Chapter 5 found that an infinite well's levels grow as

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The harmonic oscillator

Problem

E_n = n^2h^2/8ma^2

, so

E_n\propto n^2

. This chapter finds the oscillator's levels grow as

E_n=(n+\tfrac12)\hbar\omega

, so

E_n\propto n

for large

n

. For each potential, compute the relative spacing

(E_{n+1}-E_n)/E_n

n=10

and at

n=1000

, and explain why the correspondence principle, that levels appear continuous at large

n

, holds for both despite their different growth laws.