Introduction: The Eigenvalue Problem
The TISE is an eigenvalue equation for the Hamiltonian. Solving it gives the allowed energies and corresponding wavefunctions—the building blocks for all quantum dynamics.
The Equation
\[\hat{H}\psi = E\psi\]In position representation for one particle:
\[-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi\]Structure of Solutions
- Bound states: \(E < V(\pm\infty)\), discrete spectrum, normalizable
- Scattering states: \(E > V(\pm\infty)\), continuous spectrum, not normalizable
- Classically forbidden regions: \(E < V(x)\), \(\psi\) decays exponentially
Worked Example: Strategy
To solve the TISE:
- Identify classically allowed (\(E > V\)) and forbidden (\(E < V\)) regions
- Write general solution in each region
- Apply boundary conditions (continuity of \(\psi\) and \(\psi'\))
- Apply normalization or scattering boundary conditions
- Find allowed values of \(E\)
The Quantum Connection
The TISE determines the energy spectrum—the "fingerprint" of a quantum system. Atomic spectra, molecular bonds, and semiconductor band gaps all come from solving the TISE with appropriate potentials. We'll work through the classic examples in coming lessons.