Lesson 298: Boundary Conditions for the Wavefunction

Introduction: Matching at Interfaces

At boundaries where the potential changes (or at physical walls), the wavefunction must satisfy boundary conditions. These conditions are what quantize energy levels.

Standard Conditions

1. \(\psi\) is continuous: No jumps in probability amplitude
2. \(\psi'\) is continuous: (for finite \(V\)) Ensures finite kinetic energy
3. \(\psi \to 0\) as \(|x| \to \infty\): For bound states (normalizability)

Special Cases

• Infinite wall: \(\psi = 0\) at the wall (can't be inside infinite potential)
• Delta function: \(\psi\) continuous, but \(\psi'\) has a discontinuity
• Periodic boundary: \(\psi(x + L) = \psi(x)\) (for crystals)

Worked Example: Why Energy is Quantized

In infinite square well from 0 to \(L\):

• Boundary conditions: \(\psi(0) = 0\) and \(\psi(L) = 0\)
• General solution in the well: \(\psi = A\sin(kx) + B\cos(kx)\)
• \(\psi(0) = 0 \Rightarrow B = 0\)
• \(\psi(L) = 0 \Rightarrow A\sin(kL) = 0 \Rightarrow kL = n\pi\)
• Only specific \(k\) values allowed → energy quantized!

The Quantum Connection

Boundary conditions are the origin of quantization. The requirement that wavefunctions be well-behaved (continuous, normalizable) restricts the allowed solutions to a discrete set. Different boundary conditions give different physics: hard walls, soft walls, periodic structures, etc.