Magnetic Moments of Hydrogen Atom in 2T Magnetic Field

[Brewer]

Homework Statement

Consider the Hydrogren atom in a magnetic field of 2T. If the atom is in the ground state (orbital angular momentum L=0)
a) Write down the magnetic moments of the proton and the spinning electron
b) What is the splitting of the energy of the ground state in eV due to the electron?
c) Same question for the proton.

Homework Equations

[tex]\mu = -g\frac{e\hbar}{2m_e}m_j[/tex]
[tex]E = -\mu B[/tex]

The Attempt at a Solution

Well generally I think that its ok - its not that hard to do, except for the g-factor, and the [tex]m_j[/tex] bit of the magnetic moment equation.

I've got that [tex]g = 1 + \frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)}[/tex], but I'm a little confused at how to use this.

I can gleam from my textbook and notes that s=spin and in this case s=1/2, and I think that l=0, but then what is j? I think its something to do with both s and l, but I can't work out what it is.

Following from that what's this [tex]m_j[/tex] component thing. In all the examples in textbooks it just seems to disappear, and I don't follow why.

After this - the whole question is a doddle, so any hints would be welcomed.

Thanks

 

[nrqed]

Homework Statement

Consider the Hydrogren atom in a magnetic field of 2T. If the atom is in the ground state (orbital angular momentum L=0)
a) Write down the magnetic moments of the proton and the spinning electron
b) What is the splitting of the energy of the ground state in eV due to the electron?
c) Same question for the proton.

Homework Equations

[tex]\mu = -g\frac{e\hbar}{2m_e}m_j[/tex]
[tex]E = -\mu B[/tex]

The Attempt at a Solution

Well generally I think that its ok - its not that hard to do, except for the g-factor, and the [tex]m_j[/tex] bit of the magnetic moment equation.

I've got that [tex]g = 1 + \frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)}[/tex], but I'm a little confused at how to use this.

I can gleam from my textbook and notes that s=spin and in this case s=1/2, and I think that l=0, but then what is j? I think its something to do with both s and l, but I can't work out what it is.

"j" is the quantum number associated to the total angular momentum. The value of j ranges from |l-s| to l+s in steps of 1 (here |l-s| means the absolute value of l-s). In your case, l=0 and s=1/2 so j may only take the value j=1/2.

Following from that what's this [tex]m_j[/tex] component thing. In all the examples in textbooks it just seems to disappear, and I don't follow why.

After this - the whole question is a doddle, so any hints would be welcomed.

Thanks

[itex] m_j [/itex] is simply the m quantum number associated to j, so it may range from -j to +j in steps of 1. If j=1/2 as in your example, there are two possible m_j and two possible energies (depending is the angular momentum is "aligned" with the B field or opposite to it).

Patrick