In the harmonic oscillator, Chapter 7's ladder operators give in level

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Observables, commutators and the uncertainty relations

Problem

In the harmonic oscillator, Chapter 7's ladder operators give

\Delta x\,\Delta p = (n + \tfrac12)\hbar

in level

n

. Using the bound

\Delta x\,\Delta p \ge \hbar/2

proved in this chapter, find the smallest level

n

for which the product first exceeds twenty times that minimum,

\Delta x\,\Delta p > 10\hbar