Particle in a potential.

Let us consider a general situation where there is a particle in a potential $V(x)$. We have seen that there exist wavefunctions of the form

$$\psi= e^{-i E t/\hbar} X(x)$$ (20.40)

where $X(x)$ is a solution to the equation

$$\frac{\hbar^2}{2m} \frac{d^2 X}{dx^2} = - [E-V(x)] X\,.$$ (20.41)

What happens if we measure the energy of a particle with such a wavefunction? This is an eigenfunction of the Hamiltonian operator with eigenvalue $E_1$. This means that this state has a well defined energy $E$. It is called an energy eigenstate. The probability density of such a state is time independent.

$$\rho(x) = \psi^*(x,t) \psi (x,t) = X^* (x) X(x)$$ (20.42)

For this reason a state with a wave function of this form is also called a stationary state.