Introduction: Why Only Certain Energies?
Energy quantization isn't mysterious—it's a consequence of boundary conditions. Just as a guitar string has fixed frequencies, a particle in a box has fixed energies. The key is the requirement that wavefunctions "fit" in the box.
The Node Theorem
For one-dimensional bound states in any potential:
- Ground state (\(n = 1\)): 0 nodes
- \(n\)-th excited state: \(n - 1\) nodes
- Higher energy → more oscillations → more nodes
Physical Intuition
More nodes = shorter wavelength = higher momentum = higher kinetic energy
The de Broglie relation \(p = h/\lambda\) connects nodes to energy.
Worked Example: Counting Nodes
For infinite well \(\psi_n = \sin(n\pi x/L)\):
- \(\psi_1\): Half sine wave, 0 interior nodes
- \(\psi_2\): Full sine wave, 1 node at \(x = L/2\)
- \(\psi_3\): 1.5 sine waves, 2 nodes at \(x = L/3, 2L/3\)
The Quantum Connection
The node theorem is universal: it applies to any 1D bound state problem. You can estimate energy ordering just by counting nodes. In atomic orbitals, nodal structure determines chemical bonding—electrons in higher energy orbitals have more nodes and are less tightly bound.