I Schrödinger equation and classical wave equation

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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?
 
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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.
 
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Demystifier said:
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?
 
Meden Agan said:
AFAIK, Schrödinger's equation is quite different from the classical wave equation.
Yes, Schrödinger looks more like a diffusion equation with imaginary prefactors.

Meden Agan said:
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##.
Meden Agan said:
AFAIK the probabilistic interpretation of the wave function came two years after his equation?
Yes.
 
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I always liked this discussion.
 
pines-demon said:
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?
 
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