Time dependent Schrödinger equation

We want to describe an electron wavefunction by a wavepacket $\psi (x,t)$ that is a function of position $x$ and time $t$. We assume that the electron is initially localized around $x_0$, and model this by a Gaussian multiplying a plane wave:

$$\psi(x,t=0)=\exp{\left[-\frac{1}{2}\left(\frac{x-x_0}{\sigma _0} \right)^2\right ]} e^{ik_0x}$$ (79)

This wave function does not correspond to an electron with a well defined momentum. However, if the width of the Gaussian $\sigma _0$ is made very large, the electron gets spread over a sufficiently large region of space and can be considered as a plane wave with momentum $k_0$ with a slowly varying amplitude.

The behavior of this wave packet as a function of time is described by the time-dependent Schröedinger equation (here in 1d):

$$i\frac{\partial \psi}{\partial t}=H\psi(x,t).$$ (80)

$H$ is the Hamiltonian operator:

$$H=-\frac{1}{2m}\frac{\partial ^2}{\partial x^2}+V(x),$$ (81)

where $V(x)$ is a time independent potential. The Hamiltonian is chosen to be real. we have picked teh energy units such that $\hbar=1$, and from now on, we will pick mass units such that $m=1$ to make equations simpler.

Schrödinger's equation is obviously a PDE, and we can use generalizations of the techniques learned in previous sections to solve it. The main observation is that this time we have to deal with complex numbers, and the function $\psi (x,y)$ has real and imaginary parts:

$$\psi (x,t) = R(x,t)+iI(x,t).$$

However, is this section we will present an alternative method that makes the quantum mechanical nature of this problem more transparent.