### A.17 The virial theorem

The virial theorem says that the expectation value of the kinetic energy of stationary states is given by

 (A.75)

Now for the potential of a harmonic oscillator, produces . So for energy eigenstates of the harmonic oscillator, the expectation value of kinetic energy equals the one of the potential energy. And since their sum is the total energy , each must be .

For the constant potential of the hydrogen atom, note that according to the calculus rule for directional derivatives, . Therefore produces , So the expectation value of kinetic energy equals minus one half the one of the potential energy. And since their sum is the total energy , and . Note that is negative, so that the kinetic energy is positive as it should be.

To prove the virial theorem, work out the commutator in

using the formulae in chapter 4.5.4,

and then note that the left hand side above is zero for stationary states, (in other words, states with a precise total energy).