Expansion of Hydrogen GS (Relativistic)

In summary, the person is struggling to verify equations 16 and 17 on a paper and is not getting the same values as shown. They have tried using Mathematica and noticed a missing factor of 1/2 in front of equation 16. They have also checked their output using WolframAlpha but are still not getting the correct values. They are unsure how the author of the paper obtained their values and are seeking help to understand where they went wrong.
  • #1
sean_mp
20
0

Homework Statement


I'm trying to verify equations 16 and 17 on the attached paper, but I'm just not getting the same values they are. I've used mathematica and it just isn't happening. For one thing, there should be a factor of 1/2 in front of equation 16, but I'm really not seeing how they got this from a typical series expansion.

Homework Equations


I need help getting the same values as 16 and 17 in the paper. I'm just not getting it in my output.


The Attempt at a Solution

 

Attachments

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  • #2
I don't have a lot to add, other than I plugged (14) into WolframAlpha and reproduced your factor of 1/2 (http://www.wolframalpha.com/input/?i=series+sqrt%281-+g%2F%28sqrt%28k^2%2Bg%29%2Bn-k%29^2%29+in+g+to+order+2). I have no idea why he gets something else.
 
  • #3
Wow... I'm embarrassed that I didn't get this myself... I must have had a typo in my equation or something.

Thanks a lot, you're a lifesaver.
 

FAQ: Expansion of Hydrogen GS (Relativistic)

What is the Expansion of Hydrogen GS (Relativistic)?

The Expansion of Hydrogen GS (Relativistic) refers to the phenomenon where the energy levels of hydrogen atoms are shifted due to the effects of relativity, specifically the increase in mass and decrease in size of the electron when moving at high speeds.

How does Relativity affect the energy levels of hydrogen atoms?

The theory of relativity predicts that as an electron moves at high speeds, its mass and size will increase, leading to a decrease in energy levels. This is known as the relativistic effect, which is responsible for the expansion of hydrogen GS.

What is the significance of the Expansion of Hydrogen GS (Relativistic)?

Understanding the Expansion of Hydrogen GS (Relativistic) is important in accurately predicting and measuring the energy levels of hydrogen atoms. It also has implications in fields such as quantum mechanics and atomic physics.

How is the Expansion of Hydrogen GS (Relativistic) observed?

The Expansion of Hydrogen GS (Relativistic) can be observed through spectroscopy, where the energy levels of hydrogen atoms are measured and compared to theoretical predictions. The observed differences are attributed to the relativistic effects.

Can the Expansion of Hydrogen GS (Relativistic) be applied to other atoms?

Yes, the relativistic effect can also be observed in other atoms, but it is most significant in hydrogen due to its simple structure and lone electron. The effect becomes less pronounced as the atomic number increases.

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