0, \quad \sum_n
\lambda_n = 1.$$

For mixed states, the WDF is thus

$$P_W(x,v) = \int e^{\frac{i}{\hbar}vy}\sum_n \lambda_n\psi^*_n\left(x+\frac{y}{2}\right)\psi_n\left(x-\frac{y}{2}\right)
d^3y,
\label{eq:WDF}$$

while for a pure state, it reads

$$P_W(x,v) = \int e^{\frac{i}{\hbar}vy}\psi^*\left(x+\frac{y}{2}\right)\psi\left(x-\frac{y}{2}\right)
d^3y.$$

To simplify the expressions, we will use the notation $x_\pm = x \pm \frac{y}{2}$ and $\psi_{n\pm} = \psi_n(x_\pm)$ in what follows. The WDF has many similarities to the classical distribution function: $P_W(x,v)$ is a real function, as can be seen by taking the conjugate and performing the change of variable $y\rightarrow-y$. It is also normalized to $1$ in the following sense

$$\int d^3x\int \frac{d^3v}{(2\pi \hbar)^3} P_W(x,v) = 1.$$

It has similar marginal distributions as can be seen by integrating over all velocities: $$\begin{aligned}
\int\frac{d^3v}{(2\pi \hbar)^3} P_W &=& \sum_n
\lambda_n\int
\delta^3\left(y\right)\psi^*_n\left(x_+\right)\psi_n\left(x_-\right)d^3y\nonumber  \\
&=& \sum_n \lambda_n \left| \psi_n\left(x\right) \right|^2,
\label{eq:density}\end{aligned}$$ or over all space

$$\begin{aligned}
 \int d^3x P_W &=& \sum_n \lambda_n \int d^3x_-
 d^3x_+e^{\frac{i}{\hbar}vx_+}e^{-\frac{i}{\hbar}vx_-}\psi^*_{n+}\psi_{n-} \nonumber \\
  &=& \sum_n \lambda_n\left|\tilde\psi_n\left(\frac{p}{\hbar}\right) \right|^2.\end{aligned}$$

In both cases the non-negative property of these marginal distributions is a property of the wavefunctions in quantum mechanics.

The Wigner distribution function does, however, have the peculiar property that it may assume negative values. For this reason, it is called a *quasi-probability distribution* and cannot be interpreted as a 