Canonical transforms and the efficient integration of quantum mechanical wave equations

The integration of time-dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry. The wave-function’s energy spectrum as well as its momentum spectrum impose fundamental limits on the performance of numerical algorithms for the solution of wave equations. We demonstrate how canonical transforms may be applied to negotiate these limitations and to increase the performance of numerical algorithms by up to several orders of magnitude. Our approach includes the so-called Kramers-Henneberger transform as a special case and puts forward modifications toward an improved numerical efficiency.

Figure 1

(Color online) Reduction in time-spatial oscillations by the spectrum-guided integration. (a) A one-dimensional Gaussian wave packet $\Psi \left(x,t\right)$ in a harmonic potential with initial momentum $p=4$ and width $\sigma =1$ oscillates rapidly in time and space. (b) Spectrum-guided integration in the energy domain reduces temporal oscillations, while the (c) spectrum-guided integration in the momentum domain reduces spatial oscillations. (d) The combination of spectrum-guided integration in the energy domain and the spectrum-guided integration in the momentum domain smoothes the wave function and removes all fast oscillations. Figures display real (left) and imaginary (right) parts of the wave function.

Figure 2

(Color online) Motion of a wave packet in a harmonically oscillating homogeneous electric field (49) with field strength $ϵ=16$ and frequency $\omega =1$. The figure compares the wave-packet’s motion in the (a) laboratory frame, in the (b) Kramers-Henneberger frame, and in the (c) center-of-mass frame. The initial wave function in the laboratory frame is a Gaussian wave packet (29) of width $\sigma =1$ and momentum $p=8$. Figures display real (left) and imaginary (right) parts of the wave function.

Figure 3

Numerical errors (60a, 60b) as a function of the temporal step width $\Delta t$ for the motion of a Gaussian wave packet in a harmonic potential. The figure shows the maximal deviation of the modulus (circles) and the phase (squares) of the numerically calculated wave function at time $t=\pi$ (half period of the harmonic oscillator) from the exact solution. Open symbols correspond to $\Psi \left(x,t\right)$; full symbols correspond to ${\Psi }^{\prime }\left(x,t\right)$. The wave function was initially given by a Gaussian wave packet (29) of width $\sigma =1$ and initial momentum $p=8$. The inset shows the discrete energy spectra of $\Psi \left(x,t\right)$ and ${\Psi }^{\prime }\left(x,t\right)$; $c_n$ denotes the expansion coefficient related to the $n\text{th}$ eigenfunction.

Figure 4

Numerical errors (60a, 60b) as a function of the mean energy $⟨E⟩$ for the motion of a wave packet (61) in a harmonic potential. The figure shows the maximal deviation of the modulus (circles) and the phase (squares) of the numerically calculated wave function at time $t=\pi$ (half period of the harmonic oscillator) from the exact solution. Open symbols correspond to $\Psi \left(x,t\right)$; full symbols correspond to ${\Psi }^{\prime }\left(x,t\right)$.

Figure 5

Numerical errors (60a, 60b) as a function of the temporal step width $\Delta t$ for the motion of a free Gaussian Dirac wave packet. Meaning of the symbols as in Fig. 3.

Figure 6

Numerical errors (60a, 60b) as a function of the spatial step width $\Delta x$ for the motion of a Gaussian wave packet in a harmonic potential. The figure shows the maximal deviation of the modulus (circles) and the phase (squares) of the numerically calculated wave function at time $t=2\pi$ (full period of the harmonic oscillator) from the exact solution. Open symbols correspond to $\Psi \left(x,t\right)$; full symbols correspond to ${\Psi }^{\prime }\left(x,t\right)$. The wave function was initially given by a Gaussian wave packet (29) of width $\sigma =1$ and initial momentum $p=7$.

Figure 7

Numerical errors (60a, 60b) as a function of time $t$ at fixed $\Delta t$ and $\Delta x$ for the motion of a Gaussian wave packet in a harmonically oscillating potential. Dashed lines and left scale correspond to the wave function $\Psi \left(x,t\right)$ in the laboratory frame, dot-dashed lines and left scale correspond to the wave function ${\Psi }^{\prime }\left(x,t\right)$ in the Kramers-Henneberger frame, and full lines and right scale correspond to ${\Psi }^{\prime }\left(x,t\right)$ in the center-of-mass frame. Same parameters and initial conditions as in Fig. 2.

Figure 8

CPU time as a function of the numerical accuracy for the simulation of free wave packets with and without the application of the spectrum-guided integration in the energy domain.