Showing that the Virial Theorem holds

In summary, the expectation value for the interaction potential for hydrogenic atoms is -(uZ^2e^4)/((hbar^2)*n^2).
  • #1
Ed Quanta
297
0
So I have already calculated correctly that the expectation of the interaction potential for hydrogenic atoms is

<nlm|V(r)|nlm>=-(uZ^2e^4)/((hbar^2)*n^2)

Note that u= mass,and V(r)=-Ze^2/r

I now have to calculate <nlm|T|nlm> where T is the kinetic energy operator, and

T=p^2/(2u) + L^2/(2ur^2)

Note that p is the radial momentum operator and L is the angular momentum operator

I know hbar^2l(l+1)=L^2 and I know (pretty sure) that <1/r^2>=e^2/(2n^3hbar^2).

However, I am unsure how to find the expectation value of the p^2/(2u) term, and don't see how <T> is going to equal <V> which the book hints at being true since <T> and <V> are said to satisfy the Virial Theorem.

Help anyone?
 
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  • #2
Ed Quanta said:
However, I am unsure how to find the expectation value of the p^2/(2u) term,

You can do it by brute force (that's what I call working in function space). Insert the differential operator for p into the expression for T, sandwich it between &psi;n'l'm'* and &psi;nlm, and integrate.
 
  • #3
There's a piece of genius on my behalf:

[tex] \hat{H}=\hat{T}+\hat{V} [/tex] (1)

[tex] \langle nlm|\hat{H}|nlm\rangle = E_{n} =-\frac{Z^{2}\mu e^{4}}{2\hbar^{2}(4\pi\epsilon_{0})} \frac{1}{n^{2}} [/tex] (2)

[tex] \langle nlm|\hat{V}|nlm\rangle =-\frac{Ze^{2}}{(4\pi\epsilon_{0})}\langle \frac{1}{r}\rangle _{|nlm\rangle} [/tex] (3)

Compute (3) using the average of 1/r.

Then:

[tex] \langle nlm|\hat{T}|nlm\rangle =\langle nlm|\hat{H}-\hat{V}|nlm\rangle=E_{n}+\frac{Ze^{2}}{(4\pi\epsilon_{0})}\langle \frac{1}{r}\rangle _{|nlm\rangle} [/tex] (4)

Tell where u get stuck.

Daniel.
 
Last edited:
  • #4
My book seems to leave out the 4pi*epsilon term in the denominator for some reason. But yeah thanks, I am cool now. I was missing the expectation value for the Hamiltonian. Thanks for your genius my man.
 
  • #5
Ed Quanta said:
My book seems to leave out the 4pi*epsilon term in the denominator for some reason. But yeah thanks, I am cool now. I was missing the expectation value for the Hamiltonian. Thanks for your genius my man.

It must use the cgs unit system.. in cgs a factor of [tex] ( 4 \pi \epsilon_0 )^{\frac{1}2} [/tex] is "absorbed" into the unit of charge, so that Coulomb's law can be written as [tex] \vec{F} = \frac{e^2}{r^2}\hat{r} [/tex]
 

1. What is the Virial Theorem?

The Virial Theorem is a mathematical relationship that describes the equilibrium state of a system of particles. It states that the average kinetic energy of a system is equal to the negative of its average potential energy.

2. How is the Virial Theorem derived?

The Virial Theorem is derived using statistical mechanics and the equations of motion for a system of particles. It involves taking the time average of the kinetic energy and potential energy terms and equating them to each other.

3. What evidence do we have that the Virial Theorem holds?

There is strong observational evidence that the Virial Theorem holds in many physical systems, such as stars, galaxies, and clusters of galaxies. This is supported by the fact that these systems are observed to be in equilibrium, with a balance between their kinetic and potential energies.

4. How is the Virial Theorem used in astronomy and astrophysics?

The Virial Theorem is used to study the properties of large-scale structures in the universe, such as galaxies and galaxy clusters. It can also be used to estimate the masses of these structures and to understand their evolutionary processes.

5. Are there any exceptions to the Virial Theorem?

While the Virial Theorem is a powerful tool in understanding the equilibrium states of physical systems, there are some exceptions to its applicability. These include systems with strong magnetic fields, systems with non-Newtonian dynamics, and systems with strong interactions between particles.

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