Introduction: The Analytical Method
The harmonic oscillator equation can be solved by power series expansion. The requirement that the series terminates (for normalizability) leads to energy quantization.
Asymptotic Behavior
For large \(|\xi|\): \(\psi'' \approx \xi^2\psi\) → \(\psi \sim e^{\pm\xi^2/2}\)
For normalizability, choose \(\psi \sim e^{-\xi^2/2}\) at infinity.
Factor out: \(\psi(\xi) = h(\xi)e^{-\xi^2/2}\)
Equation for h(ξ)
Substituting gives:
\[h'' - 2\xi h' + (\epsilon - 1)h = 0\]This is Hermite's equation when \(\epsilon - 1 = 2n\).
Energy Quantization
The power series for \(h(\xi)\) must terminate to avoid divergence. This requires:
\[\epsilon = 2n + 1, \quad n = 0, 1, 2, \ldots\] \[E_n = \hbar\omega\left(n + \frac{1}{2}\right)\]The Quantum Connection
The energy levels are equally spaced by \(\hbar\omega\). The ground state energy \(E_0 = \frac{1}{2}\hbar\omega\) is non-zero—the particle can never be perfectly at rest. This zero-point energy has measurable consequences: the Casimir effect, helium remaining liquid at T = 0, and quantum fluctuations in fields.