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From what I understand I think if I construct the phase space by plotting $x$ component of momentum against $x$ for a quantum mechanical particle, I cannot specifically put a single point to mark the particle's state in that phase space. For that would mean I can state the position and momenta of the quantum particle with arbitrary precision, which is forbidden by the uncertainty principle. All I can do, at best, is to mark an area in the phase space.

Is this understanding of phase space of quantum system correct?

If it is, then my other question is:

Does the marked area of phase space (to specify the state of the particle) have an area of the order of $h$?

Qmechanic
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Samapan Bhadury
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1 Answers1

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Your surmise is basically right.

Indeed, spikey δ-function distributions (perfectly localized) in phase space cannot describe quantum states. The narrowest possible quantum state has to extend to an area broader than h, as dictated by the uncertainty principle. Classical mechanics is precise and spikey, but quantum mechanics is uncertain and fuzzy.

In fact, for a phase-space quasi-probability distribution (Wigner function) describing a state, if it is normalized to one, $\int dx dp f(x,p)=1$, you may easily prove that is is of finite height, $|f|\leq 2/h$. It must extend to an h-base or broader to produce unit volume. This is in sharp contrast to classical localized states, and segues to them in the classical limit.

Much of this discussion is expanded and illustrated in our book, namely ISBN 978-981-4520-43-0, World Scientific Publishing 2014, A Concise Treatise on Quantum Mechanics in Phase Space, by Curtright, Fairlie, and myself.

Now, to be sure, δ(x)δ(p), a point/spike in phase space is meaningful in QM, but not as a specifier of a state: instead, it specifies the parity operator, in this Weyl-correspondence mapping of operators to phase-space functions; but this is highly technical and might not be of interest to you. There are also associated peculiarities of f actually going negative, but also in small phase-space regions of area smaller than h, so the uncertainty principle protects them from observation too--it works almost miraculously to fix things and eliminate paradoxes!

The takeaway is that quantum distributions are fluffy/fuzzy: think of them as marshmallows. They only look sharp and classical for systems with huge actions, on the scale of such huge actions themselves. (Scaling the phase-space variables down and f up to preserve the unit normalization ultimately collapses the base of the pillbox we considered above to an apparent "point" in phase space; and leads to a divergent height for f, so an apparently localized classical particle. However, the entropy has increased: several different quantum configurations reduce to this same limit, obliterating information.)

Cosmas Zachos
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