Electron spin - energy difference between split levels?

[accountkiller]

I found the interaction energy U of an electron in an atom with orbital quantum number l=0 with a uniform 2.00-T magnetic field to be 1.159E-4 eV, now it asks what is the energy difference between the two split states?

We have short 3-point quizzes each day before lecture just to make sure that we read the section in the book because we will start talking about this new material in class. However, I don't see anything with energy difference information in this section in the book or in the MasteringPhysics side note.

The only thing close to is that there is an example for a 3p level of a sodium atom where you find energy E = hc/lambda of the two photonsbut then would my lambda be? I don't think that helps either.

I'd appreciate any help, thanks!

 

[fzero]

Was the interaction energy you found due to the electron spin? Does the spin up state have the same energy as the spin down state?

 

[zhermes]

Yeah... MasteringPhysics is amazingly bad.

The split states:
When you have an electron in an atom, it has some energy E_0 (lets say). In a magnetic field, because of the intrinsic spins of the electrons, they can either gain or lose the interaction energy with the magnetic field (in your case \Delta E \approx 1E-4), thus the final split energies will be E = E_0 - \Delta E and E = E_0 + \Delta E, what's the difference in energies between those two split states?

 

[accountkiller]

Ah, thank you for those equations. But what is my E₀? Using deltaE = 1.159*10^4, I just picked a random number for E₀ (I used 5) then calculated 5 - deltaE and 5 + deltaE and took the difference, and I got the correct answer, 2.318*10^-4. I take it this is probably no the scientific way to do it. What is?

 

[zhermes]

Ah, thank you for those equations. But what is my E₀? Using deltaE = 1.159*10^4, I just picked a random number for E₀ (I used 5) then calculated 5 - deltaE and 5 + deltaE and took the difference, and I got the correct answer, 2.318*10^-4. I take it this is probably no the scientific way to do it. What is?

Great question. Especially in physics, its best to set up your equations before plugging in values. This way you can see what aspects of the equations 'drop out' or cancel. Using the equations I gave you, try solving for the difference in energy levels (i.e. before plugging values in). See what you find