Principles of Quantum Mechanics · Problem 7.4.3

The Path Integral Formulation of Quantum Theory — R. Shankar

Physics — Quantum Mechanics · virial-theorem

Problem

*(Virial Theorem). The virial theorem in classical mechanics states that for a particle bound by a potential

V(r) = ar^k

, the average (over the orbit) kinetic and potential energies are related by

\bar{T} = c(k)\bar{V}

when

c(k)

depends only on

k

. Show that

c(k) = k/2

by considering a circular orbit. Using results from the previous exercise show that for the oscillator (

k=2

)

\langle T\rangle = \langle V\rangle

in the quantum state

|n\rangle