Quantum Mechanics and Classical Mechanics: Is there a connection?

[Amok]

What is the connection between QM and classical mechanics? This has never been obvious to me, but it seems that it would be desirable that we should be able to derive the laws of CM from QM, but I've never really seen that done convincingly. Can you actually show that a very large quantum system will follow the laws of CM, just like you can show that CM is an approximation of relativity at low speeds? Can you give me your opinion on the subject?

 

[Jazzdude]

The quantum to classical transition is still problematic. It is intimately related to the quantum measurement problem, which deals with the appearance of definite outcomes of measurements from quantum superpositions.

You will find many handwaving arguments for a classical to quantum transition, based on a phase space with the limit h->0 or the time evolution of expectation values, the classicality of coherent states, etc. All those only describe certain aspects of the transition, without really being able to derive classical mechanics from quantum mechanics.

If you're really interested in this question then I can recommend "Decoherence and the appearance of a classical world in quantum theory" by Zee/Joos/et al. But its scope is also limited as it focuses on the questions that can be answered by using decoherence. But at least they discuss in depth what aspects of the transition still remain unsolved.

 

[Amok]

You will find many handwaving arguments for a classical to quantum transition, based on a phase space with the limit h->0 or the time evolution of expectation values, the classicality of coherent states, etc. All those only describe certain aspects of the transition, without really being able to derive classical mechanics from quantum mechanics.

You see, the arguments with h->0 always seemed extremely flimsy to me. Taking the limit of h goes to zero just shows that in a world with an arbitrarily small Planck's constant there would be no quantum effects (at any scale, including atomic, which means solid objects as described by Newton's laws would probably not even exist). The thing with Ehrenfest's theorem is cool, but that you have:

$$\frac{d \left< p \right>}{dt} = \left< - \nabla V \right>$$

And not:

$$\frac{d \left< p \right>}{dt} = \left - \nabla< V \right>$$

Is already a problem. I guess, as you said, this is all connected to the interpretation of QM. And yes, I am interested in the question because I think it's very important question that is still open and people seem to neglect this fact. I'll give that book a look if I have the time, thank for the referrence btw.

 

[Jazzdude]

I guess, as you said, this is all connected to the interpretation of QM. And yes, I am interested in the question because I think it's very important question that is still open and people seem to neglect this fact. I'll give that book a look if I have the time, thank for the referrence btw.

The best correspondence actually comes in the statistical limit if quantum theory is formulated on phase space. That is related to deformation quantization and non-commutative geometry. The transition is done by blurring the quantum phase space to get a classical phase space of an ensemble.

Another related approach is that of Hamilton Jacobi which is a classical limit of the Schroedinger equation. This has conceptional issues however, because the Hamilton-Jacobi does not describe classical states but something more abstract.

You're right that you won't get very far with Ehrenfest. This gets somewhat better if you have coherent states (states of minimum uncertainty in phase space) and a process that stabilizes coherent states (like decoherence can). Then Ehrenfest is a good description of the classical behavior of well phasespace-localized quantum state.

I don't think that this problem is neglected, it's just very hard. Without understanding quantum measurement one of the most important steps of the classical transition is missing, and physicists have been working for 80 years on this problem. So it's not a matter of not trying but rather not succeeding.

But if you think you can contribute to this field then I would recommend you start doing your own research. The chances that you will discover something relevant are not great, but if you do you will surely be rewarded.