A.17 The virial theorem

The virial theorem relates the expectation kinetic energy of a quantum system to the potential. That is of theoretical interest, as well as important for computational methods like “density functional theory.”

Consider a quantum system in a state of definite energy

To keep it simple, for now assume that there is a single particle with
position vector

Then the virial theorem relates the expectation kinetic energy

(A.75) |

For example, consider the harmonic oscillator. There

so in Cartesian coordinates

Then according to the virial theorem

Also consider the hydrogen atom. There

so in polar coordinates

Then according to the virial theorem the expectation potential energy is minus twice the expectation kinetic energy. And their sum, the total energy

The virial theorem does not apply to the particle in a pipe, as that
particle is in a bounded space. (You can assume infinite space if
you take the potential infinite outside the pipe, but obviously by
itself that does not help much. You could assume infinite space
with a potential

if you then take the limit

But the virial theorem does apply to any number of particles, not just
to one. Just sum over all the particles:

where i is the particle number.

For example, consider the hydrogen molecule, where there are four
particles, two protons and two electrons. Here

where

so the total expectation potential energy of the system is still twice the total energy

In some computations you might need to assume that the electrons are
in a state of definite energy, like in the ground state, but the
nuclei are not. In such computations the nuclei are at an assumed
position and you will only compute the state of the electrons. So the
summation in

now extends only over the electrons. But this summation does includes potentials of the electrons due to the attraction by the nuclei, and those terms are no longer equal to minus the corresponding potentials. You may need to evaluate these terms explicitly. But that is not too bad, as these potentials are now known functions of the individual electron positions only. The difficult term, due to the electron-electron interaction, is still given by minus the corresponding potential.

Finally, you might wonder where the virial theorem comes from. Well,
one way to prove the virial theorem, as found in quantum textbooks and
on Wikipedia, is to work out the commutator in

using the formulae in chapter 4.5.4, to give

and then note that the left hand side above is zero for stationary states, (in other words, for states with a precise total energy). This follows the classical way of deriving the classical virial theorem, but requires a messy purely mathematical derivation. The theorem then pops up out of the complex mathematics without any plausible physical reason why there would be such a theorem in the first place.

The original derivation by Fock in 1930 is much more physically
appealing and more instructive. The idea is to slightly stretch the
given quantum system: replace every position coordinate coordinate

First however, recall that the square magnitude of the wave function
gives the probability of that state, and that all probabilities must
integrate together to 1, certainty. Phrased differently, the
expectation value of one must be one;

Next, the expectation kinetic energy consists of terms like

For the potential energy we can use a linear Taylor series to figure
out how it changes:

where in the right hand side

From the above expressions, it is seen that compared to the
unstretched system, in the stretched system the sum of expectation
kinetic and potential energies is different by an amount

But, as described in {A.7}, if you mess up an energy wave function by an amount of order