To implement boundary conditions which distinguish between particles flowing into and those flowing out of a system, we must reexpress the Liouville equation (2.3) in terms of the classical phase space , where q in this case corresponds to the position x and p is the momentum. This is naturally done by the Wigner-Weyl transformation, which transforms the density operator into the Wigner distribution function (Wigner, 1932; Heller, 1976; Berry, 1977; Carruthers and Zachariasen, 1983). For the present purposes, the Wigner-Weyl transformation consists of a change of independent coordinates to the diagonal and cross-diagonal coordinates:
followed by a Fourier transformation with respect to r. The variables x and may be expressed in terms of q and r by
Thus the Wigner distribution can be expressed as
The Liouville equation becomes
where the kernel of the potential operator is given by
These expressions are derived under the assumption that the domain is unbounded.
Let us consider the interpretation of the terms of the Liouville equation (4.33). The first term on the right-hand side is derived from the kinetic energy operator and is of the form known as a drift, streaming, or advection term (in various nomenclatures). This term is exactly the same as the corresponding term of the classical Liouville equation with force F:
The correspondence between the classical and quantum drift terms will be exploited in defining the open-system boundary conditions.
Quantum-interference effects enter the Wigner-Weyl representation via the nonlocal potential term of (4.33). The kernel of this operator in effect redistributes the Wigner function among different p's at each position q. The extent to which it does so depends upon the potential at positions remote from q (4.34). This is the way that interference between alternative paths is incorporated into the equation. Thus, a rough intuitive image of the action of is that it represents particles which have scattered off the potential at some point and upon returning, interfere with the particles propagating over other paths. This image will be invoked to interpret the effects of cutting off the integral in (4.34) at some finite value, which is required in practical computations.
Let us now consider a model in which the domain is bounded by q = 0 and q = l. To address the question of boundary conditions, first note that in the Wigner-Weyl representation, the Liouville equation (4.33) is of first order with respect to q and does not contain derivatives with respect to p. The characteristics of the derivative term are lines of constant p, and we must supply one and only one boundary value at some point on each characteristic, because the equation is of first order in q. The kinds of boundary conditions which are appropriate are illustrated in Fig. 7. To implement the picture described above, that the particles entering the device depend only upon the state of the reservoirs and that the particles leaving the device depend only upon the state of the device, we should apply the boundary conditions illustrated in Fig. 7(c). That is, we set
where is the distribution function of the reservoir to the left of the system and is the distribution function of the reservoir to the right. These boundary conditions are not invariant under time-reversal, because time-reversal would change the problem of Fig. 7(c) into that of Fig. 7(d).
Figure 7. Possible boundary conditions for the Liouville equation (4.33) in phase space. The points at which the boundary values are specified (indicated by a heavy line) can be at q = 0 as in (a), at q = l as in (b), or divided between the two boundaries, depending upon the sign of p, as shown in (c) and (d). The boundary conditions (c) are, in fact, the appropriate ones for an open system.
Conceptually, the boundary conditions (4.36) are identical to those employed in the conventional tunneling theory (see Appendices 9 and 12), in the Landauer approach (Landauer, 1957, 1970; Büttiker et al., 1985; Stone and Szafer, 1988), and in solutions of the Boltzmann equation for nonuniform systems (see Appendix 11 and Duderstadt and Martin, 1979). However, some care must be taken in this identification. It is true that the variable p goes over into the classical momentum appearing in the Boltzmann equation, by the correspondence principle. However, it is not true that p is the same quantity as the operator , or its eigenvalue. In particular, as will be discussed in Sec. 6.1, the traveling-wave boundary conditions actually depend upon the energy of the state, rather than p. Thus, the boundary conditions (4.36) are conceptually identical to but mathematically different from those employed in the tunneling and Landauer approaches.
Let us call the Liouville superoperator which results from the boundary conditions (4.36) (for open-system, irreversible). For the purposes of the present discussion, it will be separated into two terms:
where is the superoperator derived from the kinetic energy term of the Hamiltonian:
and is the superoperator derived from the potential term:
Let us note in passing that can be written in two other forms. One is Groenewold's expression (Groenewold, 1946):
The other is the Wigner-Moyal expansion (Moyal, 1949):
where in the last expression it is understood that only acts upon . The utility of both of these expressions depends upon the existence of a rapidly-converging series expansion for . Such an expansion is not available for the abrupt energy-barrier structures which originally motivated the present study, so the integral form of (4.39) is preferred for practical computations.