Introduction: A Powerful Technique
For time-independent Hamiltonians, we can separate the TDSE into a spatial part (the time-independent Schrödinger equation) and a temporal part. This reduces a PDE to an ODE—much easier to solve.
The Method
Assume \(\psi(x,t) = \phi(x)T(t)\). Substitute into TDSE:
\[i\hbar\phi\frac{dT}{dt} = T\hat{H}\phi\]Divide by \(\phi T\):
\[\frac{i\hbar}{T}\frac{dT}{dt} = \frac{\hat{H}\phi}{\phi} = E\]Each side depends on different variables → both equal constant \(E\).
The Two Equations
Time equation: \(\frac{dT}{dt} = -\frac{iE}{\hbar}T\) → \(T(t) = e^{-iEt/\hbar}\)
Space equation: \(\hat{H}\phi = E\phi\) (Time-Independent Schrödinger Equation)
Worked Example
For a particle in a box, we find \(\phi_n(x)\) and \(E_n\) from the TISE.
The separable solutions are:
\[\psi_n(x,t) = \phi_n(x)e^{-iE_nt/\hbar}\]General solution: any superposition \(\psi = \sum_n c_n\phi_n e^{-iE_nt/\hbar}\).
The Quantum Connection
Separation of variables is why we focus on the TISE (time-independent equation). Once we find energy eigenstates, time evolution is trivial: each component rotates at its own frequency. The hard work is finding the spatial solutions \(\phi_n(x)\).