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## 6.2.6 A Feasible Phase Space Distribution

There are numerical issues with the Wigner transform so that we would like to avoid this task. Instead we will now define a distribution which in one dimension classically approximates the phase space distribution. We want our distribution to reproduce exactly the marginal expectation values for , and . Other local expectation values are not necessarily reproduced correctly.

Carrier, current, and energy marginal densities can be directly calculated from the Schrödinger function respectively the hydrodynamical quantities with . Instead of the densities we prefer to work with the marginal -distributions for the in the case of constant mass equivalent set of local'' observables , where denotes the identity operator.

Introducing the variable (6.65)

we get the marginal expectation values (using the split of the energy operator as in Equation 6.63) for a pure state :  (6.66)  (6.67)  (6.68)

To define a corresponding classical phase space distribution function we assume that for a wave function the corresponding distribution w(x,p) for each consists of left (up) and right (down) going modes, i.e.: (6.69)

with .

This assumption makes sense in cases, in which the classical problem has a similar property. Such is the case for the scattering problems and the open Schrödinger equation which we consider in Section 7.1. In this case particles are injected with a fixed momentum from one electrode and condition 6.70 is fulfilled exactly. In the electrodes we have a superposition of transmitted and reflected modes.

From the ansatz 6.70 we can calculate the classical quantities:  (6.70)  (6.71)  (6.72)

Classically and are always positive quantities. Equating the marginal expectation values from the wave function and from the classical'' distribution function we get the system of equations: (6.73) (6.74) (6.75)

From the last equation we get (6.76)

which reduces the system of equations to  (6.77)  (6.78)

with the solution (6.79) (6.80)

with (6.81)

The coefficients and are positive values between 0 and as has to be the case for proper probabilities. To sum up: Our (1+1)-dimensional distribution function has the property that it is positive and reproduces the moments up to an order correctly.