The virial theorem says that the expectation value of the kinetic
energy of stationary states is given by
(A.75) |
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