imported>Paul Wormer |
imported>Paul Wormer |
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| The quantum mechanical virial theorem follows | | The quantum mechanical virial theorem follows |
| :<math> | | :<math> |
| 2\langle T\rangle = -\sum_{j=1}^n \langle \mathbf{r}_j \cdot\mathbf{F}_j \rangle
| | \langle T\rangle = -\tfrac{1}{2} \sum_{j=1}^n \langle \mathbf{r}_j \cdot\mathbf{F}_j \rangle |
| </math> | | </math> |
| where ⟨ … ⟩ stands for an expectation value with respect to the exact eigenfunction Ψ of ''H''. | | where ⟨ … ⟩ stands for an expectation value with respect to the exact eigenfunction Ψ of ''H''. |
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| For instance, for a stable atom (consisting of charged particles with Coulomb interaction): ''k'' = −1, and hence 2⟨''T'' ⟩ = −⟨''V'' ⟩. | | For instance, for a stable atom (consisting of charged particles with Coulomb interaction): ''k'' = −1, and hence 2⟨''T'' ⟩ = −⟨''V'' ⟩. |
| | |
| ==Reference== | | ==Reference== |
| <references /> | | <references /> |
Revision as of 16:22, 15 February 2010
In mechanics, a virial of a stable system of n particles is a quantity proposed by Rudolf Clausius in 1870.[1]
The virial (from the Latin vis, force) 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,[2]
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 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 the potential V is of the form
where ai 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
Hence
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,
In order to obtain the virial theorem, we consider
Use
Define
Use
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
From this:
For instance, for a stable atom (consisting of charged particles with Coulomb interaction): k = −1, and hence 2⟨T ⟩ = −⟨V ⟩.
Reference
- ↑ 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.