Stationary States in One Dimension

In this section, we study the stationary Schrödinger equation for an important class of potentials $V({\bf x})=V(x)$ that only depend on one spatial coordinate $x$ and are independent of $y$ and $z$. Such potentials could be generated, for example, by electric fields that only vary in one direction $x$ of space and are constant in the other directions.

We make a separation ansatz for $\psi({\bf x})$,

$$\psi({\bf x}) = \phi(x) e^{i(k_yy+k_zz)}.$$ (85)

Inserting into the stationary Schrödinger equation yields

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

$$\left[-\frac{\hbar^2\partial_x^2}{2m}+ \frac{\hbar^2}{2m}(k_y^2+k_z^2) + V({x}) \right]\phi(x) e^{i(k_yy+k_zz)} = E \phi({x})e^{i(k_yy+k_zz)}$$

$$\left[-\frac{\hbar^2\partial_x^2}{2m} + V({x}) \right]\phi(x) = \tilde{E}\phi({x})$$ (86)

$$\tilde{E} := E-\frac{\hbar^2}{2m}(k_y^2+k_z^2).$$

The general solution therefore can be written as the product of two plane waves running in the $y-z$-plane, and a wave function $\phi(x)$ which is the solution of (2.9). This means we have just to solve the one-dimensional stationary Schrödinger equation, with the energy $E$ replaced by $E-{\hbar^2}/{2m}(k_y^2+k_z^2)$ which is the energy for the one-dimensional motion in the $x$-direction. Together with ${\hbar^2}/{2m}(k_y^2+k_z^2)$, the kinetic energy of the motion in the $y$-$z$-plane, this is just the total energy $E$. With the potential $V({\bf x})=V(x)$, the particle sees no potential change when moving in only $y$- or $z$-direction: there is no force acting on the particle in this direction which is why its free motion turns out to be described by plane waves in the $y$- and $z$-direction.

In the following, we will therefore only consider the rest of the problem, i.e. the motion in $x$-direction.