Mathematical Properties
1. P(x, p) is real
2. The x and p probability distributions are given by the marginals:
- If the system can be described by a pure state, one gets
- . If the system can be described by a pure state, one has
- Typically the trace of ρ is equal to 1.
3. P(x, p) has the following reflection symmetries:
- Time symmetry:
- Space symmetry:
4. P(x, p) is Galilei-covariant:
- It is not Lorentz covariant.
5. The equation of motion for each point in the phase space is classical in the absence of forces:
In fact, it is classical even in the presence of harmonic forces.
6. State overlap is calculated as:
7. Operator expectation values (averages) are calculated as phase-space averages of the respective Wigner transforms:
8. In order that P(x, p) represent physical (positive) density matrices:
where |θ> is a pure state.
9. By virtue of the Cauchy–Schwarz inequality, for a pure state, it is constrained to be bounded,
This bound disappears in the classical limit, ħ → 0. In this limit, P(x, p) reduces to the probability density in coordinate space x, usually highly localized, multiplied by δ-functions in momentum: the classical limit is "spiky". Thus, this quantum-mechanical bound precludes a Wigner function which is a perfectly localized delta function in phase space, as a reflection of the uncertainty principle.
Read more about this topic: Wigner Quasiprobability Distribution
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