Schrödinger equation and classical wave equation

[PeterDonis]

I can't conclude that Schrödinger's equation is completely different from the classical wave equation.

What does "completely different" even mean? And why do you care?

From this point of view, I don't find the petition of principle that Schrödinger's equation is completely different from classical wave equations to be very well-founded.

Who is even making this claim? And again, why do you care?

All this makes me think that, technically, after all it is a wave equation.

Okay, suppose we say it is "technically" a wave equation (whatever that means). So what? Why do you care?

The reason I keep asking "why do you care" is that none of this has anything to do with actually using the Schrödinger Equation to make predictions. And physics is about making predictions that match experiments. Physics is not about arguing over how we label things, whether the label "wave equation" makes sense, etc., etc. So I'm really struggling to see why you are belaboring all this.

Isn't that a bit contradictory?

I don't see a contradiction. I see you throwing around vague terms and manufacturing imaginary issues out of nowhere. Sorry to be blunt, but that's how I see it. I don't see any actual issue here that needs to be resolved.

 

[Meden Agan]

I don't see a contradiction. I see you throwing around vague terms and manufacturing imaginary issues out of nowhere. Sorry to be blunt, but that's how I see it. I don't see any actual issue here that needs to be resolved.

I think you're right.
So, although my original question was different and vague, I'd like to analyze some technicalities of Schrödinger's equation.
Is what I said in post #30 correct?

 

[PeterDonis]

Is what I said in post #30 correct?

You said a lot in post #30. What particular point or points are you asking about?

 

[Meden Agan]

You said a lot in post #30. What particular point or points are you asking about?

These paragraphs.

What we know since Lagrange is that the eikonal equation governs the geometric limit of wave optics. At the same time, the equation of mechanical "rays" is governed by a similar equation, the Hamilton-Jacobi equation, in which, however, we have a first derivative with respect to time. The difference between the two types of equations in contemporary terms can be attributed to the following.
Light is propagated by individually massless particles, leading to a homogeneous Lagrangian of degree $1$, which supports a presymplectic formalism and freedom of parameterization with respect to time (from which Fermat's principle derives). Dynamics of material particles, on the other hand, derive from a principle of minimum in which time is not a free parameter.
Schrödinger's intuition was to reverse the process by which we derive the eikonal equation for geometric optics from wave optics, but starting from the Hamilton-Jacobi equation.

Schrödinger was really looking for the oscillations postulated by De Broglie, which accounted for Bohr and Sommerfeld's atomic model. And at first, he thought he had found them. The problem that emerged, however, was that the hypothesis that these were oscillations of a classical type of field was not compatible with Born's law, which was based on solid experimental evidence. At the same time, the problem of explaining the interactions between matter and the electromagnetic field had arisen. Focusing on this problem, Heisenberg developed matrix mechanics. From this, a much broader picture emerged, in which Schrödinger's equation was only one aspect. It was discovered that it was also involved in the phenomenology of electromagnetic waves, which culminated in the Copenhagen interpretation and the unification carried out by Jordan between matrix mechanics and Schrödinger mechanics.
Jordan proved that they were two different but equivalent representations. However, this unification required a change in the paradigm of matter waves initially assumed by Schrödinger, in favor of the probability waves that we now learn about together with the equation.

 

[PeterDonis]

These paragraphs.

Still a lot, and I'm not sure where it's all coming from or why you're asking about it. Some references would be helpful.

If your main point is that the quantity $\Psi$ that appears in the Schrödinger Equation describes a probability amplitude, not an actual "matter wave" or anything like that, yes, I agree with that.

 

[Meden Agan]

Still a lot, and I'm not sure where it's all coming from or why you're asking about it. Some references would be helpful.

My primary reference is Variational Principles in Dynamics and Quantum Theory by Yourgrau and Mandelstam.

It's possible to see an outline of Schrödinger's original arguments for the time-independent Schrödinger equation in section 8 of Field's paper Derivation of the Schrödinger equation from the Hamilton-Jacobi equation in Feynman's path integral formulation of quantum mechanics.

If your main point is that the quantity $\Psi$ that appears in the Schrödinger Equation describes a probability amplitude, not an actual "matter wave" or anything like that, yes, I agree with that.

I was trying to answer my question in post #1

When deriving the classical wave equation for entities such as the electric and magnetic fields, we consider the oscillation of the electric and magnetic fields. Similarly, why did Schrödinger not derive the wave equation for the physical oscillation of an electron? How did he know that there is no physical oscillation in the electron and that it is only a matter of probability, given that AFAIK the probabilistic interpretation of the wave function came two years after his equation?

and I was looking for your confirmation, which I believe I have received.

 

[PeterDonis]

I was trying to answer my question in post #1

Again, there's a lot in your post #1, and I'm not sure which question you're talking about.

I was looking for your confirmation, which I believe I have received.

I only confirmed that $\Psi$ is a probability amplitude. I didn't say anything about what Schrödinger was thinking. If the main question you are concerned about is this...

How did he know that there is no physical oscillation in the electron and that it is only a matter of probability

...then the answer, which is a matter of history, not physics, is that, when he came up with his equation, he didn't know that $\Psi$ was a probability amplitude. He thought it was describing some kind of physical "matter wave" associated with the electron. He only found out that $\Psi$ was a probability amplitude when Born came up with the probability interpretation of $\Psi$ two years later. @Demystifier already gave you that answer in post #2 of the thread.

 

[pines-demon]

Feel free to go over the arguments again—sometimes repeating them makes things easier to understand.

I will repeat this once again. I am not the only one that does not get what the main question is.

 

[Meden Agan]

I will repeat this once again. I am not the only one that does not get what the main question is.

Sorry, I wasn't clear from the very start.
In my academic studies, I've heard some professors say that Schrödinger's equation is a wave equation, and others say that Schrödinger's equation is a diffusion equation.

I was confused by the terminology. I think @PeterDonis is right when he asserts this:

The reason I keep asking "why do you care" is that none of this has anything to do with actually using the Schrödinger Equation to make predictions. And physics is about making predictions that match experiments. Physics is not about arguing over how we label things, whether the label "wave equation" makes sense, etc., etc. So I'm really struggling to see why you are belaboring all this.

 

[PeterDonis]

In my academic studies, I've heard some professors say that Schrödinger's equation is a wave equation, and others say that Schrödinger's equation is a diffusion equation.

"I've heard" is not a valid reference. If you can give us references to actual textbooks or peer-reviewed papers that make statements like these, we can discuss them. But we can't discuss vague allusions to something somebody heard.

I'm closing the thread since it appears that all valid points have been addressed.