Wigner Quasiprobability Distribution - Mathematical Properties

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:

  • \frac{\partial P(x,p)}{\partial t}=\frac{-p}{m}\frac{\partial P(x,p)}{\partial x}

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:

  • \langle \psi|\hat{G}|\psi\rangle=Tr(\hat{\rho}\hat{G})=\int_{-\infty}^\infty dx\, \int_{-\infty}^\infty dp P(x,p)g(x,p).

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.

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