Physical interpretation of Schrodinger equation

[Mayan Fung]

Schrödinger Equation is the very first step when we start learning QM. However, I never learned about the physical meaning of it. I have read a number of articles and discussion online. Regarding the ones I understand, there are generally two points of view.

1. Fundamental physical laws are not understandable but only accepted. Just like F=ma, we generally accept the definition of this physical quantity and develop our classical mechanical theory. The same applies to SE. We can only accept it and to develop our physics on it.

2. The Schrödinger Equation try to describe things (e.g. electrons) in a way similar to the classical wave theory. SE provides us a way to calculate the wave function.

If 1 is the way we see SE, I am wondering how Schrödinger could come up with that equation which is so much complicated than F=ma. Or in other words, I can believe that Newton can invent F=ma only by imagination but not SE. If we interpret SE like 2 does, what is the physical meaning of each detail variable and constants? I saw a lot of articles only give qualitative arguments. How is the wave function relates to ħ, and how do we relate the SE to classical wave theory?

Sometimes it is discouraging if I only focus on the math. Solving the PDE and ODE for several hours to obtain the wave functions of, say Hydrogen atom, seems to do nth with physics, if I don't even have a sense of the SE.

 

[ShayanJ]

Its actually a combination of 1 and 2. Yes, Schrödinger was searching for a wave equation. No, what we're talking about is not actually classical waves!
At first de Broglie proposed that particles show wave-like behavior and have a corresponding wave-length given by $\lambda=\frac h p$. Schrödinger wanted to find out what wave equation those waves should satisfy. His reasoning was that just like geometric optics which is the long wave-length limit of wave optics(Maxwell's equations), there should be a wave equation whose long wave-length limit gives classical mechanics(Hamilton-Jacobi equation). But it was just an analogy, wave-functions in wave mechanics don't describe classical waves like sound waves or EM waves or etc.

 

[jtbell]

I am wondering how Schrödinger could come up with that equation

At that time, the idea of "particles as waves" was in the air, thanks to de Broglie. Schrödinger set out to find a wave equation for these waves. Basically, he made the following analogies:

Classical mechanics <--> geometrical optics (based on Fermat's principle)
Quantum mechanics <--> wave optics (based on Huygens's principle)

https://www.physicsforums.com/threads/schrodingers-equation.57867/#post-418069

 

[Mayan Fung]

Things become not that obvious in quantum mechanics. I can hardly get a feel of it. In classical mechanics, in many cases, I can visualise the physical picture. But in quantum mechanics, I don't even know whether my calculation result make sense.

 

[vanhees71]

The Schrödinger equation describes the evolution of the quantum state of a single- or many-body system in the case that the particle number is strictly conserved. The meaning of the wave function is that its square is the probability distribution to find the particle(s) at (a) position(s). There is no wave-particle duality thanks to this Born rule, and this resolves all the contradictions and quibbles of the old-fashioned quantum theory which is simply outdated and should not be taught anymore except in lectures on the history of science.

 

[bhobba]

Schrödinger Equation is the very first step when we start learning QM. However, I never learned about the physical meaning of it

Read the first 3 chapters of Ballentine.

Its physical meaning is symmetry, but that revelation you must discover for yourself.

One professor posts when he teaches students he gets stunned silence - its that profound.

Strangely philosophy types don't discuss it - yet it can be argued as physics greatest insight.

Thanks
Bill

 

[vanhees71]

Philosophers rather engage in useless ideas about the "meaning" of quantum theory rather than to study the theory itself, let alone the symmetry principles behind all of physics. Ironically symmetry principles are longer a topic in philosophy than in the natural sciences. Take e.g., Platon's Timaios.