Phase space: a one-to-one mapping with all quantum dynamics?

Loren Booda

Does the history of wave packets translate exactly onto infinite phase space, or is phase space incompletely (or redundantly) covered by quantum mechanics?

Haelfix

Can you rephrase your question with a little more precision?

Phase space is a classical mechanics concept (specifically its the cotangent bundle of the configuration space). "Quantum Wave packets" is a very quantum notion.

Which map are you refering too? I suspect you are asking a question about the quantization procedure, but I cannot venture a guess as to which part.

vanesch

I can only agree with Haelfix that the terms used in the original question, in their usual meaning, seem to be syntactically corrrect and semantically meaningless, which leaves the reader up to guessing what the original poster could possibly have meant. For instance, "quantum mechanics" does not correspond to a SET, so there's no map going from quantum mechanics anywhere. But, the reader can GUESS that what is meant is the state space of a certain quantum mechanical system (the rays in hilbert space).
As haelfix pointed out, "phase space" only has a classical meaning (the set of all positions and momenta of the system - more nicely put: the cotangent bundle of configuration space).
So we can GUESS again that we have to do with a classical system and its quantized counterpart.
Only, I'm at loss what might mean "the history of wave packets" as a map from Hilbert space into phase space. Well, I could guess one thing. If we drop "history", with some grain of salt we can take "wave packets" as more or less coherent states (special elements of hilbert space).
So exists there a map from the coherent states (special quantum states) into phase space ?
Answer: more or less. To each point in phase space (classical state of motion) corresponds a coherent state that is "the best possible quantum description of the classical state". But the hilbert space is much bigger. There exist a lot of quantum states that do not have a classical counterpart.

Loren Booda

vanesch,
Thanks for your tolerating my incomplete knowledge of quantum mechanics. You translated my question admirably. How would you define a special quantum state (a wave packet or coherent state) in relation to Hilbert space? Are there a finitude or infinity of quantum states (special or otherwise) compared to classical ones?

vanesch

Well, for some systems (harmonic oscillators) there is a precise mathematical meaning to it: they are the eigenstates of the annihilation operator. For other quantum systems, it can be more involved to find the best mapping between the quantum states and the classical states.

Have a look here:
http://en.wikipedia.org/wiki/Coherent_state