What’s the relation between the title of my blog (reading Einstein) and Schrödinger’s equation? Two things, I’d say. First, all of my reading on quantum mechanics is because I think Einstein’s intuition in regard to the wavefunction—that it, somehow, must represent something real—must be right, even if mainstream physicists seem to have given up on trying to find what it could be.
Why? Assuming it is just a mathematical construct is absurd: if an electron (or its wavefunction) interferes with itself when going through a slit, something must be interfering, and that something must be as real as the diffraction pattern. I also believe the quantum-mechanical operators (such as the position, energy, linear or angular momentum operator) have to operate on something real, because they give us real (observable) information. I don’t want to waste words on this because I prefer to use my time to think about how it could be real. 🙂
The other thing is that, when thinking about a possible physical dimension for the wavefunction (or for its two components, I should say), we have two obvious candidates: force per unit mass (newton per kg), or force per unit charge (newton per coulomb). In the first interpretation—newton per kg, which reduces to m/s2, so that’s an acceleration (or a deceleration)—we’re basically thinking of a wavefunction as a two-dimensional gravitational wave. I’ve abandoned that idea now for a much simpler assumption: the wavefunction for an electron and an electromagnetic wave have a similar interpretation, but they differ in their geometry—their orientation in space, basically. In addition, the matter-wave does push and pull some pointlike electric charge around which explains why “when you do find the electron some place, the entire charge is there.” (Feynman, III-21-4) In short, the interpretation of the wavefunction of an electron that is offered here combines (1) the idea of an orbital and (2) the pointlike charge that is whizzing around in it.
But I am getting ahead of myself here. Let me start by explaining what I mean with a two-dimensional wave.
Electromagnetic waves are two-dimensional oscillations, as you can see from the animations below, which show a right-handed versus a left-handed wave respectively. [I don’t make my own animations, so I should credit open sources such as Wikipedia here.]
Is this a two-dimensional oscillation? It is. The rotating vector can be analyzed in terms of two perpendicular components: look at the green and blue wave below, which oscillate in one dimension only but, taken together, create the (circularly polarized) waves above.The green and blue components above correspond to the electric and magnetic field vectors E and B in electromagnetics. It’s one phenomenon (or one force, I should say)—as evidenced by the force formula: F = qE + qv×B—but the propagation mechanism (shown below) can only be explained in terms of its components, as shown below.
A propagation mechanism? Yes. I’ve written about this before, and so I won’t repeat myself here. Just note that the ∇× operator: it is the curl operator. When it operates on a vector (as it does here), it will give us the (infinitesimal) rotation of a vector field. It captures the idea of circulation. It is not easy to visualize what happens here, exactly, but we get the basic idea: linear and circular concepts are combined in a rather wonderful and mysterious perpetuum mobile mechanism. The analysis becomes even more interesting when we combine it with the concept of the Poynting vector, which models an energy flow.
Now, can we analyze wavefunctions, or Schrödinger’s equation itself, in a similar way? Let us think about it.
Feynman’s summary interpretation of Schrödinger’s equation is the following:
“We can think of Schrödinger’s equation as describing the diffusion of the probability amplitude from one point to the next. […] But the imaginary coefficient in front of the derivative makes the behavior completely different from the ordinary diffusion such as you would have for a gas spreading out along a thin tube. Ordinary diffusion gives rise to real exponential solutions, whereas the solutions of Schrödinger’s equation are complex waves.”
Feynman further formalizes this in his Lecture on Superconductivity, in which he refers to Schrödinger’s equation as the “equation for continuity of probabilities”. His analysis there is centered on the local conservation of energy, which makes me think Schrödinger’s equation might be an energy diffusion equation. I’ve written about this ad nauseam in the past, and so I’ll just refer you to one of my other papers here for the details, and limit this post to the basics, which are as follows.
The wave equation (so that’s Schrödinger’s equation in its non-relativistic form, which is an approximation that is good enough) is written as:
The resemblance with the standard diffusion equation (shown below) is obvious:
As Feynman notes, it is just that imaginary coefficient that makes the behavior quite different. How exactly? Well… You know we get all of those complicated electron orbitals (i.e. the various wave functions that satisfy the equation) out of Schrödinger’s differential equation. We can think of these solutions as (complex) standing waves. They basically represent some equilibrium situation, and the main characteristic of each is their energy level. I won’t dwell on this because – as mentioned above – I assume you master the math. Now, you know that – if we would want to interpret these wavefunctions as something real (which is surely what I want to do!) – the real and imaginary component of a wavefunction will be perpendicular to each other. A general geometric representation of the elementary wavefunction ψ(θ) = a·e−i∙θ = a·e−i∙(E/ħ)·t = a·cos[(E/ħ)∙t] − i·a·sin[(E/ħ)∙t] is shown below:
The 90° angle makes me think of the similarity with the mathematical description of an electromagnetic wave. Let me quickly show you why. For a particle moving in free space – with no external force fields acting on it – there is no potential (U = 0) and, therefore, the Vψ term – which is just the equivalent of the the sink or source term S in the diffusion equation – disappears. Therefore, Schrödinger’s equation reduces to:
∂ψ(x, t)/∂t = i·(1/2)·(ħ/meff)·∇2ψ(x, t)
Now, the key difference with the diffusion equation – let me write it for you once again: ∂φ(x, t)/∂t = D·∇2φ(x, t) – is that Schrödinger’s equation gives us two equations for the price of one. Indeed, because ψ is a complex-valued function, with a real and an imaginary part, we get the following equations:
- Re(∂ψ/∂t) = −(1/2)·(ħ/meff)·Im(∇2ψ)
- Im(∂ψ/∂t) = (1/2)·(ħ/meff)·Re(∇2ψ)
In case you would wonder where these equations come from, they can be easily derived from noting that two complex numbers a + i∙b and c + i∙d are equal if, and only if, their real and imaginary parts are the same. Now, the ∂ψ/∂t = i∙(ħ/meff)∙∇2ψ equation amounts to writing something like this: a + i∙b = i∙(c + i∙d). Now, remembering that i2 = −1, you can easily figure out that i∙(c + i∙d) = i∙c + i2∙d = − d + i∙c. [Now that we’re getting a bit technical, let me note that the meff is the effective mass of the particle, which depends on the medium. For example, an electron traveling in a solid (a transistor, for example) will have a different effective mass than in an atom. atom. In free space, we can drop the subscript and write meff = m.]
Can we interpret the equations above as representing a similar propagation mechanism as the equations for an electromagnetic wave in free space (no stationary charges or currents), as shown below?
I am not sure, but it is very tempting to think so. 🙂 The implications are obvious. We know the wavefunction consist of a sine and a cosine: the cosine is the real component, and the sine is the imaginary component. Could they be equally real? Could each represent half of the total energy of our particle? I firmly believe they do. The obvious question then is the following: why wouldn’t we represent them as vectors, just like E and B? I mean… Representing them as vectors (I mean real vectors – something with a magnitude and a direction, in a real vector space – as opposed to these state vectors from a Hilbert space) would show they are real, and the transformation matrices we need to go from one (representational) base to another might become more intuitive. In fact, that’s why vector notation was invented (sort of): we don’t need to worry about the coordinate frame. It’s much easier to write physical laws in vector notation because… Well… They’re the real thing, aren’t they?
What about dimensions? Well… I am not sure. However, because we are – arguably – talking about some pointlike charge moving around in those oscillating fields, I would suspect the dimension of the real and imaginary component of the wavefunction will be the same as that of the electric and magnetic field vectors E and B. We may want to recall these:
- E is measured in newton per coulomb (N/C).
- B is measured in newton per coulomb divided by m/s, so that’s (N/C)/(m/s).
The weird dimension of B is because of the weird force law, which involves v×B cross-product. It involves a vector cross product, as shown by Lorentz’ formula:
F = qE + q(v×B)
Hence, we may associate the (1/c)∙ex× operator, which amounts to a rotation by 90 degrees, with the s/m dimension. Now, multiplication by i also amounts to a rotation by 90° degrees. Hence, if we can agree on a suitable convention for the direction of rotation here, we may boldly write:
B = (1/c)∙ex×E = (1/c)∙i∙E
This is, in fact, what triggered my geometric interpretation of Schrödinger’s equation about a year ago now. I have had little time to work on it, but think I am on the right track.
Having said that, it is obvious that the Laplace operator (∇2) is quite different from the curl operator, and we’d need to think about what it represents really.
 Richard Feynman, Lectures on Physics, Volume III (1966), p. 16-4
 Richard Feynman, Lectures on Physics, Volume III (1966), p. 21-3 and 21-4
 See: http://vixra.org/author/jean_louis_van_belle_maec_baec_bphil, accessed on 12 June 2018.
 The convention would also need to include our understanding of the directions of the real and imaginary component of the wavefunction vis-á-vis the line of sight between the observer and the object or, possibly, its direction of motion.