Stationary States

Consider separable solutions to Schrödinger's equation of the form

$$\psi(x,t) = \psi(x)\,{\rm e}^{-{\rm i}\,\omega\,t}.$$ (4.68)

According to Equation (4.17), such solutions have definite energies $E=\hbar\,\omega$. For this reason, they are usually written

$$\psi(x,t) = \psi(x)\,{\rm e}^{-{\rm i}\,(E/\hbar)\,t}.$$ (4.69)

The probability of finding the particle between $x$ and $x+dx$ at time $t$ is

$$P(x,t) = \vert\psi(x,t)\vert^{2}\,dx = \vert\psi(x)\vert^{2}\,dx.$$ (4.70)

This probability is time independent. For this reason, states whose wavefunctions are of the form (4.69) are known as stationary states. Moreover, $\psi(x)$ is called a stationary wavefunction. Substituting (4.69) into Schrödinger's equation, (4.22), we obtain the following differential equation for $\psi(x)$;

$$-\frac{\hbar^{2}}{2\,m}\,\frac{d^{2}\psi}{d x^{2}} + U(x)\,\psi = E\,\psi.$$ (4.71)

This equation is called the time-independent Schrödinger equation.