Revision as of 09:16, 15 February 2010 by imported>Paul Wormer
In mechanics, a virial of a stable system of n particles is a quantity proposed by Rudolf Clausius in 1870. The virial is defined by

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:
![{\displaystyle {\frac {dG}{dt}}=\sum _{i=1}^{n}\left[{\frac {d\mathbf {r} _{i}}{dt}}\cdot \mathbf {p} _{i}+\mathbf {r} _{i}\cdot {\frac {d\mathbf {p} _{i}}{dt}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0507e3de450ad74d28e52655ef118e8036ae0c28)
Use Newtons's second law and the definition of kinetic energy:

and it follows that

Averaging over time gives:
![{\displaystyle \left\langle {\frac {dG}{dt}}\right\rangle \equiv {\frac {1}{T}}\int _{0}^{T}{\frac {dG}{dt}}dt={\frac {1}{T}}\left[G(T)-G(0)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1557f88d7e99f2ff2e4e8d38f5dfa015b34a9c)
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 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.
Application
An interesting application arises when each particle experiences a potential V of the form

where A is some constant (independent of space and time).
An example of such potential is given by Hooke's law with k = 2 and Coulomb's law with k = −1.
The force derived from a potential is

Consider

Then applying this for i = 1, … n,

For instance, for a system of charged particles interacting through a Coulomb interaction:

Quantum mechanics
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. The theorem will be proved and 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

Using the self-adjointness of H and the definition of a commutator one has for an arbitrary operator G,
![{\displaystyle 0=\langle \Psi |[G,H]|\Psi \rangle \quad {\hbox{with}}\quad H|\Psi \rangle =E|\Psi \rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56f4b13d045b7ea141546ab1196c0b3d7a567ef1)
In order to obtain the virial theorem, we consider

Use
![{\displaystyle [\mathbf {r} _{k}\cdot \mathbf {p} _{k},H]=[\mathbf {r} _{k},T]\cdot \mathbf {p} _{k}+\mathbf {r} _{k}\cdot [\mathbf {p} _{k},V]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91461abed3221bbe78fa28a3a87b4b62274ee85e)
Define
![{\displaystyle \mathbf {F} _{k}\equiv -i[\mathbf {p} _{k},V]=-[{\boldsymbol {\nabla }}_{k},V].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e287a143cbf259ba321222f19da27ae8c06d4ec)
Use
![{\displaystyle [r_{k\alpha },p_{j\beta }^{2}]=\delta _{kj}\delta _{\alpha \beta }2ip_{k\alpha },\quad \alpha ,\beta =x,y,z;\quad k,j=1,,2,\ldots ,n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c16a3f3e08c1f719475c5f78a7fb0bb9010652a)
and we find
![{\displaystyle [G,H]=i{\big (}2T+\sum _{j=1}^{n}\mathbf {r} _{j}\cdot \mathbf {F} _{j}{\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c723376beae04e51e917dd2ccc2bc06db10d8346)
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
![{\displaystyle \mathbf {F} _{j}=-[{\boldsymbol {\nabla }}_{j},V]=-a_{j}\,k\mathbf {r} _{j}\,(r_{j})^{k-2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8eb3032df63034b1697afc8e1dea13b33d2ca24)
From this:

For instance, for a stable atom (consisting of charged particles with Coulomb interaction): k = −1, and hence 2⟨T ⟩ = −⟨V ⟩.