Piecewise Constant one-Dimensional Potentials

We consider the one-dimensional stationary Schrödinger equation

$$\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V({x}) \right]\psi(x) = {E}\psi({x}),$$ (87)

where for the sake of a nicer notation we again write $E$ instead of $\tilde{E}$ and $\psi(x)$ instead of $\phi(x)$. Furthermore, there is only one variable $x$ so that the partial derivative $\partial_x =d/dx$.

In the following, we will concentrate on the important case where $V(x)$ is piecewise constant, i.e.

$$V(x)=\left\{ \begin{array}{cc} V_1, & -\infty<x\le x_1 \\ V_2, & ... ... ... \\ V_N & x_{N-1}< x\le x_{N}\\ V_{N+1} & x_N<x< \infty \end{array} \right.$$ (88)

Let us look at (2.10) on an interval where $V(x)$ is constant, say $[x_1,x_2]$ with $V=V_2$. The Schrödinger equation

$$\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V \right]\psi(x) = {E}\psi({x})$$ (89)

on this interval is a second order ordinary differential equation with constant coefficients. There are two independent solutions

$$\psi_+(x)=e^{ikx},\quad \psi_-(x)=e^{-ikx},\quad k:=\sqrt{\frac{2m}{\hbar^2}\left(E-V\right)}.$$ (90)

1. If $E>V$, the wave vector $k$ is a real quantity and the two solutions $\psi_{\pm}(x)$ are plane waves running in the positive and the negative $x$-direction. Such solutions are called oscillatory solutions.

2. If $E<V$, $k$ becomes imaginary and we write

$$k = i\kappa := i \sqrt{\frac{2m}{\hbar^2}\left(V-E\right)}$$ (91)

with the real quantity $\kappa$. The two independent solutions then become exponential functions $e^{\pm\kappa x}$. Such solutions are called exponential solutions.

For fixed energy $E$, the general solution $\psi(x)$ will be a superposition, that is a linear combination

$$\psi(x)=a e^{ikx}+ b e^{-ikx}$$ (92)

with $k$ either real or imaginary, $k=i\kappa$. Since the wave function in general is a complex function, the coefficients $a$, $b$ can be complex numbers. Note that we can not have linear combinations with one real and one imaginary term in the exponential like $a e^{ikx}+ b e^{-\kappa x}, a,b \ne 0$.

The solution of our Schrödinger equation with the piece-wise constant potential (2.11) must fulfill it everywhere on the $x$-axis. Therefore, it can be written as

$$\psi(x)=\left\{ \begin{array}{cc} a_1e^{ik_1x}+b_1e^{-ik_1x}, & -... ...a_{N+1}e^{ik_{N+1}x}+b_{N+1}e^{-ik_{N+1}x}, & x_N<x< \infty \end{array} \right.$$ (93)

with complex constants $a_j$, $b_j$ and $k_j=\sqrt{({2m}/{\hbar^2})\left(E-V_j\right)}$ either real or complex.