱ^{(kx − ωt)}= −iω·[cos(kx − ωt) + i·sin(kx − ωt)] = ω·sin(kx − ωt) − iω·cos(kx − ωt)

The second-order derivative on the right-hand side is equal to:

∇²ψ_E= ∂²ψ_E/∂x²= −k²·eⁱ^{(kx − ωt)}= −k²·cos(kx − ωt) − ik²·sin(kx − ωt)

So the two equations above are equivalent to writing:

Re(∂ψ_E/∂t) = −(ħ/2m)·Im(∇²ψ_E) ⇔ ω·sin(kx − ωt) = k²·(ħ/2m)·sin(kx − ωt)
Im(∂ψ_E/∂t) = (ħ/2m)·Re(∇²ψ_E) ⇔ −ω·cos(kx − ωt) = −k²·(ħ/2m)·cos(kx − ωt)

Both conditions are fulfilled if, and only if, ω = k²·(ħ/2m). Now, assuming we measure time and distance in equivalent units (c = 1), we can calculate the phase velocity of the electromagnetic wave as being equal to c = ω/k = 1. We also have the de Broglie equation for the matter-wave, even if we’re not quite sure whether or not we should apply that to an electromagnetic wave. In any case, the de Broglie equation tells us that k = p/ħ. So we can re-write this condition as:

ω/k = 1 = k·(ħ/2m) = (p/ħ)·(ħ/2m) = p/2m ⇔ p = 2m ⇔ m = p/2

So that’s different from the E = m = p equality we imposed when discussing the wavefunction of the zero-mass particle: we’ve got that 1/2 factor which bothered us so much once again! And it’s causing us the same trouble: how do we interpret that m = p/2 equation? It leads to nonsense once more! E = m·c²= m, but E is also supposed to be equal to p·c = p. Here, however, we find that E = p/2! We also get strange results when calculating the group and phase velocity. So… Well… What’s going on here?

I am not quite sure. It’s that damn 1/2 factor. Perhaps it’s got something to do with our definition of mass. The m in the Schrödinger equation was referred to as the effective or reduced mass of the electron wavefunction that it was supposed to model. Now that concept is something funny: it sure allows for some gymnastics, as you’ll see when going through the Wikipedia article on it! I promise I’ll dig into it—but not now and here, as I’ve got no time for that. 😦

However, the good news is that we also get a magnetic field vector with an electromagnetic wave: B. We know B is always orthogonal to E, and in the direction that’s given by the right-hand rule for the vector cross-product. Indeed, we can write B as B = e_x×E/c, with e_x the unit vector pointing in the x-direction (i.e. the direction of propagation), as shown below.

E and b

So we can do the same analysis: we just substitute E for B everywhere, and we’ll find the same condition: m = p/2. To distinguish the two wavefunctions, we used the E and B subscripts for our wavefunctions, so we wrote ψ_Eand ψ_B. We can do the same for that m = p/2 condition:

m_E= p_E/2
m_B= p_B/2

Should we just add m_Eand m_E to get a total momentum and, hence, a total energy, that’s equal to E = m = p for the whole wave? I believe we should, but I haven’t quite figured out how we should interpret that summation!

So… Well… Sorry to disappoint you. I haven’t got the answer here. But I do believe my instinct tells me the truth: the wavefunction for an electromagnetic wave—so that’s the wavefunction for a photon, basically—is essentially the same as our wavefunction for a zero-mass particle. It’s just that we get two wavefunctions for the price of one. That’s what distinguishes bosons from fermions! And so I need to figure out how they differ exactly! And… Well… Yes. That might take me a while!

In the meanwhile, we should play some more with those E and B vectors, as that’s going to help us to solve the riddle—no doubt!

Fiddling with E and B

The B = e_x×E/c equation is equivalent to saying that we’ll get B when rotating E by 90 degrees which, in turn, is equivalent to multiplication by the imaginary unit i. Huh? Yes. Sorry. Just google the meaning of the vector cross product and multiplication by i. So we can write B = i·E, which amounts to writing:

B = i·E = eⁱ^(π/2)·eⁱ^{(kx − ωt)} = eⁱ^{(kx − ωt + π/2)} = cos(kx − ωt + π/2) + i·sin(kx − ωt + π/2)

So we can now associate a wavefunction ψ_B with the field magnetic field vector B, which is the same wavefunction as ψ_E except for a phase shift equal to π/2. You’ll say: so what? Well… Nothing much. I guess this observation just concludes this long digression on the wavefunction of a photon: it’s the same wavefunction as that of a zero-mass particle—except that we get two for the price of one!

It’s an interesting way of looking at things. Let’s look at the equations we started this post with, i.e. Maxwell’s equations in free space—i.e. no stationary charges, and no currents (i.e. moving charges) either! So we’re talking those ∂B/∂t = –∇×E and ∂E/∂t = ∇×B equations now.

Note that they actually give you four equations, because they’re vector equations:

∂B/∂t = –∇×E ⇔ ∂B_y/∂t = –(∇×E)_y and ∂B_z/∂t = –(∇×E)_z
∂E/∂t = ∇×B ⇔ ∂E_y/∂t = (∇×B)_y and ∂E_z/∂t = (∇×B)_z

To figure out what that means, we need to remind ourselves of the definition of the curl operator, i.e. the ∇× operator. For E, the components of ∇×E are the following:

(∇×E)_z = ∇_xE_y– ∇_yE_x= ∂E_y/∂x – ∂E_x/∂y
(∇×E)_x = ∇_yE_z– ∇_zE_y= ∂E_z/∂y – ∂E_y/∂z
(∇×E)_y = ∇_zE_x– ∇_xE_z= ∂E_x/∂z – ∂E_z/∂x

So the four equations above can now be written as:

∂B_y/∂t = –(∇×E)_y = –∂E_x/∂z + ∂E_z/∂x
∂B_z/∂t = –(∇×E)_z = –∂E_y/∂x + ∂E_x/∂y
∂E_y/∂t = (∇×B)_y = ∂B_x/∂z – ∂B_z/∂x
∂E_z/∂t = (∇×B)_z= ∂B_y/∂x – ∂B_x/∂y

What can we do with this? Well… The x-component of E and B is zero, so one of the two terms in the equations simply disappears. We get:

∂B_y/∂t = –(∇×E)_y = ∂E_z/∂x
∂B_z/∂t = –(∇×E)_z = – ∂E_y/∂x
∂E_y/∂t = (∇×B)_y = – ∂B_z/∂x
∂E_z/∂t = (∇×B)_z= ∂B_y/∂x

Interesting: only the derivatives with respect to x remain! Let’s calculate them:

∂B_y/∂t = –(∇×E)_y = ∂E_z/∂x = ∂[sin(kx − ωt)]/∂x = k·cos(kx − ωt) = k·E_y
∂B_z/∂t = –(∇×E)_z = – ∂E_y/∂x = – ∂[cos(kx − ωt)]/∂x = k·sin(kx − ωt) = k·E_z
∂E_y/∂t = (∇×B)_y = – ∂B_z/∂x = – ∂[sin(kx − ωt + π/2)]/∂x = – k·cos(kx − ωt + π/2) = – k·B_y
∂E_z/∂t = (∇×B)_z= ∂B_y/∂x = ∂[cos(kx − ωt + π/2)]/∂x = − k·sin(kx − ωt + π/2) = – k·B_z

What wonderful results! The time derivatives of the components of B and E are equal to ±k times the components of E and B respectively! So everything is related to everything, indeed! 🙂

Let’s play some more. Using the cos(θ + π/2) = −sin(θ) and sin(θ + π/2) = cos(θ) identities, we know that B_y and B_z= sin(kx − ωt + π/2) are equal to:

B_y= cos(kx − ωt + π/2) = −sin(kx − ωt) = −E_z
B_z= sin(kx − ωt + π/2) = cos(kx − ωt) = E_y

Let’s calculate those derivatives once more now:

∂B_y/∂t = −∂E_z/∂t = −∂sin(kx − ωt)/∂t = ω·cos(kx − ωt) = ω·E_y
∂B_z/∂t = ∂E_y/∂t = ∂cos(kx − ωt)/∂t = −ω·sin(kx − ωt) = −ω·E_z

This result can, obviously, be true only if ω = k, which we assume to be the case, as we’re measuring time and distance in equivalent units, so the phase velocity is c = 1 = ω/k.

Hmm… I am sure it won’t be long before I’ll be able to prove what I want to prove. I just need to figure out the math. It’s pretty obvious now that the wavefunction—any wavefunction, really—models the flow of energy. I just need to show how it works for the zero-mass particle—and then I mean: how it works exactly. We must be able to apply the concept of the Poynting vector to wavefunctions. We must be. I’ll find how. One day. 🙂

As for now, however, I feel we’ve played enough with those wavefunctions now. It’s time to do what we promised to do a long time ago, and that is to use Schrödinger’s equation to calculate electron orbitals—and other stuff, of course! Like… Well… We hardly ever talked about spin, did we? That comes with huge complexities. But we’ll get through it. Trust me. 🙂