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.:
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.
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