$$\require{cancel}$$

# 3.10: Stationary States

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

${\rm i}\,\hbar\,\frac{\partial \psi_E(x,t,E_i)}{\partial t} = E_i\,\psi_E(x,t,E_i).$ It is evident that this equation can be solved by writing $\psi_E(x,t,E_i) = \psi_i(x)\,{\rm e}^{-{\rm i}\,E_i\,t/\hbar},$ where $$\psi_i(x)$$ is a properly normalized stationary (i.e., non-time-varying) wavefunction. The wavefunction $$\psi_E(x,t,E_i)$$ corresponds to a so-called stationary state, because the probability density $$|\psi_E|^{\,2}$$ is non-time-varying. Note that a stationary state is associated with a unique value for the energy. Substitution of the previous expression into Schrödinger’s equation ([e3.1]) yields the equation satisfied by the stationary wavefunction:

$\label{etimeii} \frac{\hbar^{\,2}}{2\,m}\,\frac{d^{\,2} \psi_i}{d x^{\,2}} = \left[V(x)-E_i\right]\psi_i.$ This is known as the time-independent Schrödinger equation. More generally, this equation takes the form $\label{etimei} H\,\psi_i = E_i\,\psi_i,$ where $$H$$ is assumed not to be an explicit function of $$t$$. Of course, the $$\psi_i$$ satisfy the usual orthonormality condition:

$\label{e4.157} \int_{-\infty}^\infty \psi_i^\ast\,\psi_j\,dx = \delta_{ij}.$

Moreover, we can express a general wavefunction as a linear combination of energy eigenstates:

$\label{e4.158} \psi(x,t) = \sum_i c_i\,\psi_i(x)\,{\rm e}^{-{\rm i}\,E_i\,t/\hbar},$ where $c_i = \int_{-\infty}^{\infty} \psi_i^\ast(x)\,\psi(x,0)\,dx.$ Here, $$|c_i|^{\,2}$$ 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 previous results to the case where $$H$$ has continuous eigenvalues is straightforward.

If a dynamical variable is represented by some Hermitian operator $$A$$ that commutes with $$H$$ (so that it has simultaneous eigenstates with $$H$$), and contains no specific time dependence, then it is evident from Equations ([e4.157]) and ([e4.158]) 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.

# Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
