Analogy to Real-Time Wave Packet Molecular Dynamics

Wave packet molecular dynamics (WPMD) was first used by Heller (1975) and later applied to scattering processes in nuclear physics (Feldmeier, 1990) and plasma physics (Klakow et al., 1994b; Ebeling and Militzer, 1997). An ansatz for the wave function $\psi({q_\nu})$ is made and the equation of motions for the parameters ${q_\nu}$ in real time can be derived from the principle of stationary action (Feldmeier, 1990),

$$\delta \int dt \: L = 0 \quad,\quad L \left({q_\nu}(t),{\dot... ...i {\partial}_t - {\mathcal{H}}\right\vert \psi \right\rangle$$ (142)

This leads to a set of first order equations, which provides an approximate solution of the Schrödinger equation. However, this principle cannot be directly applied to the Bloch equation because there is no imaginary part in the density matrix. For this reason, we followed in our derivation in section 3.2 the principle by McLachlan (1964), which minimizes the quantity

$$\int \vert {\mathcal{H}}\psi - i \Theta \vert ^2 \, dt,\quad\Theta = \frac{\partial \psi}{\partial t}\;.$$ (143)

This method was employed by Singer and Smith (1986) to obtain the dynamical equations in real time.

The VDM approach and WPMD method share the zero temperate limit, which is given by the Rayleigh-Ritz principle (see section 3.5.1). At high temperature, the width of wave packets in WPMD grows without limits, which is a known problem of this method (Knaup et al., 1999; Militzer, 1996). In the VDM approach, the correct high temperature limit of free particles is included. The average width shown in Fig. 3.11 can be used to verify the attempts to correct the dynamics of the real time wave packets by Knaup et al. (1999).