Solving Spin-Orbit Coupling in Hydrogen & Li+2

[hhhmortal]

Homework Statement

One of the n=5 states of hydrogen is split by spin-orbit coupling into two levels with an energy difference of 0.0039 cm^-1 . Determine the 'l' quantum number for this state and predict the analogous splitting for doubly ionised Li .

Homework Equations

The fine structure constant is 0.007 297 3.

The Attempt at a Solution

Ok I did the first part and I got l=3 basically I use the spin orbit coupling energy equation and set j= l + 1/2 and j= l-1/2 find the difference between both of them and then work out the value of 'l' knowing that the wave number is 0.0039.

What I don't understand is the second part. Do I use n=5 and l=3 and do the same thing?

Thanks.

 

[vela]

Yup, except this time you're talking about an electron in lithium instead of hydrogen.

 

[hhhmortal]

Yup, except this time you're talking about an electron in lithium instead of hydrogen.

So I first need to find the reduced electron mass of doubly ionized lithium. Then using this I can find the Rydberg constant for Lithium, and then I can actually get the difference in wave number?

I got a value of 0.3133 cm^-1 ..which seemed much bigger than that for Hydrogen.

 

[vela]

Did you take into account the different charge of the nucleus?

 

[hhhmortal]

Did you take into account the different charge of the nucleus?

Yes, I took this into account when calculating the reduced electron mass of Lithium and also when calculating the difference in wave number afterwards. Is this value remotely incorrect?

 

[vela]

I don't understand what you mean by that. The charge of the nucleus doesn't have anything to do with calculating the reduced mass of the electron. Are you referring to the mass of the nucleus? I'm talking about the electric charge.

 

[hhhmortal]

Oh ok! I think I know where I've gone wrong. When calculating the reduced electron mass I used:

5(m_e).7(m_n)/[5(m_e) + 7(m_n)]

But I don't think this is so..