Introduction: Building Any State
Given an arbitrary initial state \(\psi(x, 0)\), we expand it in energy eigenstates to solve the time evolution. This is Fourier analysis applied to quantum mechanics.
The Expansion
\[\psi(x, 0) = \sum_{n=1}^{\infty} c_n \psi_n(x)\]Coefficients found using orthonormality:
\[c_n = \langle\psi_n|\psi\rangle = \int_0^L \psi_n^*(x)\psi(x, 0)\, dx\]Time Evolution
\[\psi(x, t) = \sum_{n=1}^{\infty} c_n \psi_n(x) e^{-iE_n t/\hbar}\]Each component oscillates at its own frequency \(\omega_n = E_n/\hbar\).
Worked Example
Initial state: uniform \(\psi(x, 0) = \sqrt{1/L}\) for \(0 < x < L\).
\[c_n = \sqrt{\frac{2}{L}} \cdot \sqrt{\frac{1}{L}} \int_0^L \sin\left(\frac{n\pi x}{L}\right) dx = \frac{\sqrt{2}}{n\pi}(1 - \cos(n\pi))\]Only odd \(n\) contribute (even \(n\) integrate to zero): \(c_n = \frac{2\sqrt{2}}{n\pi}\) for odd \(n\).
The Quantum Connection
The probability of measuring energy \(E_n\) is \(|c_n|^2\). For the uniform initial state, higher energies are suppressed as \(1/n^2\). The time evolution shows "quantum revivals"—the wavefunction periodically returns to (near) its initial shape due to the discrete spectrum.