Stationary states

An eigenstate of the energy operator $H\equiv{\rm i}\,\hbar\,\partial/\partial t$ corresponding to the eigenvalue $E_i$ satisfies

$${\rm i}\,\hbar\,\frac{\partial \psi_E(x,t,E_i)}{\partial t} = E_i\,\psi_E(x,t,E_i).$$ (275)

It is evident that this equation can be solved by writing

$$\psi_E(x,t,E_i) = \psi_i(x)\,{\rm e}^{-{\rm i}\,E_i\,t/\hbar},$$ (276)

where $\psi_i(x)$ is a properly normalized stationary (i.e., non-time-varying) wave-function. The wave-function $\psi_E(x,t,E_i)$ corresponds to a so-called stationary state, since the probability density $\vert\psi_E\vert^{\,2}$ is non-time-varying. Note that a stationary state is associated with a unique value for the energy. Substitution of the above expression into Schrödinger's equation (119) yields the equation satisfied by the stationary wave-function:

$$\frac{\hbar^2}{2\,m}\,\frac{d^2 \psi_i}{d x^2} = \left[V(x)-E_i\right]\psi_i.$$ (277)

This is known as the time-independent Schrödinger equation. More generally, this equation takes the form

$$H\,\psi_i = E_i\,\psi_i,$$ (278)

where $H$ is assumed not to be an explicit function of $t$. Of course, the $\psi_i$ satisfy the usual orthonormality condition:

$$\int_{-\infty}^\infty \psi_i^\ast\,\psi_j \,dx = \delta_{ij}.$$ (279)

Moreover, we can express a general wave-function as a linear combination of energy eigenstates:

$$\psi(x,t) = \sum_i c_i\,\psi_i(x)\,{\rm e}^{-{\rm i}\,E_i\,t/\hbar},$$ (280)

where

$$c_i = \int_{-\infty}^{\infty} \psi_i^\ast(x)\,\psi(x,0)\,dx.$$ (281)

Here, $\vert c_i\vert^{\,2}$ is the probability that a measurement of the energy will yield the eigenvalue $E_i$. Furthermore, immediately after such a measurement, the system is left in the corresponding energy eigenstate. The generalization of the above results to the case where $H$ has continuous eigenvalues is straightforward.

If a dynamical variable is represented by some Hermitian operator $A$ which commutes with $H$ (so that it has simultaneous eigenstates with $H$), and contains no specific time dependence, then it is evident from Eqs. (279) and (280) that the expectation value and variance of $A$ are time independent. In this sense, the dynamical variable in question is a constant of the motion.