10.3: Stationary States

Even though, for a given \(V (r)\) term, the Schrödinger equation may be impossible to solve in closed form, much can be said about the nature of its solutions without knowing them in detail. This is done by expressing \(\psi (r, t)\) as a sum of functions known as stationary states.

Stationary states are by definition solutions of the Schrödinger equation that are of a particular form, namely the product of a function of space times another function of time. It can be easily shown from the Schrödinger equation that the most general form stationary states can have is

\(\psi (r, t) = \phi (r)e^{-iEt/ \hbar} \tag{10.5}\)

for some real constant \(E\) (real because otherwise \(\psi (r, t)\) would grow without bound for very large or very small time), where \(\phi (r)\) obeys the equation (not involving time)

and where the integral over all space of \(|\phi(r)|^2\) is 1. This technique of separating the dependence of \(\psi (r, t)\) on its two variables \(r\) and \(t\) is sometimes called “separation of variables.”

Nonzero solutions for \(\phi (r)\) cannot be obtained for all values of \(E\). There may be some ranges in which any value of \(E\) is OK and other ranges in which only specific discrete values of \(E\) lead to nonzero wave functions. Generally speaking, solutions corresponding to discrete values of \(E\) become small far away (i.e., they “vanish at infinity”) and are therefore localized in space, although their “probability blobs” may have large values at several places and might therefore be thought of as representing two or more particles.

These solutions are called “stationary states” because the magnitude of the wave function (and therefore the probability density as well) does not change in time; it is only a function of space.

For these stationary states, \(E\) has an interesting interpretation. If we multiply each side of Equation 10.6 by \(\phi ^*(r)\) and integrate over space, we see that (just as in the previous section) E is the sum of two terms from the right-hand side, interpreted as the kinetic and potential energies of the object. Thus \(E\) is the total energy associated with that solution.

Of course in general solutions to the Schrödinger equation with this potential \(V (r)\) are not stationary states, i.e., do not have the special form of Equation 10.5. But remember that any linear combination of solutions to the Schrödinger equation is also a solution. We can use these stationary states as building blocks to generate more general solutions.

We are most interested in stationary states that are localized in space, so that the allowed values of \(E\) are discrete, although there could be many of them (perhaps even a countable infinite number). If we let \(j\) be an index over the stationary states, then it is possible to define the resulting wave functions \(\psi_j (r, t)\) so that they are both “normalized” in the sense that the space integral of the magnitude of each squared is 1 and “orthogonal” in the sense that the product of any one with the complex conjugate of another is zero when integrated over all space. We will then denote the values of \(E\), which we have interpreted as the energy associated with that state, by \(e_j\).

Then the general solutions to the Schrödinger equation are written as a linear combination of stationary states

\(\psi (r, t) = \displaystyle \sum_{j} a_j \phi_j (r) e^{-ie_jt/\hbar} \tag{10.7}\)

where \(a_j\) are known as expansion coefficients, and may be complex. If the wave function \(\psi(r, t)\) is normalized then it is easily shown that

\(1 = \displaystyle \sum_{j} | a_j |^2 \tag{10.8}\)

and that the energy associated with the function can be written in terms of the \(e_j\) as

\(\displaystyle \sum_{j} e_j| a_j |^2 \tag{10.9}\)

From these relationships we observe that \(|a_j |^2\) behaves like a probability distribution over the events consisting of the various states being occupied, and that this distribution can be used to calculate the average energy associated with the object.

The conclusion of our brief excursion into quantum mechanics is to justify the multi-state model given in the next section. Those readers who were willing to accept this model without any explanation have skipped over the past two sections and are now rejoining us.