Stationary States

When the potential $V(x)$ does not depend on time, the Schrödinger equation admits solutions of the form $\psi(x,t) = \phi(x)\,f(t)$. Substituting this into the time-dependent Schrödinger equation and dividing by $\phi f$ gives:

The left side depends only on $t$, while the right side depends only on $x$. The only way this can hold for all $x$ and $t$ is if both sides equal a constant, which we call $E$ (the energy). This yields two ordinary differential equations:

The time equation has the simple solution $f(t) = e^{-iEt/\hbar}$. The spatial equation is called the time-independent Schrödinger equation (TISE).

A state of the form $\psi(x,t) = \phi(x)\,e^{-iEt/\hbar}$ is called a stationary state. Despite the name, the wave function itself does depend on time through the phase factor. However, the probability density $|\psi|^2 = |\phi|^2$ is time-independent.

All expectation values of time-independent operators are constant in a stationary state. For example, $\langle x\rangle$, $\langle p\rangle$, and $\langle H\rangle$ do not change with time. The only thing that evolves is the overall phase.

Stationary states are energy eigenstates: $\hat{H}\phi = E\phi$. The possible values of $E$ are called the energy spectrum. For bound states in a confined potential, the spectrum is discrete. For scattering states, the spectrum is continuous.

The general solution to the time-dependent Schrödinger equation is a superposition of stationary states: $\psi(x,t) = \sum_n c_n \phi_n(x)\,e^{-iE_n t/\hbar}$. Once we know the stationary states and their energies, we can construct any solution.

The time-independent Schrodinger equation is an eigenvalue equation for the Hamiltonian operator. Because the Hamiltonian is Hermitian, its eigenvalues (the energies) are real numbers, and its eigenfunctions corresponding to distinct eigenvalues are orthogonal.

The orthogonality of eigenfunctions means:

For bound states, the eigenfunctions can be normalized. For continuous spectra (like the free particle), normalization requires $\delta$ functions.

The completeness of the eigenfunctions means that any reasonable function can be expanded as a sum (discrete spectrum) or integral (continuous spectrum) of the eigenfunctions. This is the mathematical foundation of the generalized statistical interpretation.

Given an initial wave function $\psi(x,0)$, we can expand it in the energy eigenbasis:

The expansion coefficients are found by projecting the initial state onto each eigenfunction:

Once the $c_n$ are known, the full time-dependent solution is obtained by attaching the time-dependent phase to each term:

The probability of measuring energy $E_n$ is $|c_n|^2$. This follows from the generalized statistical interpretation: the eigenfunctions of the Hamiltonian form a complete orthonormal set, and $|c_n|^2$ is the squared overlap between the state and the eigenfunction.

A particle in an infinite square well of width $a$ has initial wave function $\psi(x,0) = A\,x(a-x)$ for $0 < x < a$, and $\psi = 0$ otherwise.

Step 1 — Find the normalization constant $A$. We require $\int_0^a |\psi|^2\,dx = 1$. Computing $\int_0^a x^2(a-x)^2\,dx = a^5/30$. Thus $A = \sqrt{30/a^5}$.

Step 2 — Find the expansion coefficients $c_n$. The eigenfunctions are $\phi_n(x) = \sqrt{2/a}\,\sin(n\pi x/a)$. Then $c_n = \int_0^a \phi_n^*(x)\,\psi(x,0)\,dx$. After two integrations by parts, $c_n = \frac{2\sqrt{15}}{n^3\pi^3}[1-(-1)^n]$. Only odd $n$ contribute because the initial state is symmetric about $x = a/2$.

Step 3 — Probability of ground state. $P(E_1) = |c_1|^2 = 960/\pi^6 \approx 0.9986$. The particle is almost certainly in the ground state because $\psi(x,0) = x(a-x)$ has a very similar shape to $\sin(\pi x/a)$.

Step 4 — Expectation value of energy. $\langle H\rangle = \sum_n |c_n|^2 E_n = \sum_{n\text{ odd}} \frac{960}{n^6\pi^6} \cdot \frac{n^2\pi^2\hbar^2}{2ma^2} = \frac{5\hbar^2}{ma^2}$. This can be verified directly from the initial state.