Physical intrepretation of Sr.E.

[astro2cosmos]

as we know whole Quantum mechanics is based on Schrondinger Equation,
but how for a wave we can physically vizualize/interprete this eq. in the context of wave function?

 

[Gerenuk]

The
http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics
is not fully resolved yet.

Actually I once asked a question about Schrödinger, but didn't get answers:
https://www.physicsforums.com/showthread.php?t=390005
There I was trying to figure out if the equation is just the results of some axiomatic approach, where complex number are basically the only things satisfying it.

 

[alxm]

There I was trying to figure out if the equation is just the results of some axiomatic approach, where complex number are basically the only things satisfying it.

Well, postulates rather than axioms. There was another thread on the complex number thing, but the answer is simply no. Neither quantum mechanical wave functions or classical mechanical waves require the use of complex numbers. It's just much more convenient to use them. (also, time-independent solutions to the S.E., with no external field, are real-valued)

 

[Gerenuk]

Neither quantum mechanical wave functions or classical mechanical waves require the use of complex numbers. It's just much more convenient to use them. (also, time-independent solutions to the S.E., with no external field, are real-valued)

That was not the question. When some talks about the complex wavefunction he means all it representations. This could be real valued matrices or anything. A difference in representation is completely transparent to mathematicians.
The question was what sort of intuitive rules do you need to derive the mathematics of quantum mechanics. I once read about axioms like "the is only one vacuum state" and "it looks the same to all observers". Now one could make up more axioms to finally single out the quantum mechanics as it is today.

 

[alxm]

That was not the question. When some talks about the complex wavefunction he means all it representations. This could be real valued matrices or anything.

You asked why 'complex numbers are the only thing satisfying it'. Now you're saying this implied that you can have a real-valued formalism?

The question was what sort of intuitive rules do you need to derive the mathematics of quantum mechanics. I once read about axioms like "the is only one vacuum state" and "it looks the same to all observers". Now one could make up more axioms to finally single out the quantum mechanics as it is today.

And most textbooks derive QM more or less rigorously (depending on the textbook) in exactly this fashion. And if you want to go deeper there are many good books on the topic, starting with Dirac's and von Neumann's early books, and many more in the same vein since.

 

[Gerenuk]

You asked why 'complex numbers are the only thing satisfying it'. Now you're saying this implied that you can have a real-valued formalism?

Yes, this includes all other notations. May it be real valued matrices or whatever. In Maths it's the same as changing the font of the letters. It doesn't make a different and only the logic and connection behind the algebra matters.

And most textbooks derive QM more or less rigorously (depending on the textbook) in exactly this fashion. And if you want to go deeper there are many good books on the topic, starting with Dirac's and von Neumann's early books, and many more in the same vein since.

Can you name a method? Because most books actually start from the Schrödinger guess "We assume it's a wave. So let's take complex numbers. And hey, if energy conservation works for a single wave why not use it for a superposition of waves?"
It's a vague guess with no justification.

 

[alxm]

Can you name a method? Because most books actually start from the Schrödinger guess "We assume it's a wave. So let's take complex numbers. And hey, if energy conservation works for a single wave why not use it for a superposition of waves?"
It's a vague guess with no justification.

What book are you talking about? I don't have a single QM book that follows any such ad-hoc assertions to derive basic QM.

If you can't be bothered to look up the books I mentioned, try starting http://en.wikipedia.org/wiki/Mathematical_formulations_of_quantum_mechanics" .

 

[Gerenuk]

If you can't be bothered to look up the books I mentioned, try starting http://en.wikipedia.org/wiki/Mathematical_formulations_of_quantum_mechanics" .

OK, so if you know how to introduce QM without postulating mathematics, please name the method. Because in the link you give it's the usual undergrad stuff where they postulate that QM has to be a Hilbert space, and observables are operators and so. They usually don't give any justification whatsoever.
I don't have access to the early books now, but I did some extensive research for book some time ago and hardly any book took a different approach.

 

[George Jones]

It's a vague guess with no justification.

Why does F = dp/dt?

 

[Gerenuk]

Why does F = dp/dt?

That's no answer. It's the same logic as saying "I can get killed in a car accident, then I might as well kill myself with smoking"
The point is that you don't have to define more than necessary. A good physicist tries to minimize postulates. It's possible and I've seen attempts with quantum logic and so on. But they too are quite involved and not very natural. At least these guys understood the question.

And if you really want to know: Your equation is the definition of force. That's nowhere comparable to a physical law! You can invent as many variables as you wish as long as you don't give a physical meaning to them.

 

[ytuab]

I think the physical interpretation of Schrodinger equation is "vague", which causes the discrepancies in the discussion.

Basically, the next two things can harmonize well with each other ?

1 [tex]|\psi|^2[/tex] means the probability density of the particles (Ex. electrons).

2 The Schrodinger equation uses the de Broglie's theory and the Coulomb potential (V), so
[tex]E \psi = -\frac{\hbar^2}{2m} \Delta \psi + V \psi[/tex]

For example, in the two-slit or Davisson-Germer experiments, the above 1 and 2 conditions harmonize well.
(The probability density is [tex]|\phi_1 + \phi_2|^2[/tex], And the de Broglie's theory (wave) is also satisfied.)
[though the probability density near infinity is not zero...]

But how about the case of the Schroedinger equation of the hydrogen atom ?

If we choose the condition 1, we have to give up the condition 2 ?
For example, in the Bohmian mechanics (I don't intend to attack the BM theory personally, but this is a good example of the "real" physical meaning.), if we try to obey the theories of the Coulomb force and the de Broglie's waves, it is impossible that the electrons behave obeying the probability density of the Schroedinger equations.
(If the electron is moving by the Coulomb force between the electron and nuclues, it can't move obeying the probability density of the hydrogen Schrodinger equation.)

So the idea of "Shut up and calculate" mainly remains, I think.

 

[alxm]

Basically, the next two things can harmonize well with each other ?

Yes. Psi in (2) is defined as being what it is in (1). These are not two separate definitions of the wave function.

it is impossible that the electrons behave obeying the probability density of the Schroedinger equations.
(If the electron is moving by the Coulomb force between the electron and nuclues, it can't move obeying the probability density of the hydrogen Schrodinger equation.)

That's exactly what they do. To within experimental accuracy, which is something like 10-12 digits in the case of hydrogen. There are few physical phenomena which are as theoretically well-verified as the quantum mechanics of the hydrogen atom.

 

[The_Duck]

OK, so if you know how to introduce QM without postulating mathematics, please name the method. Because in the link you give it's the usual undergrad stuff where they postulate that QM has to be a Hilbert space, and observables are operators and so. They usually don't give any justification whatsoever.
I don't have access to the early books now, but I did some extensive research for book some time ago and hardly any book took a different approach.

I guess very basic things like "the state is a vector in Hilbert space" are probably always taken as assumptions. But I believe Ballentine, for example, derives the Schrodinger equation mostly from that assumption and the symmetries of (Galilean) spacetime, which seems fairly deep and minimalist and elegant to me, but perhaps that's because I don't understand it that well.

 

[tom.stoer]

QM allows different but "isomorphic" or equivalent representations.

Heisenberg started with the idea to introduce only observable objects. Therefore he avoided to talk about position and momentum but introduced his matrices (operators). The matrices are of course not observable, but the marix elements (e.g. energy) are.

Schrödinger introduced the wave function based on the deBroglie idea of assigning waves (wavelength) to particles. His idea was that the wave function and therefore QM as a whole will show "smooth and predictable behaviour" just as classicsl mechanics (no jumps). We know that he failed.

Then it was shown that both reopresentations are equivalent.

Heisenberg made the experience that representation and interpretation was an issue in Europe, but not so much in the U.S. When he presented his ideas to the U.S. physicists for the first time, they were confident that it's "right" if "you can calculate things correctly". They didn't care so much what things "are".

The abstract representations by Dirac was a step towards simplification, it has nothing to do with new interpretations on the ontological level. The concept of Hilbert spaces, states / rays and operators was an observation made after the theory was formulated; it was not "planned" from the very beginning.

 

[astro2cosmos]

you all discuss so much but my confusion remain same that how we can intrepretate the Sr.E. (which is actual a complex function),
Is there any expremental evidence or proof of this complex function in order to see in the nature?

 

[tom.stoer]

The fact that the Shrödinger equation uses a complex wave function is due to the specific representation; in other representations (Heisenberg, Dirac) there is no complex wave function; there is no wave function at all, but other mathematical objects. In addition one can rewrite the wave function (like any other complex number) in polar representation with to real functions.

It's like asking why the electromagnetic 4-potential is a 4-vector; it's simply because it works!

There is one deeper fact, namely that quantum mechanics has a global symmetry which is just an U(1) rotation, i.e. the global phase of the wave function. That's why it's natural to find a complex one-dim. representation, i.e. the Schrödinger wave function.

Now you can ask "why this global U(1) symmetry?". Again, because it works.

 

[zenith8]

Someone was talking about the 'Bohmian mechanics' point of view (albeit completely wrongly). Let me expand on this.

In that theory, particles have trajectories guided by the objectively-existing wave.

The Hilbert space is complex because the Schroedinger equation is a complex equation and its solutions are complex functions.

Why is the Schroedinger equation complex? Because we are using a standard mathematical trick to write two real equations as one complex equation. The two real ones are the continuity equation (which keeps the probability distribution of the particles normalized over time) and the quantum Hamilton-Jacobi equation (which describes the dynamics). You can replace the latter with the ordinary classical Hamilton-Jacobi equation if you want to describe classical mechanics with a Schroedinger-like equation.

So there is nothing fundamental about complex Hilbert spaces, or indeed in the use of a Hilbert space at all. It just turns out to be precisely the right object to provide a compact summary of the statistics of particle trajectories in a de Broglie universe.

 

[tom.stoer]

Please don't mix Bohmian mechanics and rewriting the wave function the using polar decomposition:

Rewriting the wave function is a new mathematical formulation which always works, regardless if you believe in Bohmian mechanics, Kopenhagen, Many-Worlds, ...; you can do that, it will work, it does not require any special interpretation; sometimes it's useful, sometimes it's strange, but that's calculation only.

Believing in Bohmian mechanics requires to rewritre the wave function as described, but nobody urges you to believe in Bohmian mechanics.

Let's ask another question: why is the wave function complex and not quaternionic?

 

[ytuab]

Yes. Psi in (2) is defined as being what it is in (1). These are not two separate definitions of the wave function.

That's exactly what they do. To within experimental accuracy, which is something like 10-12 digits in the case of hydrogen. There are few physical phenomena which are as theoretically well-verified as the quantum mechanics of the hydrogen atom.

You mean the probability density and the energy calculation based on the wavefunction are inseparable ?

As you know, in the energy calculation of the hydrogen atom, the Dirac equation is more correct than the Schrodinger equation.

But in the textbook, the probability density of the Dirac equation is not written instead of the Schrodinger equation, though the above two things are inseparable. Why?

-------
As shown in the two-slit or Davisson-Germer experiment, the electrons are moving obeying the direction and the magnitude of the "momentum" expressed by its wavefunction, which means this wavefunction is almost same as de Broglie's wave itself.

But in the wavefunction of the hydrogen atom, if the electrons are moving obeying the direction and the magnitude of the "momentum" expressed by its wavefunction at each point, its movement becomes "weird".
(For expample, if the momentum operator acting on the 1S (or 2P) hydrogen wavefunction means the "real" direction and the magnitude of the electron's momentum at each point, the electron is not oscillating (or rotating) around the nucleus, when connecting each momentum vector.)

This means that the hydrogen wavefunction is not de Broglie's wave itself though it uses the de Broglie's theory.
(So it is impossible that we consider the hydrogen wavefunction is "real wave" which is "compatible" with the de Broglie's theory.)

This is why the name "de Borglie's wave" is not used so much in QM ?
(As shown in the other threads, the "de Broglie's waves" and the "pilot waves" of de Broglie-Bohm theoy are different things, aren't they?)