A particle of mass kg moves on a line in the V-shaped potential

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

A particle of mass

m = 0.50

kg moves on a line in the V-shaped potential

V(x) = F|x|

, with

F = 3.0

N, and has energy

E = 1.5

J. ((a)) Find its action variable

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

as a function of

E

, and evaluate it. ((b)) Invert the result to get

H(I)

, and use

\nu = \partial H/\partial I

to find the angular frequency, first as a function of

E

and then as a number. Check it against the period found directly from the motion under a force of constant magnitude

F

. ((c)) Apply the Bohr-Sommerfeld rule

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

to find how the allowed energies depend on

n

, and evaluate

E_2/E_1

. Compare with the oscillator, where

E\propto n

, and with a particle between hard walls, where

E\propto n^2