Checkpoint: Stationary States

A stationary state is a solution of the Schrödinger equation that has a single, definite energy. They arise when the potential $V$ does not depend on time, so that the Schrödinger equation separates: write $\Psi(x, t) = \psi(x)\,T(t)$ and substitute into

Dividing both sides by $\psi(x)\,T(t)$ yields a left side that depends only on $t$ and a right side that depends only on $x$. Both must equal the same constant $E$, giving two ordinary differential equations. The time equation $i\hbar\,\dot{T} = E\,T$ is solved immediately:

This oscillatory factor carries no physical probability weight on its own — $|T(t)|^2 = 1$ — which is why a stationary state has a time-independent probability density:

The spatial part $\psi(x)$ satisfies the time-independent Schrödinger equation (TISE):

The infinite square well

The simplest non-trivial potential traps a particle inside a box of width $L$ with infinitely hard walls: $V(x) = 0$ for $0 \le x \le L$ and $V = \infty$ elsewhere. The infinite walls force $\psi(0) = \psi(L) = 0$. Inside the box the TISE reduces to

The general solution is $\psi(x) = A\sin(kx) + B\cos(kx)$. The boundary condition $\psi(0) = 0$ forces $B = 0$. The condition $\psi(L) = 0$ then requires $kL = n\pi$ for a positive integer $n$, giving the quantization condition

Substituting back yields the discrete energy levels:

where the ground-state energy is

The key result is that energies grow as $n^2$: the second level is four times the ground state, the third level is nine times, and so on. This $n^2$ scaling is a fingerprint of the infinite square well.

The normalized wave functions

Normalizing $A\sin(k_n x)$ over $[0, L]$ gives $A = \sqrt{2/L}$, so

Each $\psi_n$ has $n-1$ nodes (zeros strictly inside the well) — $\psi_1$ has no internal nodes, $\psi_2$ has one, and so on.

Phase evolution of a stationary state

Each stationary state acquires a phase at a rate $\omega_n = E_n/\hbar$:

Multiplying by a pure phase factor does not change any probability. That is the precise sense in which these are "stationary" states — the physics (all expectation values, all probabilities) is frozen in time, even though the complex amplitude rotates.

When two stationary states are superposed, the two phase factors rotate at different rates $\omega_1$ and $\omega_2$, and their interference pattern oscillates at the beat frequency $(\omega_2 - \omega_1) = (E_2 - E_1)/\hbar$. This is how the time-dependence reappears in non-stationary states.

Numerical scale: an electron in a 1 nm box

For an electron ($m = 9.109 \times 10^{-31}\,\text{kg}$) in a well of width $L = 1.0\,\text{nm}$,

The third level therefore sits at $E_3 = 9\,E_1 \approx 3.38\,\text{eV}$ — a few electron-volts, the same scale as electronic energies in atoms and molecules. What spectroscopy actually measures is the difference between levels: here $E_3 - E_1 \approx 3.01\,\text{eV}$, a photon of wavelength $\approx 412\,\text{nm}$ at the violet edge of the visible range. Nanometre-scale confinement is what puts these gaps into the optical band, the effect exploited in quantum-dot spectroscopy.

Try it

This is a numerical exercise — return a number. Compute the energy of the $n = 3$ stationary state of an infinite square well of width $L = 1.0\,\text{nm}$, for an electron, in eV. Build it from first principles: calculate $E_1$ from the formula above, then multiply by $n^2$.

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