How to solve Schrodinger Equation - Tunnelling

Question

I have to solve analitically the Schrodinger equation in one-dimension with a barrier of potential (tunnel effect):

$$ih \frac{d}{dt} U(x,t) = \left[ \left(-h^2 \frac{d^2}{dx^2} \right) + q V(x) \right] U(x,t)$$

where: $i$ is the imaginary unit, ($d/dt$) is the time derivative, $h$ is the Plank constant, ($d^2/dx^2$) is the second derivative in space, $V(x)$ is an external potential function of $x$, $U(x,t)$ is the wave function of time and place. The barrier of potential is:

$$V(x) = \begin{cases} 0, & \mbox{if } -d<x<-L \\ V_0, & \mbox{if } -L<x<L \\ 0, & \mbox{if } L<x<d \end{cases} $$

with $d=10L$ and $V_0>0$; Also the boundary condition are: $U(-d,t)=U(d,t)=0$; and the initial condition is

$$U(x,t_{0})=\frac{1}{\sqrt{Dx}} exp \left(i P_0 \frac{x}{h} \right)$$

if $-d<x<-d+Dx$ and $U(x,t_{0})=0$ if $-d+Dx<x<d$; where $Dx<<L$ and $P_0$ is the quantum moment at t0. Also I know that at time $t_0$, the Fourier Transform of $U(x,t_{0})$ is a sinc centered in $P_0$, the aspectated value of position is $-d+Dx/2$ and the aspectated value of velocity is $P_0/m$, where $m$ is the mass of the particle.

Then I have to compare analytical results with results from FINITE ELEMENTS and FINITE DIFFERENCE method.

I hope that someone can help me to solve this problem.

Answer 1

Without complete solution, just a road map. In principle, there is a standard way how this kind of problems is solved.

First solve stationary problem: $$ \left[ \left(-h^2 \frac{d^2}{dx^2} \right) + q V(x) \right] U_i(x) = E_i U(x).$$ As long as the potential is symmetric, I would recommend use this and search for even and odd solutions. For $E<V_0$ $$U_{i+}(x) = \begin{cases} A\cos{k(|x|-x_0)}, & \mbox{if } L<|x|<d \\ B\mathrm{ch}{\varkappa x}, & \mbox{if } -L<x<L \\ \end{cases}, $$ $$ U_{i-}(x) = \begin{cases} A\sin{k(|x|-x_0)}, & \mbox{if } L<|x|<d \\ B\mathrm{sh}{\varkappa x}, & \mbox{if } -L<x<L \\ \end{cases} $$ with $k=\sqrt{\frac{2mE}{\hbar^2}}$, $\varkappa = \sqrt{\frac{2m(V_0-E)}{\hbar^2}}$

For energies above $V_0$ replace hyperbolic functions with $\cos{k_2x}$, $\sin{k_2x}$ with $k=\sqrt{\frac{2m(E-V_0)}{\hbar^2}}$.

If you substitute these functions into equation, you'll get equation for energy. Note that as long as this form of functions automatically satisfies equation in the regions of constant potential, you should just take care of boundary conditions. Solve this equation and get a spectrum $\left\{ E_{i\pm} \right\}_{i=1}^{\infty}$ and wavefunctions $U_i(x)$. Obviously, these solutions have very nice feature when you consider non-stationary problem: $$ U_i(x,t)=e^{-i\frac{\hbar}{E_i} t}U_i(x), $$ which means that if you decompose your initial condition into these solutions (which you may do because eigenfunctions of a Hermitian operator is a complete set) $$ U(x,t_0) = \sum_1^{\infty} C_i U_i(x) $$ then time evolution of wavefunction is given by $$ U(x,t) = \sum_1^{\infty} C_i U_i(x) e^{-i\frac{\hbar}{E_i} t} $$