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Physical Review A

Exact space-time propagator for the step potential

T. O. de Carvalho

Published Apr 1, 1993• 26 citations
Publisher

Abstract

We devise and apply a method for the calculation of the exact propagator for the Schr\"odinger equation for the one-dimensional Hamiltonian scrH(p,x)=${\mathit{p}}^{2}$/2m+${\mathit{V}}_{0}$\ensuremath{\Theta}(x). The limiting cases ${\mathit{V}}_{0}$\ensuremath{\rightarrow}0 (free particle) and ${\mathit{V}}_{0}$\ensuremath{\rightarrow}\ensuremath{\infty} (opaque wall) are checked. The method is also applicable to any piecewise-constant potential.

DOI
10.1103/physreva.47.2562

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We devise and apply a method for the calculation of the exact propagator for the Schr"odinger equation for the one-dimensional Hamiltonian scrH(p,x)=p2{\mathit{p}}^{2}p2/2m+V0{\mathit{V}}_{0}V0​\ensuremath{\Theta}(x). The limiting cases V0{\mathit{V}}_{0}V0​\ensuremath{\rightarrow}0 (free particle) and V0{\mathit{V}}_{0}V0​\ensuremath{\rightarrow}\ensuremath{\infty} (opaque wall) are checked. The method is also applicable to any piecewise-constant potential.
A propagator for the one dimensional time-dependent Schrödinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and transmission processes as the particle scattering at the potential jump (in contrast to the conventional wave-like picture) and for accounting for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to the particle. This propagator completely resolves the corresponding time-dependent Schrödinger equation (defines the wave function ψ(x,t)ψ(x,t)ψ(x,t)) and allows for considering the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of time. These results are related to fundamental issues such as measuring time in quantum mechanics (tunneling time, time of arrival, dwell time). For imaginary time, which represents an inverse temperature (t→−iℏ/kBT)(t\rightarrow-i{\hbar}/k_{B}T)(t→−iℏ/kB​T), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to a rectangular potential. If one shifts to imaginary time (t→−it)(t\rightarrow-it)(t→−it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals from elementary functions and thus allow for a numerical visualization of the probability density ∣ψ(x,t)∣2|ψ(x,t)|^2∣ψ(x,t)∣2, the density matrix and the solution of the diffusion-like equation. The results obtained may also be useful for spintronics applications due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.
One-dimensional wave package propagation through the potential step is considered. Several approaches are tested for the ability to obtain transmission coefficients. Consideration includes direct numeric solution of the time-dependent Schrödinger Equation with the step potential, usage of the spectral decomposition of the propagator. An asymptotic formula is proposed for calculation of the transmission coefficient, which produces accurate results and is much easier to apply then the others considered approaches.
In a series of previous notes (i.e. Parts II and III) we considered three methods for computing the propagator for the problem: id/dt W = -1/2m d/dx d/dx W + k(t)xxW + f(t)x (-id/dx) W. The first method is based on postulating a form Q(t) exp(-i{ b(t)XX + g(t)YY +g2(t)XY}), inserting into the time-dependent Schrodinger equation and equating coefficients of XX etc The other two methods are based on classical Lagrangians. We argued in Parts II and III that for f(t)=0 there is agreement between the methods, but issues arise with the presence of f(t)x (-id/dx) W. In this note, we try to resolve this issue by using H= p dx/dt - L ((1)) where L is the Lagrangian. In method 2, a Lagrangian with f(t)x dx/dt is considered, but this term is converted to d/dt (.5xx f) - .5xx df/dt. Thus, the time-dependent oscillator potential is modified and the d/dt [] piece thrown away because it does not affect the classical equations of motion. Method 3 (3) computes the propagator using Q(t)exp(i Action), so d/dt [] is included and leads to a phase. By using ((1)) we argue that if f(t)x dx/dt exists in the Lagrangian, one must consider a corresponding Hamiltonian of the form: H= [p+f(t)x)][p+f(t)x)] (1/2m) + k(t)xx. This is equivalent to: pp/2m + f(t)f(t)xx (1/2m) + k(t)xx + f(t)(1/m)xp -i f/2m. Thus, we see an extra f(t)f(t)xx/2m in the Hamiltonian in addition to the f(t)xp type term. This “fixes” the problem of “”ff”” terms not canceling from the Schrodinger equation in parts II and III. Thus, one must be careful when dealing with velocity dependent potential, but also note that p=mdx/dt -f(t)x now so p as in the electromagnetic case is not simply the momentum of the particle.
A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.