# Can we make a phase space plot for any quantum mechanical system?

Can we make a phase space plot for any quantum mechanical system? If we can then lets say at a particular point represented by $$(q,p)$$ does not it violate the Heisenberg Uncertainity as we simultaneously measure the momentum and position.

It is possible to make phase space plots in quantum mechanics, but they are not based on the usual Schrödinger wavefunctions $$\psi(x)$$ or $$\phi(p)$$, both of which depend on only one variable. Rather, Wigner functions, which for $$x$$ and $$p$$ are obtained as \begin{align} W(x,p)=\frac{1}{2\pi\hbar}\int_{-\infty}^\infty dy e^{-ipy/\hbar} \psi(x+\textstyle\frac{1}{2}y)\psi^* (x-\frac{1}{2}y) \end{align} and play the role of a phase space distribution, in the sense that average values are phase space integrals of the variable multiplied by the probability distribution $$W(x,p)$$. Thus for instance \begin{align} \langle x\rangle &= \int_{-\infty}^\infty \int_{-\infty}^\infty dx dp \,x \, W(x,p)\, , \\ \langle p\rangle &= \int_{-\infty}^\infty\int_{-\infty}^\infty dx dp \, p\, W(x,p)\, . \end{align}

Nothing in this violates the uncertainty principle as one can show that it is not possible for the Wigner function be arbitrarily sharply peaked, meaning it's not possible to have a WF that have very sharply defined position and momentum. Indeed, as discussed near Eq.(23) of this paper the maximum amplitude of the WF at a point satisfies \begin{align} \vert W(x,p)\vert\le 2/\hbar\, . \end{align}

The Wigner function for the ground state of the harmonic oscillator is given below.

One of the most intriguing part of this approach is that Wigner functions can have regions where they are negative, i.e. parts of the probability distribution is negative. For instance the Wigner function of the $$n=1$$ state of the harmonic oscillator yields

This has been verified experimentally by David Wineland and his group in

Leibfried, Dietrich, D. M. Meekhof, B. E. King, C. H. Monroe, Wayne M. Itano, and David J. Wineland. "Experimental determination of the motional quantum state of a trapped atom." Physical Review Letters 77, no. 21 (1996): 4281.

The figure below is extracted from this paper, where parts of the region of negative probability density are very apparent.

The negative regions in the Wigner functions are a considerable source of discussion. In particular it has been argued in

Kenfack, A. and Życzkowski, K., 2004. Negativity of the Wigner function as an indicator of non-classicality. Journal of Optics B: Quantum and Semiclassical Optics, 6(10), p.396.

that are tied to non-classical features of the associated quantum states.

You can never locate a particle at one point in phase space, as that would violate the uncertainty principle. You could plot the expected value of $$q$$ and $$p$$ in time, but you would clearly be missing a lot of information about the wave function.

There are some phase space formulations of quantum mechanics, which use quasi-probability distributions on phase space (which are like probability distributions, but can be negative), like the Wigner distribution or the Glauber-Sudarashan P representation.