Energy of different energy levels of hydrogen atom

[leroyjenkens]

Homework Statement

A hydrogen atom is in 2p state, how many different energy levels are there if a magnetic field of 0.10 T is applied to it, and what are their energies? Make sure spin is included.

Homework Equations

U = -gm_sμ_zB

g is the Lande-g factor, which is about -2 for electrons
m_s is spin, which is +/- 1/2
μ_z is -μ_bm, which is the Bohr magneton times mass.
Bohr magneton is equal to about 5.78x10^-5
The mass is, I assume, the mass of the electron, which is about 9.109x10^-31

And the initial energy of the ground state is E₀ is -13.6/n^2

n is 2.

The Attempt at a Solution

I get 6 energy levels, because m_l has -1,0,1 and each of those has a spin of +/- 1/2.

So what I did was just make the equation E₀ + U, which gives me the initial energy, plus the energy that U provides. But I'm getting extremely small numbers for U. For example, for m_l = 1, and the +1/2 spin, I'm getting, for U, 5.27x10^-36. Added to E₀, that's miniscule. Plus I'm always going to have a negative answer, since E₀ is negative.

So I don't know why it's negative, and I'm not sure if my answers are correct, because they seem extremely small.

Thanks.

 

[DrClaude]

I get 6 energy levels, because m_l has -1,0,1 and each of those has a spin of +/- 1/2.

Because of spin-orbit coupling, $m_l$ and $m_s$ are not good quantum numbers. You need to look at the total angular momentum of the electron $j$ and its projection $m_j$.

 

[leroyjenkens]

Because of spin-orbit coupling, $m_l$ and $m_s$ are not good quantum numbers. You need to look at the total angular momentum of the electron $j$ and its projection $m_j$.

In what way are they not good quantum numbers?

For total angular momentum of the electron, do I use \frac{\sqrt{3}}{2}\hbar

Thanks.