Boundary conditions for the Schrödinger equation in the numerical simulation of quantum systems

Abstract

We study the problem of the boundary conditions in the numerical simulation of closed and open quantum systems, described by a Schrödinger equation. On one hand, we show that a closed quantum system is defined by local boundary conditions. On the other hand, we argue that, because of the uncertainty principle, no local boundary condition can be defined for open quantum systems. For this reason plane waves or wave packet trains cannot be simulated on a finite numerical lattice with the usual procedures. We suggest a method that avoids these difficulties by using only a small numerical lattice and maintains the correspondence with the physical picture, in which the incident and scattered waves may be infinitely extended.

Figures (4)

• Figure 1: Propagationof the plane wave $\Phi_{0}(x, t) = A \exp \left(i k x-i k^{2} t\right)$, with $A=1$ and $k=2.4$, injected at $x_{s}=-15$. (a) $P(x, t)=|\psi(x, t)|^{2}$ vs $x$ at $t=0,2.5,5, \infty$ (by " $\infty$" we mean a large time $t \simeq 20$). The initial configuration is $\psi(x, 0)=0$ for $x \leq x_{s}$ and $\psi(x, 0)=\Phi_{0}(x, 0) g(x)$ for $x>x_{s}$, where $g(x)$ is given by (9). The dashed line represents the absolute value (in arbitrary units) of the imaginary potential $i V_{i}(x)=-i c\left(x-x_{i}\right)^{2} \Theta\left(x-x_{i}\right)$, where $\Theta$ is the Heaviside function, $x_{i}=20$, and $c=0.1$, plotted only at $t=0$ for clarity. (b) Quantum current calculated numerically from Eq. (6) at $x=10$.
• Figure 2: Same wave of Fig. $1$, scattering off the square potential barrier $V(x)=V_{0} \Theta(x-a) \Theta(x-b)$, where $V_{0}=5, a=-1$$\boldsymbol{b}=+1$. (a) $|\psi(\boldsymbol{x}, t)|^{2}$ at $t=0,2.5,5, \infty .$ For the sake of clarity, the real potential $V(x)$ (continuous line) and the absolute value of the imaginary potential $V_{i}(x)$ (dashed line) are plotted only at $t=0$. The oscillations between the point of injection and the barrier are due to the interference between incident and reflected waves. (b) Reflected (negative) current $J^{R}(t)$ and transmitted (positive) current $J^{T}(t)$, evaluated at $x=-20$ and $10$, respectively.
• Figure 3: Reflection (dashed line) and transmission (continuous line) coefficients $T$ and $R$ of the square potential barrier of Fig. 2(a), calculated from Eqs. (12) and (13), are compared with the values obtained by numerical simulations: numerical values of $R$ ($\blacksquare$) and numerical values of $T$ ("$\blacktriangle$"). The integration steps used in the numerical simulation are $\Delta t=0.01$ and $\Delta x=0.05$.
• Figure 4: Same plane wave of Figs. 1 and $2$, scattering off the time-dependent potential $V(x, t) = V_{0} [1 + \alpha \cos (\omega t)] \Theta(x-a) \Theta(b-x)$, where $V_{0} = 5$, $\alpha = 1/2$, $v = \omega / 2 \pi = 1$, $a = -1$, $b = +1$. (a) $|\psi(x, t)|^{2}$ at $t=0,3,6,9$. Note the oscillations in space. (b) Reflected and transmitted currents, calculated at $x=-20$ and $x=10$, respectively. Notice the oscillations in time.