Introduction to Quantum Mechanics · Problem 3.37

Formalism — David J. Griffiths

Physics — Quantum Mechanics · virial_theorem

Problem

Virial theorem. Use Equation 3.73 to show that

\frac{d}{dt}\langle xp \rangle = 2\langle T \rangle - \left\langle x \frac{\partial V}{\partial x} \right\rangle, \tag{3.112}

where

T

is the kinetic energy (

H = T + V

). In a stationary state the left side is zero (why?) so

2\langle T \rangle = \left\langle x \frac{dV}{dx} \right\rangle. \tag{3.113}

This is called the virial theorem. Use it to prove that

\langle T \rangle = \langle V \rangle

for stationary states of the harmonic oscillator, and check that this is consistent with the results you got in Problems 2.11 and 2.12.