Introduction: States That Don't Change
Stationary states are solutions of the TISE. Despite the name, they do evolve in time—but only by acquiring a phase. All physical observables remain constant, hence "stationary."
Why They're Special
For stationary state \(\psi_n(x,t) = \phi_n(x)e^{-iE_nt/\hbar}\):
\[|\psi_n(x,t)|^2 = |\phi_n(x)|^2\]The probability density doesn't change in time. Expectation values of any operator also stay constant.
The Energy Spectrum
The set of allowed energies \(\{E_n\}\) is the spectrum of the Hamiltonian:
- Discrete spectrum: Bound states, typically \(E_1, E_2, E_3, \ldots\)
- Continuous spectrum: Scattering states, range of energies
Worked Examples
Example 1: Infinite Square Well
Energies: \(E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}\), \(n = 1, 2, 3, \ldots\)
All energies are discrete—no scattering states.
Example 2: Harmonic Oscillator
Energies: \(E_n = \hbar\omega(n + \frac{1}{2})\), \(n = 0, 1, 2, \ldots\)
Evenly spaced levels with zero-point energy \(E_0 = \frac{1}{2}\hbar\omega\).
Example 3: Free Particle
Energies: \(E = \frac{\hbar^2k^2}{2m}\) for any \(k\)—continuous spectrum.
The Quantum Connection
Energy quantization is the hallmark of quantum mechanics. The discrete spectrum explains atomic stability (electrons can't radiate continuously) and line spectra (transitions between discrete levels). The ground state—the lowest energy eigenstate—is where systems end up at low temperature.