Combining Fine Structure Corrections

  • Thread starter logic smogic
  • Start date
  • Tags
    Structure
In summary, we needed to combine the Relativistic Kinetic Energy, Spin-Orbit Interaction, and Darwin fine structure correction terms into a single formula for the energy shift in the Hydrogen atom. This formula must depend only on j = l +/- 1/2, but not l, and must be valid for all l, including l = 0. We were able to do this by breaking down the cases of l=0, l is not zero, and considering two subcases for j=l-1/2 and j=l+1/2. In the end, all three results turned out to be identical, simplifying the formula.
  • #1
logic smogic
56
0

Homework Statement



We are to combine the Relativistic Kinetic Energy, Spin-Orbit Interaction, and Darwin fine structure correction terms into a single formula for the energy shift in the Hydrogen atom. The formula must depend only on j = l +/- 1/2, but not l, and must be valid for all l, including l = 0.

Homework Equations



The above corrections are given as:
https://mywebspace.wisc.edu/dpfahey/web/PF01.bmp

The Attempt at a Solution



Well, [itex] \Delta {E}_{n,total} = \Delta {E}_{n,kin} + \Delta {E}_{n,so} + \Delta {E}_{n,D}[/itex]

Where [itex]<S \cdot L> = \left[j(j + 1) - l(l + 1) - s(s +1) \right ] [/itex]
So, presumably, we just add the given corrections, and collect/eliminate like terms. I began doing this until I became confused by stipulation of dependence on j only, and not l.

So my (simple) question is: If the formula will depend on j, and j depends on l, then how will the formula not depend on l?

Also, how will the resultant formula be good for all l, as one of the correction terms does not allow for l = 0?
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
logic smogic said:

Homework Statement



We are to combine the Relativistic Kinetic Energy, Spin-Orbit Interaction, and Darwin fine structure correction terms into a single formula for the energy shift in the Hydrogen atom. The formula must depend only on j = l +/- 1/2, but not l, and must be valid for all l, including l = 0.

Homework Equations



The above corrections are given as:
https://mywebspace.wisc.edu/dpfahey/web/PF01.bmp

The Attempt at a Solution



Well, [itex] \Delta {E}_{n,total} = \Delta {E}_{n,kin} + \Delta {E}_{n,so} + \Delta {E}_{n,D}[/itex]

Where [itex]<S \cdot L> = \left[j(j + 1) - l(l + 1) - s(s +1) \right ] [/itex]
So, presumably, we just add the given corrections, and collect/eliminate like terms. I began doing this until I became confused by stipulation of dependence on j only, and not l.

So my (simple) question is: If the formula will depend on j, and j depends on l, then how will the formula not depend on l?

Also, how will the resultant formula be good for all l, as one of the correction terms does not allow for l = 0?


You will have to break it down into three cases.

First consider l=0 (in which case, j is obviously l+1/2 =1/2). Add the kinetic and darwin corrections

Now consider l is not zero. Break this up into two subcases. First consider j=l-1/2. So replace all the "l"s by j+1/2 and add the kinetic and spin-orbit.

Now do j=l+1/2, repeat as above.

If I recall correctly, something quite miraculous happens. I think that all three results end up identical. But don't quote me on that.

Patrick
 
Last edited by a moderator:
  • #3
Of course all 3 turn equal, else the formula would be much more complicated.
 
  • #4
Ah, thanks to both of you for the advice. It's great to see it turn out!
 

FAQ: Combining Fine Structure Corrections

What is the purpose of combining fine structure corrections?

The purpose of combining fine structure corrections is to more accurately describe the behavior of atoms and molecules at the quantum level. These corrections take into account the effects of relativity and electron-electron interactions, which are not accounted for in simpler models. By combining these corrections, scientists can better understand and predict the behavior of matter.

How do scientists combine fine structure corrections?

Scientists combine fine structure corrections by using mathematical formulas and computational methods. These corrections are added to the basic equations of quantum mechanics, such as the Schrodinger equation, to account for the effects of relativity and electron-electron interactions.

What are the main challenges in combining fine structure corrections?

One of the main challenges in combining fine structure corrections is the complexity of the calculations involved. These corrections require advanced mathematical techniques and powerful computational tools. Additionally, fine structure corrections can vary depending on the specific atoms or molecules being studied, making it difficult to develop a universal formula.

What are the benefits of combining fine structure corrections?

The benefits of combining fine structure corrections include a more accurate understanding of atomic and molecular behavior, which can lead to advancements in fields such as materials science, chemistry, and physics. These corrections also help to improve the precision of experiments and measurements, allowing for more precise predictions and applications.

What are some examples of real-world applications of combining fine structure corrections?

Some examples of real-world applications of combining fine structure corrections include the development of more efficient solar cells, better understanding of chemical reactions, and improved accuracy of atomic clocks. These corrections are also used in the design of electronic devices and in studying the behavior of atoms in extreme environments, such as in high-energy particle accelerators.

Similar threads

Replies
14
Views
2K
Replies
3
Views
3K
Replies
6
Views
2K
Replies
1
Views
2K
Replies
1
Views
1K
Replies
4
Views
2K
Replies
1
Views
1K

Back
Top