1.23: The Time Independent Schrödinger Equation

When the potential energy is constant in time we can simplify the wave equation. We assume that the spatial and time dependencies of the solution can be separated, i.e.

\[ \Psi(x,t)=\psi(x)\zeta(t) \nonumber \]

Substituting this into Equation (1.21.7) gives

\[ i\hbar\psi(x)\frac{d}{dt}\zeta(t)=-\frac{\hbar^{2}}{2m}\zeta(t)\frac{d^{2}}{dx^{2}}\psi(x)+V(x)\psi(x)\zeta(t) \nonumber \]

Dividing both sides by \(\psi(x)\zeta(t)\) yields

\[ i\hbar\frac{1}{\zeta(t)}\frac{d}{dt}\zeta(t)=-\frac{\hbar^{2}}{2m}\frac{1}{\psi(x)}\frac{d^{2}}{dx^{2}}\psi(x)+V(x). \nonumber \]

Now the left side of the equation is a function only of time while the right side is a function only of position. These are equal for all values of time and position if each side equals a constant. That constant turns out to be the energy, E, and we get two coupled equations

\[ E\zeta(t)=i\hbar\frac{d}{dt}\zeta(t) \nonumber \]

and

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

The solution to Equation (1.23.4) is

\[ \zeta(t)=\zeta(0)\text{exp}[-i\frac{E}{\hbar}t] \nonumber \]

Thus, the complete solution is

\[ \Psi(x,t)=\psi(x)\text{exp}[-i\frac{E}{\hbar}t] \nonumber \]

By separating the wavefunction into time and spatial functions, we need only solve the simplified Equation (1.23.5).

There is much more to be said about this equation, but first let's do some examples.