where Fi is the total force acting on the i th particle and ri is the position of the i th particle; the dot stands for an inner product between the two 3-vectors. Indicate long-time averages by angular brackets. The importance of the virial arises from the virial theorem, which connects the long-time average of the virial to the long-time average 〈 T 〉 of the total kinetic energy T of the n-particle system,
Proof of the virial theorem
Consider the quantity G defined by
The vector pi is the momentum of particle i. Differentiate G with respect to time:
Use Newtons's second law and the definition of kinetic energy:
and it follows that
Averaging over time gives:
If the system is stable, G(t) at time t = 0 and at time t = T is finite. Hence, if T goes to infinity, the quantity on the right hand side, being divided by infinite T, goes to zero. Alternatively, if the system is periodic with period T, G(T) = G(0) and the right hand side will also vanish. Whatever the cause, we assume that the time average of the time derivative of G is zero, and hence
which proves the virial theorem.
An interesting application arises when the potential V is of the form
where ai is some constant (independent of space and time).
Then applying this for i = 1, … n,
For instance, for a system of charged particles interacting through a Coulomb interaction:
The virial theorem holds also in quantum mechanics. Quantum mechanically the angular brackets do not indicate a time-average, but an expectation value with respect to an exact stationary eigenstate of the Hamiltonian of the system. First the theorem will be proved and then it is applied to the special case of a potential that has a rk-like dependence. Everywhere Planck's constant ℏ is taken to be one.
Let us consider a n-particle Hamiltonian of the form
where mj is the mass of the j-th particle. The momentum operator is
In order to obtain the virial theorem, we consider
and we find
The quantum mechanical virial theorem follows
where 〈 … 〉 stands for an expectation value with respect to the exact eigenfunction Ψ of H.
If V is of the form
it follows that
For instance, for a stable atom (consisting of charged particles with Coulomb interaction): k = −1, and hence 2〈T 〉 = −〈V 〉.
- R. Clausius, On a Mechanical Theorem applicable to Heat, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 40, 4th series, pp. 122 – 127 (1870). Google books. Note that Clausius still uses the term vis viva for kinetic energy, but does include the factor ½ in its definition, following Coriolis.
- Clausius states this result as: the mean vis viva of the system is equal to its virial.