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