Energy difference between energy levels with same n

[Kris2456]

Homework Statement

I am somewhat confused about calculating the energy difference between different energy levels with the same principal quantum number (n). Eg. one of the questions that I have requires me to work out the energy difference between 2³P₂ state and the 2³S₁ state.

Homework Equations

I think this equation from Wikipedia may be relevant:

The Attempt at a Solution

If i interpret Hund's rules correctly, the 2³P₂ state is lower in energy since it has a higher value of L.
My attempt at getting the value of the energy difference between the levels involved using the "fine structure" part of the above equation to work out the difference using 1 & 2 for the values of J.

The value I get is 2.42e^-15 eV which corresponds to a frequency of 5.84 Ghz (the question is about metastable helium being cooled by a laser). I have tried to find correct values for this on the internet but have had little luck :(

 

[Jhero]

Hi there,

Calculating the energy difference between energy levels with the same principal quantum number (n) can be a bit confusing, but it's important to understand in order to fully understand the energy levels of an atom.

To calculate the energy difference between the 23P2 state and the 23S1 state, you can use the Rydberg formula:

E = -Rhc/n^2

where R is the Rydberg constant (which can be found in a reference table), h is Planck's constant, c is the speed of light, and n is the principal quantum number.

In this case, n = 23 for both states. So, plugging in the values, we get:

E(23P2) = -Rhc/23^2 = -Rhc/529
E(23S1) = -Rhc/23^2 = -Rhc/529

To find the energy difference, we can subtract these two values:

E(23P2) - E(23S1) = -Rhc/529 - (-Rhc/529) = 0

This means that the energy difference between the 23P2 state and the 23S1 state is zero. This makes sense because these two states have the same principal quantum number, so they are part of the same energy level.

I hope this helps clear up any confusion and gives you a better understanding of how to calculate energy differences between energy levels. Let me know if you have any further questions. Good luck with your studies!