Introduction: Building General States
Energy eigenstates form a complete basis. Any state can be written as a superposition of them, making the general time evolution automatic once we know the energy spectrum.
The General Solution
Given initial condition \(\psi(x, 0)\):
\[\psi(x, 0) = \sum_n c_n \phi_n(x)\]The time evolution is:
\[\psi(x, t) = \sum_n c_n \phi_n(x) e^{-iE_n t/\hbar}\]Finding Coefficients
Using orthonormality of eigenstates:
\[c_n = \langle\phi_n|\psi(0)\rangle = \int_{-\infty}^{\infty} \phi_n^*(x)\psi(x, 0)\, dx\]Worked Example
Initial state in infinite well: \(\psi(x, 0) = Ax(L-x)\) for \(0 < x < L\).
Expand in eigenstates \(\phi_n = \sqrt{2/L}\sin(n\pi x/L)\):
\[c_n = \sqrt{\frac{2}{L}}\int_0^L x(L-x)\sin\left(\frac{n\pi x}{L}\right)dx\]The integral gives coefficients for odd \(n\) only (by symmetry). The state evolves as:
\[\psi(x,t) = \sum_{\text{odd } n} c_n \phi_n(x) e^{-iE_n t/\hbar}\]The Quantum Connection
This is the power of linear algebra: once you solve the TISE (the hard part), time evolution is just multiplication by phase factors. The different frequencies \(E_n/\hbar\) cause "quantum beats"—the probability distribution oscillates and breathes. This is how wave packets move and spread.