Apply the Bohr-Sommerfeld condition (with ) to two one-dimensional systems

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Quantum Mechanics · Origins of quantum theory

Action and Symmetry, Ch Hamilton-Jacobi theory and action-angle variables

Problem

Apply the Bohr-Sommerfeld condition

I = nh_{\rm P}/2\pi

(with

n = 0,1,2,\ldots

) to two one-dimensional systems, taking

h_{\rm P} = 6.626\times10^{-34}

J s. ((a)) An oscillator of angular frequency

\omega = 3.0\times10^{15}\,\mathrm{rad\,s^{-1}}

has action variable

I=E/\omega

. Find the Bohr-Sommerfeld energy of its

n=2

level, in eV, and compare it with the exact quantum-mechanical level

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

for the same

n

. ((b)) An electron (

m_{\rm e}=9.109\times10^{-31}

kg) is confined between infinite walls a distance

L=0.50

nm apart (

V=0

inside,

V=\infty

outside). Its action variable is

I=\dfrac{1}{2\pi}\oint p\,\mathrm{d}x = \dfrac{L}{\pi}\sqrt{2mE}

. Find the Bohr-Sommerfeld energy of its

n=2

level, in eV, and compare it with the exact quantum-mechanical level

E_n=n^2h_{\rm P}^2/8mL^2

. ((c)) State which case the Bohr-Sommerfeld rule reproduces exactly, and by how much, as a multiple of

\hbar\omega

, it misses the other at every

n