Franz Schwabl Quantum Mechanics 4th Edit · Problem 3.3

The Uncertainty Relation

Physics — Quantum Mechanics · classical-limit-harmonic-oscillator

Problem

The classical limit of the harmonic oscillator (see end of Sect. 3.1). (a) Show using the Stirling formula

n! \approx \sqrt{2\pi n}\, n^n \mathrm{e}^{-n} \quad \text{for } n \gg 1

that the absolute value of the amplitude

\alpha^n / \sqrt{n!}

is maximal for

n_0 = |\alpha|^2 \ .

(b) Show that the expectation value

\langle \hat{n} \rangle

of the operator

\hat{n}

in the state

\varphi_\alpha

is equal to

n_0

. (c) Calculate

\Delta n / \langle \hat{n} \rangle

. (d) Show that in this limit the energy expectation value is

(\varphi_\alpha, H \varphi_\alpha) = \hbar\omega |\alpha|^2 \ .

Compare this answer with the classical result.