Re: Quantum states from classical states?

himog asked:
> Is there a legitimate way to define a quantum state as some
> kind of superposition of classical states?

Dr. Paul Kinsler replied:
> You might find it interesting to have a look at the
> Wigner distribution.

"jarek korbicz" wrote:
> Arnold Neumaier wrote:
> > markwh04@xxxxxxxxx wrote:
> > > The quantum equivalent of a classical state is a coherent state; the
> > > process of arriving at a quantum theory which has the given classical
> > > state space as its classical limit is known more generally as Berezin
> > > quantization.
> >
> > And every quantum state can be written as a superposition of coherent
> > states, though not in a unique way. Thus the view of quantums states as
> > superposition of classical (i.e., coherent) states is fully valid.
>
> The distinction between classical and quantum states can be very nicely
> formalized when one looks at the general, i.e. mixed states of a
> canonically quantized system. Then one finds that indeed there are
> states which behave like classical (the only ad hoc assumption being
> that one uses normally ordered operators when calculating averages). If
> anybody wants to learn more about how to distinct those states, he may
> want to take a look at Phys. Rev. Lett. 94, 153601 (2005)

This sounds interesting. If you can give an equivalent Arxiv preprint
reference that would be appreciated.

In the meantime, allow me to point out that with a different ad-hoc
assumption -- namely, that one uses symmetrically ordered operators when
calculating averages -- one finds that the quantum states that "behave
clasically" are precisely those with positive Wigner distributions.

> > Of course, under a sufficiemtly nontrivial quantum dynamics,
> > a coherent state does not remain coherent as time develops;
> > this explains the departure from classicality in quantum mechanics.
>
> To be more precise, only hamiltonians at most quadratic in x, p leave
> coherent states invariant.

And -- for pure states -- it is precisely these Hamiltonians that
generate unitary transformations that leave the property of having
a positive Wigner distribution unaltered. These transformations are
just linear canonical transformations. The pure states with positive
Wigner distributions include coherent states, squeezed coherent states,
and (if one allows the term "state" to be abused somewhat) states with
a definite value of Ux + Vp where U and V are real c-numbers.

//cmh

.

Relevant Pages

  • Re: Quantum states from classical states?
    ... > And every quantum state can be written as a superposition of coherent ... > superposition of classical ... The distinction between classical and quantum states can be very nicely ... > this explains the departure from classicality in quantum mechanics. ...
    (sci.physics.research)
  • Re: Klauder coherent states and Quantum-Classical correspondence
    ... Coherent states are kinematical objects, ... conservative dynamics. ... Decoherence theory tells what happens to systems ... > Klauder's theory coherent states are the cause of classicality. ...
    (sci.physics.research)