QM and confusing terminology, redux

I made a post earlier explaining why QM terminology can be so confusing as to prevent you (well, me) from learning it. It was probably too long, so I'm going to make it simpler.

Phase shift

Recall that light is an electromagnetic wave. This means it has (or is) an electric field (E) and a magnetic field (B) that are at right angles to each other:

If you put a measuring device anywhere along the x axis, you'd find that there's an electric field pointing up (or down) with some strength, and a magnetic field pointing left (or right) with proportional strength. They're proportional because they're "in phase." (The values are also oscillating in time, but forget about that for now.)

The electric and magnetic fields in EMR waves are always in phase and at 90 degrees to each other. -- Wikipedia

Now I want you to forget about the magnetic field and focus on the electric field. As a function of x, it is oscillating in one dimension (+z, -z, +z, ...). This is called linear polarization. You could rotate the field so that it's still linearly polarized, but instead of going up, down, up, down it's going upper-right, lower-left, upper-right, lower-left. (Apologies, the axes have been renamed here so that x is now called z.)

Since the x-y plane is two-dimensional, we can break that red wave down into two components: blue for x and green for y. Looked at in this way, the blue and green waves are "in-phase": when we're maximally on the right, we're maximally up; when we're down we're right; when one is zero so is the other. Note that the electric field components being in-phase has nothing to do with the magnetic field being in-phase with it.

We can also push these two "out of phase", so that when x is maximized y is zero, and vice versa. If we do this, we'll actually get a circularly polarized wave:

We say that the green and blue (or x and y) components are "90 degrees out of phase," or that we "phase shifted" the components of the wave with respect to each other. This is called a "relative" phase shift.

You can also "phase shift" the entire wave, by leaving the relative phase of the components the same, and shifting the entire spiral down the z axis. We might call this "absolute" phase shift.

When no context is given, it's almost always assumed the author means a relative phase shift, since that's the only one measurable by experiment.

Now for some gotchas.


Here's a claim from someone who "worked as a physicist at the Fermi National Accelerator Laboratory and the Superconducting Super Collider Laboratory," who explains that what's changing is the relative phase of the $E$ and $B$ fields:

The circularly polarized wave can be expressed as two linearly polarized waves, shifted by 90° in phase and rotated by 90° in polarization. If you pick some direction to measure the fields along, the components of E and B along that direction have a 90° phase shift with respect to each other. A phase shift of 90° means that as E peaks B becomes zero, and as B peaks E becomes zero.

As far as I can tell, this is just wrong.


Here's another: from Wikipedia, among many other sources:

"Light waves change phase by 180° when they reflect [off a mirror]."

If the relative phase of light were to change by 180°, then it would change from upper-right, lower-left... to upper-left, lower-right.... This would be easy to detect with a polarizing filter, but it doesn't happen. That's because they're talking about absolute phase. Good luck finding anyone who will explain that.



Finally, recall the two-slit experiment.

"When the two waves are in phase... the summed intensity is maximum, and when they are in anti-phase... then the two waves cancel and the summed intensity is zero. This effect is known as interference."

Which phase are they talking about here? Neither, of course. This is referring to the quantum wave function, where the "wave" is a complex-valued function representing the "probability amplitude" that helps you figure out where the photon might be found.

Which brings us to....

Amplitude

Recall that a single complex number $c$ can be written as: $c = re^{i\theta}$.

Now, amplitude can refer to three different things here:

1. The key innovation of QM is that we use "probability amplitudes." These refer to the complex number ($c$) itself.
2. But complex numbers themselves have "amplitudes", usually referring to $r$ above.
3. Some authors even use it to refer to $\theta$ ("The argument is sometimes also known as the phase or, more rarely and more confusingly, the amplitude")

To recap, a complex number $c = re^{i\theta}$ is an "amplitude," but it also has an amplitude, which can refer to either $r$ or $\theta$. One wonders if some day they'll use it to refer to $i$ and $e$, too.

Physicists can't decide whether they like (1) or (2). You commonly run into both 1:

"In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems." -- Wikipedia

And 2:

"The probability of getting any particular eigenvalue is equal to the square of the amplitude for that eigenvalue." -- Quantum Physicist Sean Carroll.

You first read (1) and when you get to (2) you think "wait, you can't just square a complex number and hope to get a real number (probability)." Amplitude must mean length. So you look up the relevant equation for "square of the amplitude":

Those bars are called "norm," or "length," or the "amplitude." So now whenever you detect usage (2) you mentally replace it with "norm," a concept from vector spaces. This begins to reinforce that terrible intuition they teach you in high school, of complex numbers being Real vectors.

How did you get that "vector" again? By taking the inner product of two complex vectors. That's funny, I thought the inner product was supposed to yield a scalar. No matter, let's just internalize this rule: the inner product of two complex vectors is another vector....

From the other side, maybe you have a hard time visualizing a vector in C^2 (i.e., a pair of complex numbers). So you mentally visualize it as a real vector. What is the inner product of two real vectors? It can be thought of as the length of the projection of one onto the other. So now you reinforce the intuition of the inner product as a (real) length.

So now you can't remember whether the inner product should yield a scalar (a + bi), a vector (a, b), or the length of that vector.

This is not a recipe for success.

Conclusion

Keep your concepts straight, or else you'll end up in an an abyss.