Trying to understand quantum mechanics

[HAMJOOP]

I got some questions about QM (not well organized)

There is wave-particle duality. To my understanding, particle can behave as billard ball and wave.
Is the wave property means that we can represent the particle by wavefunction ?


The modulus of wavefunction in Schrödinger equation is interpreted by Max Born as probability density. So I guess Schrödinger did not know the meaning of wavefunction when he first proposed it ? How did Max Born comes with the idea that |psi|^2 is the probability density ?
(any experiments to verify the probability density ?)


In learning quantum mechanics, what I know is to promote classical observable (not sure) to operator, then we formulate the QM problem. But how is it actually done ? e.g. how Schordinger writes down the equation ?

 

[Simon Bridge]

I got some questions about QM (not well organized)

There is wave-particle duality. To my understanding, particle can behave as billard ball and wave.
Is the wave property means that we can represent the particle by wavefunction ?

"wave particle duality" is the observation that whether you see particle or wave behavior depends on how you look at the thing in question.

But be clear - a "billiard ball" is not a good model for a particle.
Particle behavior would be when energy is delivered to a target in lumps (waves deliver energy continuously); and wave behavior is diffraction and interference. Note, however, that in diffraction experiments with, say, electrons, the distributon of electrons arriving at the "screen" is what exhibits the diffraction, not the individual electrons.

This is the key - it is the statistics that has the wave behavior, not the object. The statistics are described by the wave-function.

The modulus of wavefunction in Schrödinger equation is interpreted by Max Born as probability density.

The square-modulus is the probability density.

So I guess Schrödinger did not know the meaning of wavefunction when he first proposed it ? How did Max Born comes with the idea that |psi|^2 is the probability density ?
(any experiments to verify the probability density ?)

In reverse order - there are a great many experiments verifying that it is a good idea to treat the wavefunction as a probability amplitude.

There were historically a lot of groups working on the theory, from different angles. Schrödinger was not clear on how to interpret the wavefunction though, no.
i.e. http://en.wikipedia.org/wiki/Wave_function#Historical_background

In learning quantum mechanics, what I know is to promote classical observable (not sure) to operator, then we formulate the QM problem. But how is it actually done ? e.g. how Schordinger writes down the equation ?

Classical physics is what happens on average in the QM description.
So the classical momentum of a particle, for example, is the expectation value of the momentum of the particle in a state described by a particular wavefunction.

But if you mean "what did Schrödinger think he was doing?" ... who knows.
It is not useful to your studies though - unless you prefer to study history of course.

 

[haael]

But be clear - a "billiard ball" is not a good model for a particle.

It's the worst possible model to be precise.

 

[phinds]

It's the worst possible model to be precise.

 

[bhobba]

In learning quantum mechanics, what I know is to promote classical observable (not sure) to operator, then we formulate the QM problem. But how is it actually done ? e.g. how Schordinger writes down the equation ?

These why questions are all good questions, and have answers.

Trouble is they require advanced math and are not at the beginner level.

I will give the answers, but won't be able to explain why. Just to be sure you know what I am saying is correct its from the first 3 chapters of the following standard textbook:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

The reason behind Schroedinger's equation, the operators used etc is actually the Principle Of Relativity:
http://en.wikipedia.org/wiki/Principle_of_relativity

It turns out when you apply that to the principles of QM (without going into the detail of exactly what they are) all these equations pop out.

Pity I can't explain the detail here. But its in the reference I gave above.

How did Schroedinger come up with it?

Again it requires advanced math, but the following details it:
http://arxiv.org/abs/1204.0653

Hopefully, despite the math, you can get a bit of the gist.

Thanks
Bill

 

[Simon Bridge]

It's the worst possible model to be precise.

...it was a slight understatement yah. Oh but I can think of worse models... the "old-sock" theory of particle interactions anyone?

iirc Feynman used "bullets" as his model "classical notion of a particle".