Originally Posted by Sammywu
It seems to treat "momentum" and "position" as two basises in that both can span the complete Hilbert space and a unitary tranformation is used to translate the two basis.
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This is correct. The momentum basis alone spans the whole state space, and so does the (other) position basis. They are indeed related by a unitary transformation (which is a Fourier transformation).
I vision now the Hilbert space can be constructed with a intertwining "momentum" and "position" not exactly as a direct sum or product; they might not be orthogonal to each others but either one will not be able to span the whole Hilbert space.
Did I misunderstand QM at all?
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I guess what troubles you is the fact that in classical mechanics, the phase space (p,q) is the state space, while in quantum theory, only one of both (q or p or a mixture) seems to be necessary.
Well, that's the whole issue !
If you would construct (you are allowed so) a quantum theory where p and q are commuting observables, so that you have a basis labeled in (p,q), you would find out that ALL states are stationary states !
The simplest way to see this is in the Heisenberg picture:
i hbar d O / dt = [O,H]. Because any sensible observable, as well as the hamiltonian, are made up of p and q, and all p and q commute, the commutator [O,H] = 0. So all d/dt = 0.
So you are allowed to build such a quantum theory, but first of all it is a boring one, and second, most important: it doesn't go into the classical theory when h ->0.
Dirac worked out that when we do things the way they have to be done (using q as a basis, or p as a basis) and having the commutation relations [q,p] = i hbar, we DO find back the classical mechanics theory when h->0 (the correspondence principle).
cheers,
Patrick.