The WTE can be used to solve stationary and transient transport problems. In both cases the specification of appropriate boundary conditions and initial conditions is paramount. This will be discussed in the following.
To have a well-posed problem (2.35) must be specified with an initial condition at time t0, which specifies the Wigner function at t0 over the entire phase space. In a finite domain the initial condition is specified only in the phase space represented by the domain while the boundary conditions reflect the influence (if any) of the initial condition outside the domain under observation for all times t ≥ t0 [68, 69]. The specification of boundary conditions in open quantum systems has been the subject of intense investigation over many years and is a critical aspect, regardless of the formalism used to describe the problem, see e.g. [62, 70].
The phase-space formulation of the Wigner function makes the adoption of classical, equilibrium distributions at the boundaries an attractive proposition, foremost because these distributions are known a priori and the experience gained from semi-classical Monte Carlo simulations [71] can be built upon. However, due to the spatial correlations that exist in the WTE (through the integrand and the Wigner potential (2.36)), the boundary conditions of an equilibrium distribution should be specified infinitely far away to avoid correlations with the non-equilibrium distribution in the active region of the device. As specified in Section 2.3, based on physical and computational considerations, a finite coherence length is chosen. Therefore, once the potential changes negligibly within the coherence length, it is reasonable to assume that the distribution approaches equilibrium values within the applicable relaxation time. These conditions are given in metals (contacts), where processes are at play which destroy coherence.
Numerical studies have demonstrated that the solution of the WTE indeed depends on the size of the contact regions (extensions from the active region) [72]. What constitutes sufficiently far away for the classical distribution to be recovered depends on the device being simulated and a sensitivity analysis should be performed to obtain the optimal distance. A distance of 30nm (Si) to 60nm(GaAs) for single and double barrier structures has been suggested [73] for simulations at room temperature; at lower temperatures a larger separation is required, because the coherence length increases as scattering decreases. It is desirable to place the boundaries as close as possible to the active region to reduce the computational burden.
The use of device-specific boundary conditions has been proposed [74], where the influence of the electrostatic potential inside the device on the distribution function at the boundary is taken into consideration, thereby allowing the boundaries to be placed closer to the active region. The unbiased (equilibrium) solution is used to specify the boundary condition for the non-equilibrium case. This requires the wave function under equilibrium conditions to be obtained from analytic solutions (only possible for simple potential barriers) or a numerical solution of the Schrödinger equation, from which the Wigner function can be obtained. However, the required calculations incur additional (once-off) computational, which must be weighed against the savings gained by having a smaller domain.
It is important to note that the specification of initial and boundary conditions which are physically valid and justified is of critical importance [75]. If physical aspects are left out of consideration – as was done in [76] – the WTE can yield non-physical solutions. The question of what constitutes a physically admissible Wigner function has been investigated in [54, 75]. The appropriate initial condition for the Wigner function can be obtained from a density matrix, which can be calculated using techniques outlined in [77].
One can distinguish between three types of boundary conditions:
Absorbing: An absorbing boundary is based on the assumption that physical processes are in place, which make the boundary reflection-less. The absorbing boundary conditions for the Wigner transport equation have been mathematically derived in [78] and analysed in [79]. The implementation of absorbing boundaries in Monte Carlo simulations is straight-forward: All particles cease to exist at the boundary and the boundary does not influence the evolution of the problem.
Injecting / in-flow: The in-flow boundary conditions retain the properties of absorbing boundaries, but also inject particles into the domain according to a specified distribution. An injecting boundary functions analogously to a black-body for radiation in that it emits (injects) electrons according to a thermal equilibrium distribution (say), regardless of the electrons that are absorbed [80]. This requires the energy relaxation in the contacts to be sufficiently fast for the contact to be regarded as memory-less and treated in a Markovian manner [81], i.e. the electron is absorbed/emitted by the boundary irrespectively of the electrons that were absorbed/emitted in the time prior. The in-flow boundary conditions for the Wigner formalism were first applied in the study of RTDs [80].
Reflecting: Particles are specularly reflected from the boundary and no particles are injected from the boundary. This approximates a boundary to an infinite potential step and assumes that no particles exist outside the domain under observation, at the time of initialization, which could enter the domain through the boundary. Reflecting boundaries are useful to approximate interfaces between semiconductors and oxide, where the wave function rapidly decays towards zero. The reflecting boundary corresponds to a zero Dirichlet boundary condition.
A stationary transport problem demands a non-equilibrium solution to the transport equation (2.35) that does not change with time, i.e. 
= 0. Stationary
(steady-state) solutions are of particular interest in logic devices where the response in the ’on’ and ’off’ state is of primary interest and the time scales associated
with the transient responses are much shorter than those associated with the operating frequency.
The relative importance of the initial and boundary conditions depends on the simulation time, domain size and the specifics of the system, e.g. the potential profile. The effect of the initial condition can become insignificant after a sufficiently long simulation time, once the particles that represented the initial condition have been absorbed by the boundaries and were replaced by particles determined solely by the boundary conditions. Conversely, if the particles that constitute the initial condition are concentrated inside the domain without ever reaching the boundary, e.g. an electron trapped in a deep quantum well, the boundary conditions are not influential and the initial condition determines the stationary solution that will be achieved.
The eigenstates of a system cannot be obtained by solving the WTE, however, the equation does ’retain’ the eigensolutions, if these are correctly introduced with the initial condition.
The Wigner transport equation reduces to the Boltzmann equation for potentials of second order (quadratic) or lower. Quantum effects can be present, nonetheless,
for such potentials, e.g. a quantum harmonic oscillator with a parabolic potential displays quantization effects [82]. To correctly capture these quantum effects, the
boundary conditions must be specified appropriately and the time-derivative set to zero [77]. However, the time derivative 
term, in the WTE, can only be
discarded beforehand with valid physical reasoning [75].
A transient transport problem investigates the time-dependent solution to the transport equation (2.35). The transient solution is of interest for quantum systems subjected to time-varying boundary conditions, e.g. RF circuits, or devices stimulated by electro-magnetic fields. While the simulation of quantum systems in the stationary state has been dealt with extensively, especially using the NEGF method, the transient problems remain largely under-investigated. This can be explained by the computational challenges faced by time-resolved quantum transport simulations using NEGF and the challenges in the metrology (photoluminscence).
In the simplest case, the transient behaviour of a quantum system can be observed by following the evolution of the initial condition while keeping the boundary conditions fixed. For systems that are used for RF applications or are stimulated by electro-magnetic field, time-varying Dirichlet boundary conditions are used for the potential, which emulate the time-dependent potential or electromagnetic field1 .