I Schrödinger equation and classical wave equation

[Meden Agan]

TL;DR Summary

Implications of difference between Schrödinger equation and the classical wave equation.

Not an expert in QM.
AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order.
But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical.
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?

 

[Demystifier]

Schrodinger didn't know what is the physical meaning of the wave. He only used the nonrelativistic energy formula
$$E=\frac{p^2}{2m}+ V(x)$$
Planck's formula
$$E=\hbar\omega$$
and de Broglie's formula
$$p=\hbar k$$
From this and the ansatz $\psi=e^{-i\omega t} e^{i kx}$ for $V=0$, the Schrodinger equation for $\psi$ follows immediately. The physical meaning of $\psi$ was understood later, especially by the Born's idea that $|\psi|^2$ is the probability density.

 

[Meden Agan]

Schrodinger didn't know what is the physical meaning of the wave. He only used the nonrelativistic energy formula
$$E=\frac{p^2}{2m}+ V(x)$$
Planck's formula
$$E=\hbar\omega$$
and de Broglie's formula
$$p=\hbar k$$
From this and the ansatz $\psi=e^{-i\omega t} e^{i kx}$ for $V=0$, the Schrodinger equation for $\psi$ follows immediately. The physical meaning of $\psi$ was understood later, especially by the Born's idea that $|\psi|^2$ is the probability density.

Yes. AFAIK, Schrödinger developed his equation through an admirable reverse engineering operation, starting from the analogy between the eikonal equation of geometric optics and Hamilton-Jacobi's equation of classical mechanics. In practice, he found the theory corresponding to wave optics in mechanics, namely wave mechanics, from which he could derive classical mechanics for $h \to 0$, to construct the eigenvalue problem that would give the energy levels of the hydrogen atom as results.
Yes?

 

[pines-demon]

AFAIK, Schrödinger's equation is quite different from the classical wave equation.

Yes, Schrödinger looks more like a diffusion equation with imaginary prefactors.

As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order.

That does not matter you can render the ordinary wave equation first order in time if you want: $(\partial_x-c^{-1}\partial_t)\phi=0$.

AFAIK the probabilistic interpretation of the wave function came two years after his equation?

Yes.

 

[Meden Agan]

That does not matter you can render the ordinary wave equation first order in time if you want: $(\partial_x-c^{-1}\partial_t)\phi=0$.

Yes, but I mean: the classical wave equation has two derivatives in time and two derivatives in position; the Schrödinger equation has only one derivative in time, not two. That is different.
Yes?

 

[Meden Agan]

Sorry, haven't received any reply to posts #3 and #5.

 

[pines-demon]

Yes, but I mean: the classical wave equation has two derivatives in time and two derivatives in position; the Schrödinger equation has only one derivative in time, not two. That is different.
Yes?

I just wrote a wave equation that allows the same solutions as the usual wave equation, with one time derivative.

 

[Meden Agan]

I just wrote a wave equation that allows the same solutions as the usual wave equation, with one time derivative.

Yes. But what lies behind what I said in post #1 is that $E=\sqrt{\left[\left(mc^2 \right)^2+\left(cp \right)^2 \right]}$ in the first order becomes $E=mc^2+ \dfrac{p^2}{2m}$. By de Broglie relation, $E-mc^2$ can be identified with $\hbar \dfrac{\mathrm d}{\mathrm dt} \psi$, where the phase factor $\mathrm{e}^ \left(\mathrm{i} \, mc^2 \, t/\hbar \right)$ can be eliminated, while $p$ is identified by the gradient and squared by the Laplacian.
Starting from here, Pauli argues that the correct relativistic Hamiltonian should be obtained not by linearizing but by adding the subsequent terms. The problem is that when interactions occur, other problems arise — as set out by Pascual Jordan — which involve the extension of the position-momentum commutation rules and the role of time. The same set of problems also arises with the Dirac equation, which, in the semiclassical limit, appears to be invariant. However, to quantize correctly, Dirac developed a theory of constraints from which the BRST quantization procedure was born.

 

[pines-demon]

Yes. But what lies behind what I said in post #1 is that $E=\sqrt{\left[\left(mc^2 \right)^2+\left(cp \right)^2 \right]}$ in the first order becomes $E=mc^2+ \dfrac{p^2}{2m}$. By de Broglie relation, $E-mc^2$ can be identified with $\hbar \dfrac{\mathrm d}{\mathrm dt} \psi$, where the phase factor $\mathrm{e}^ \left(\mathrm{i} \, mc^2 \, t/\hbar \right)$ can be eliminated, while $p$ is identified by the gradient and squared by the Laplacian.
Starting from here, Pauli argues that the correct relativistic Hamiltonian should be obtained not by linearizing but by adding the subsequent terms. The problem is that when interactions occur, other problems arise — as set out by Pascual Jordan — which involve the extension of the position-momentum commutation rules and the role of time. The same set of problems also arises with the Dirac equation, which, in the semiclassical limit, appears to be invariant. However, to quantize correctly, Dirac developed a theory of constraints from which the BRST quantization procedure was born.

I don't know what this has to do with the ordinary wave equation. What is your question? What do you want to know?

 

[Meden Agan]

I don't know what this has to do with the ordinary wave equation. What is your question? What do you want to know?

I explained what the difference between Schrödinger's equation, which is first order in $t$, and the classical wave equation, which is second order in $t$, implies.
You said that this difference is not so important; I tried to explain why I think it is.
I may be wrong, as I am not an expert in QM.

 

[JimWhoKnew]

However, to quantize correctly, Dirac developed a theory of constraints from which the BRST quantization procedure was born.

as I am not an expert in QM.

As a non-expert in QM, you seem to know an awful lot about it

 

[Meden Agan]

What is your question? What do you want to know?

Well, I think I gave some context in post #1.
It's a tricky question. Is Schrödinger's equation a wave equation or not? There are similarities with the classical wave equation, but also many differences that make it a heat/diffusion equation.

Let's take it one step at a time.

In post #1, I said: 'Schrödinger's equation is a wave equation; only its interpretation makes it non-classical.' That is one of my main doubts.

Schrödinger's equation and the heat equation look very similar, even though Schrödinger's equation describes quantum systems and the heat equation describes classical systems.
Therefore, the nature of the object (i.e., whether it's quantum or classical) and the rules of measurement are not integrated into Schrödinger's equation. Schrödinger's equation is just a differential equation, and looking at it, there's nothing obviously quantum mechanical about it.

On the other hand, if we want to give an interpretation, I would say that Schrödinger's equation explicitly expresses the fact that it deals with quantum mechanics for two reasons.
The first reason is that the wave function is complex; its square modulus is real and represents a probability, not an intensity of a physical quantity. That is due precisely to its quantum nature.
The second fact is that Schrödinger's equation includes an imposed parameter, $\dfrac{h}{2 \pi}$, which clarifies the order of magnitude where the equation is essential and also why, when we move to larger objects, the equation can be approximated with a classical Hamiltonian. That is also a reason to say it deals with quantum mechanics.
IMO, $h$ is the essence of quantum mechanics: it is the parameter transforming energy into phase. As a matter of fact, it has the dimensions of an action and expresses how much the phase of the action changes as the energy varies. This is a fundamental fact in quantum mechanics and is regulated by $h$ (or rather, the scale is fixed by Planck's constant).
Therefore, Schrödinger's equation tells us that a quantum of energy corresponds to a (quantized) phase change equal to $\dfrac{h}{2 \pi}$.
Obviously, the value is arbitrary, but its physical meaning is not: it imposes the minimum scale that changes the phase of the action based on energy.

But, on the other side, $\dfrac{h}{2 \pi}$ is just a number, and that number could appear in a classical equation. Maybe we could argue that imaginary numbers cannot appear in a classical equation that describes an observable property (because observables are always real). But who said the classical equation has to describe an observable?
If we look at the Schrödinger equation, there's nothing in the form of the equation to say it is related to quantum mechanics. If we have extra information, like knowing that $h$ is related to quantized states, then we can see that the Schrödinger equation must be related to quantum mechanics, but that's extra information.
If we didn't know what $h$ was, then the Schrödinger equation would just look like any other classical equation.

How should we deal with such a contradiction?

 

[PeterDonis]

Schrödinger's equation and the heat equation look very similar

Yes, and the heat equation is not a wave equation. So this...

'Schrödinger's equation is a wave equation

...doesn't seem right.

 

[Meden Agan]

Yes, and the heat equation is not a wave equation. So this...


...doesn't seem right.

Clear, agreed.
However, could you analyze specifically all the points in message #12?
IMO, what I said involves a contradiction. Therefore, I'd like to clarify.

 

[PeterDonis]

IMO, what I said involves a contradiction.

I don't see a contradiction. A mathematical equation, in and of itself, doesn't tell you how to interpret it physically. You always have to bring in extra information for any physical interpretation. So the fact that you have to bring in extra information to interpret Schrodinger's Equation as describing a quantum wave function (as opposed to, say, heat diffusion in a solid object) doesn't look to me like a contradiction: it looks to me like physics as usual.

 

[PeterDonis]

The first reason is that the wave function is complex

The second fact is that Schrödinger's equation includes an imposed parameter, $\dfrac{h}{2 \pi}$

But you admit later on in your post that both of these things can also be true of a classical equation. So they don't explicitly tell you that the Schrodinger Equation is a quantum equation. You have to bring in extra information--like what the complex wave function describes (a probability amplitude) and what $h$ describes (the quantum of action)--in order to know that the Schrodinger Equation is a quantum equation.

 

[Meden Agan]

But you admit later on in your post that both of these things can also be true of a classical equation. So they don't explicitly tell you that the Schrodinger Equation is a quantum equation. You have to bring in extra information--like what the complex wave function describes (a probability amplitude) and what $h$ describes (the quantum of action)--in order to know that the Schrodinger Equation is a quantum equation.

I strongly agree.

 

[gentzen]

As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order.

That does not matter you can render the ordinary wave equation first order in time if you want: $(\partial_x-c^{-1}\partial_t)\phi=0$.

Yes, but I mean: the classical wave equation has two derivatives in time and two derivatives in position; the Schrödinger equation has only one derivative in time, not two. That is different.
Yes?

I just wrote a wave equation that allows the same solutions as the usual wave equation, with one time derivative.

Are you sure? It is true that you can factorize $\partial_t^2\phi=c^2\partial_x^2\phi$ (1D-WE) into $(\partial_x+c^{-1}\partial_t)(\partial_x-c^{-1}\partial_t)\phi=0$. But if you just omitt $(\partial_x+c^{-1}\partial_t)$, then also the corresponding solutions $\phi_r(t,x):=f(x-ct)$ will no longer work. So $\phi_r$ is a solution of (1D-WE), but not of your first order in time equation. Only $\phi_l(t,x):=f(x+ct)$ is a solution to both equations.

 

[pines-demon]

Are you sure? It is true that you can factorize $\partial_t^2\phi=c^2\partial_x^2\phi$ (1D-WE) into $(\partial_x+c^{-1}\partial_t)(\partial_x-c^{-1}\partial_t)\phi=0$. But if you just omitt $(\partial_x+c^{-1}\partial_t)$, then also the corresponding solutions $\phi_r(t,x):=f(x-ct)$ will no longer work. So $\phi_r$ is a solution of (1D-WE), but not of your first order in time equation. Only $\phi_l(t,x):=f(x+ct)$ is a solution to both equations.

I agree. The point is not that, the point is to ask why OP cared about the wave equation or the order of the derivative, I still do not get what does OP want to know really.

 

[gentzen]

IMO, $h$ is the essence of quantum mechanics: it is the parameter transforming energy into phase. As a matter of fact, it has the dimensions of an action and expresses how much the phase of the action changes as the energy varies. This is a fundamental fact in quantum mechanics and is regulated by $h$ (or rather, the scale is fixed by Planck's constant).

It seems to me that what changes is the rate of change of the phase, not the phase itself.

Therefore, Schrödinger's equation tells us that a quantum of energy corresponds to a (quantized) phase change equal to $\dfrac{h}{2 \pi}$.
Obviously, the value is arbitrary, but its physical meaning is not: it imposes the minimum scale that changes the phase of the action based on energy.

I guess the most direct interpretation of quantization is possible for angular momentum. A change in the quantized angular momentum can then correspond to a quantum of energy, for a specific scenario/system.
And again, there is no direct correlation between a quantum of energy and a (quantized) phase change.

(If we look at complex phase in isolation, its natural unit for quantization would be $2\pi$.)

 

[Meden Agan]

@PeterDonis Step two. In post #1, I said: 'As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order'.
Well, what does this difference in the distinction between the classical wave equation and Schrödinger's equation imply?
In particular, see posts #4, #5 and #8 (!).

The point is not that, the point is to ask why OP cared about the wave equation or the order of the derivative, I still do not get what does OP want to know really.

See above.

 

[pines-demon]

Well, what does this difference in the distinction between the classical wave equation and Schrödinger's equation imply?

Let me ask then, why do you think it makes a difference? And if you have already provided that motivation then what do you think is missing?

 

[gentzen]

Well, what does this difference in the distinction between the classical wave equation and Schrödinger's equation imply?

Let me ask then, why do you think it makes a difference? And if you have already provided that motivation then what do you think is missing?

See here for what this difference implies:

Some observations by Qiaochu Yuan are especially useful for understanding significant shortcomings of the toy models I came up with so far. The "only differences between energies are physically meaningful" remark is important:
https://qchu.wordpress.com/2012/09/...ity-the-born-rule-and-wave-function-collapse/
I noticed this in my toy models when I tried to include the time dependence. For my toy models, the absolute energy is important, but for the real quantum mechanics, only the differences of the energies are important.

My toy model itself was:

I guess that a complex superposition of polarized monochromatic plane waves traveling up or down in z-direction could be used to create a faithful optical model of a 2-qubit quantum system.

 

[Meden Agan]

It seems to me that what changes is the rate of change of the phase, not the phase itself.

I guess the most direct interpretation of quantization is possible for angular momentum. A change in the quantized angular momentum can then correspond to a quantum of energy, for a specific scenario/system.
And again, there is no direct correlation between a quantum of energy and a (quantized) phase change.

(If we look at complex phase in isolation, its natural unit for quantization would be $2\pi$.)

I agree. I was very sloppy there.

 

[Meden Agan]

Are you sure? It is true that you can factorize $\partial_t^2\phi=c^2\partial_x^2\phi$ (1D-WE) into $(\partial_x+c^{-1}\partial_t)(\partial_x-c^{-1}\partial_t)\phi=0$. But if you just omitt $(\partial_x+c^{-1}\partial_t)$, then also the corresponding solutions $\phi_r(t,x):=f(x-ct)$ will no longer work. So $\phi_r$ is a solution of (1D-WE), but not of your first order in time equation. Only $\phi_l(t,x):=f(x+ct)$ is a solution to both equations.

Yes, I agree.

Note that in the Klein-Gordon equation, mass is also present. Also note that there are three spatial components.
The spatial part is a Laplacian, while the first-order operator is a gradient: it is not true that $\left(\dfrac{\mathrm d}{\mathrm dt}-\nabla_r \right)\left(\dfrac{\mathrm d}{\mathrm dt}+\nabla_r \right)=\square$, where $\square$ is the d’Alembert operator. To obtain this equality, Dirac needed four components in the wave function and “numbers” that anticommute. The Clifford algebra of Minkowski space is just what we need: it drops the mixed spatial components (the corresponding Clifford units are anticommutative) and leaves only the second derivatives.
If we have a scalar field, though, Dirac’s trick doesn’t work.
Yes?

 

[Meden Agan]

@PeterDonis Step two. In post #1, I said: 'As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order'.
Well, what does this difference in the distinction between the classical wave equation and Schrödinger's equation imply?
In particular, see posts #4, #5 and #8 (!).

When I receive a response to this about my interpretation of post #8, I'll proceed to step three. That's what's bothering me the most.

 

[pines-demon]

When I receive a response to this about my interpretation of post #8, I'll proceed to step three. That's what's bothering me the most.

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

 

[PeterDonis]

When I receive a response to this about my interpretation of post #8, I'll proceed to step three. That's what's bothering me the most.

I'm not sure what you're asking for in post #8.

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

That might help here, yes.

The Schrodinger Equation is (writing it in one spatial dimension for simplicity, and only considering the free particle case for now)

$$
i \hbar \frac{\partial \Psi}{\partial t} = - \frac{\hbar^2}{2 m} \frac{\partial^2 \Psi}{\partial x^2}
$$

The classical wave equation is

$$
\frac{\partial^2 \Psi}{\partial t^2} = c^2 \frac{\partial^2 \Psi}{\partial x^2}
$$

Obviously these aren't the same. The Schrodinger Equation looks more like the classical heat equation:

$$
\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}
$$

However, the factor of $i$ on the LHS of the Schrodinger Equation and the minus sign on the RHS mean it's not quite the same as the heat equation either.

All this seems obvious. What I'm not clear about is what question or issue you think it raises. We know general techniques for solving the Schrodinger Equation. So what's the problem?

 

[Meden Agan]

I'm not sure what you're asking for in post #8.

IMO, the reason why Schrödinger's equation for matter waves is first order in $t$ and the classical electromagnetic wave equation is second order in $t$ is hidden in what I wrote in post #8.
It's already found in the particle counterpart: the structure of geometric optics is presymplectic, while the structure of mechanical optics is symplectic.
Do you agree?

I'd also appreciate a comment — confirming or not — on posts #3 and #25.

 

[Meden Agan]

The Schrodinger Equation is (writing it in one spatial dimension for simplicity, and only considering the free particle case for now)

$$
i \hbar \frac{\partial \Psi}{\partial t} = - \frac{\hbar^2}{2 m} \frac{\partial^2 \Psi}{\partial x^2}
$$

The classical wave equation is

$$
\frac{\partial^2 \Psi}{\partial t^2} = c^2 \frac{\partial^2 \Psi}{\partial x^2}
$$

Obviously these aren't the same. The Schrodinger Equation looks more like the classical heat equation:

$$
\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}
$$

However, the factor of $i$ on the LHS of the Schrodinger Equation and the minus sign on the RHS mean it's not quite the same as the heat equation either.

All this seems obvious. What I'm not clear about is what question or issue you think it raises. We know general techniques for solving the Schrodinger Equation. So what's the problem?

Everything you say is correct.

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

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.

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.
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.

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

Isn't that a bit contradictory?

 

[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 Schrodinger 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 Schrodinger 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 Schrodinger 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 Schrodinger 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 Schrodinger 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.