2.1. Time-independent Schrödinger equation

Here, we derive the time-independent Schrödinger equation by solving the time-dependent Schrödinger equation for a wavefunction of the special form \(\varPsi(x,t)=\psi(x)\cdot f(t)\) and a time-independent Hamiltonian.

First, inserting the above expression into the time-dependent Schrödinger equation for \(\varPsi\) gives

(2.38)\[\mathrm{i}\hbar\frac{\partial}{\partial t}\left(\psi f\right) = \hat{H}(\psi f)\]

Since \(\psi\) is time-independent by definition, we can pull it out of the derivative on the left side. Since \(\hat{H}\) is assumed to be time-independent, it does not affect \(f\), so we move \(f\) in front of the operator. This gives

(2.39)\[\mathrm{i}\hbar \psi \frac{\partial f}{\partial t} = f\hat{H}\psi\]

Dividing both sides by \(\psi f\) yields

(2.40)\[\mathrm{i}\hbar \frac{1}{f} \frac{\partial f}{\partial t} = \frac{1}{\psi}\hat{H}\psi\]

Now the left side depends only on \(t\), but not on \(x\), and the right-hand side depends only on \(x\) but not on \(t\) (since we started with the assumption that \(\hat{H}\) is time-independent). Varying either \(x\) or \(t\) would affect only one side of the equation, but not the other. Therefore, in order not to break the equality, both sides must be constant and independent of either variable. Let’s call this constant \(E\). With this, we get two equations.

(2.41)\[\mathrm{i}\hbar \frac{1}{f}\frac{\mathrm{d} f}{\mathrm{d} t} = E \qquad\qquad \frac{1}{\psi}\hat{H}\psi = E\]

The first one is an equation for \(f\) and has the general solution

(2.42)\[f(t) = C\cdot \mathrm{e}^{-\mathrm{i} E t/\hbar}\]

The factor \(C\) can be included with the time-independent part of the wavefunction, \(\psi\).

The second equation, the one for \(\psi\), is

(2.43)\[\hat{H}\psi = E\cdot\psi\]

and is called the time-independent Schrödinger equation. It cannot be solved generally, since the form of the Hamiltonian will depend on the quantum system (number and types of particles, their interactions, and external potentials).

Note that the time-independent Schrödinger equation is only valid for total wavefunctions that are a single product of \(\psi\) and \(f\), \(\varPsi(x,t)=\psi(x)\cdot f(t)\), i.e. for wavefunctions describing stationary states. It is inapplicable to non-stationary states that are described by superposition wavefunctions.