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unscientific

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## Homework Statement

Part (a): What's the origin of that expression?

Part(b): Estimate magnetic field, give quantum numbers to specify 2p and general nl-configuration

Part (c): What is the Zeeman effect on states 1s and 2s?

## Homework Equations

## The Attempt at a Solution

__Part (b)__[tex]H = -\frac{e}{m^2c^2} \frac{1}{r} \frac{\partial \phi}{\partial r} \vec S . \vec L[/tex]

[tex]-\mu . B = -\frac{e}{m^2c^2} \frac{1}{r} \frac{\partial \phi}{\partial r} \vec S . \vec L[/tex]

[tex]\frac{e}{2m}\vec S . \vec B = -\frac{e}{m^2c^2} \frac{1}{r} \frac{\partial}{\partial r} \vec S . \vec L[/tex]

[tex]|\vec B| = |\vec L| \frac{e}{m^2 c^2} \frac{m}{e} \frac{1}{r} \frac{\partial \phi}{\partial r}[/tex]

Now, ##|L| = l\hbar = \hbar## and ##\frac{\partial \phi}{\partial r} = E = \frac{e}{4\pi \epsilon_0 r^2}##.

[tex]|\vec B| = \frac{e\hbar}{mc^2 4\pi \epsilon_0 r^3}[/tex]

What value of ##r## must I use? When I use ##r = 4a_0## it gives the right answer.. as B = 0.2 T. Why can't I use ##B = a_0##?

For the 2p configuration, n =2, j = 3/2 or n=2, j = 1/2.

For general nl-configuration, ##0 < l < n, j = l \pm \frac{1}{2}##.

__Part (c)__[tex]\Delta H = -\frac{e^2}{m^2c^24\pi \epsilon_0 r^3} (\vec S . \vec L)[/tex]

We are supposed to find ##\langle \Delta H\rangle##:

##\vec S . \vec L## can be written as ##\frac{1}{2}(J^2 - S^2 - L^2)##, with eigenvalues ##\frac{l}{2}## for j = l + 1/2, and ##-\frac{1}{2}(l+1)## for j = l - 1/2.

Thus for j = l + 1/2, the splitting becomes:

[tex]\frac{e^2}{m^2c^2 4\pi \epsilon_0} \frac{1}{(l+1)(2l+1)}\left(\frac{1}{na_0}\right)^3 [/tex]

For j = l - 1/2, the splitting becomes:

[tex]\frac{e^2}{m^2c^2 4\pi \epsilon_0} \frac{1}{l(2l+1)}\left(\frac{1}{na_0}\right)^3 [/tex]

I'm not sure how to proceed from here..