Exploring the Darwin Term in Hydrogen Fine Structure

In summary, the conversation involves trying to understand the Darwin term in the fine structure of hydrogen. The perturbation from the Darwin term is given by an expression involving the momentum operator and the distance between particles. Another alternative form of the Darwin term is also mentioned. The conversation ends with a suggestion to use a well-known operator identity to simplify the expression.
  • #1
dipole
555
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I'm trying to do a HW involving the Darwin Term in the fine structure of hydrogen. I'm given that the perturbation from the Darwin term is equal to (times a constant factor which I'll ignore),

[tex] V_D = [p_i,[p_i, \frac{e^2}{r}]] =
e^2\vec{p}^2\frac{1}{r} -2e^2\vec{p}\frac{1}{r}\vec{p} +
e^2\frac{1}{r}\vec{p}^2 [/tex]

I know that the alterntive form of the Darwin term is,

[tex] V_D = 4\pi\delta(r) [/tex]

This comes from the first term in the first expression when you project the momentum operater onto real space, but there are two other terms which I can't seem to get to cancel... can anyone explain how these terms cancel? If they don't, then I have to do an intergal involving the laplacian of the wave function, and an integral involving the first derivite wrt to r of a wave function, for aribitrary n,l,m... and that is going to be a nightmare!
 
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  • #2
I think you've left out some terms. When simplifying operator expressions like this, I find it helpful to let the expressions act on an arbitrary function ##\psi##. So

[itex] V_D \psi = [p_i,[p_i, \frac{e^2}{r}]] \psi [/itex]

When expanding the commutators and then simplifying; make sure that you let the p operators act on everything to their right using the product rule.

EDIT: You can also just use the well-known operator identity: ##[p_x, f(x)] = -i\hbar \frac{df(x)}{dx}##
 
Last edited:
  • #3
TSny said:
I think you've left out some terms. When simplifying operator expressions like this, I find it helpful to let the expressions act on an arbitrary function ##\psi##. So

[itex] V_D \psi = [p_i,[p_i, \frac{e^2}{r}]] \psi [/itex]

When expanding the commutators and then simplifying; make sure that you let the p operators act on everything to their right using the product rule.

EDIT: You can also just use the well-known operator identity: ##[p_x, f(x)] = -i\hbar \frac{df(x)}{dx}##

Doh! I totally forgot about that identity! Yes I do think maybe when I operate on some wave function i should get extra terms, but it's still messy to me, since I'm trying to do everything in spherical coordinates. :(

I'll use the identity to finish the problem, then go back and try to work out how things should cancel. Thanks a bunch!
 

1. What is the Darwin term in hydrogen fine structure?

The Darwin term is a relativistic correction to the energy levels of a hydrogen atom. It takes into account the electron's motion around the nucleus at high speeds, as described by Einstein's theory of relativity.

2. Why is exploring the Darwin term important?

Exploring the Darwin term allows scientists to better understand the behavior of electrons in hydrogen atoms and their interactions with the nucleus. This can provide valuable insights into the fundamental principles of quantum mechanics and help us develop more accurate models for atomic structures.

3. How is the Darwin term calculated?

The Darwin term is calculated using the Dirac equation, which combines the principles of special relativity and quantum mechanics. It involves solving complex mathematical equations to determine the relativistic correction to the energy levels of a hydrogen atom.

4. What are the implications of the Darwin term in hydrogen fine structure?

The Darwin term has significant implications for the energy levels of hydrogen atoms, particularly in high-energy environments such as stars and other celestial bodies. It also affects the accuracy of spectroscopic measurements and plays a crucial role in understanding the fine structure of atomic spectra.

5. What are some current research areas related to exploring the Darwin term?

Scientists are currently exploring the effects of the Darwin term in other atoms and molecules, as well as investigating its role in quantum computing and other advanced technologies. There is also ongoing research on refining the calculations and measurements of the Darwin term in order to improve our understanding of atomic structure and behavior.

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