How Does the Zeeman Effect Impact Hydrogen Atom Energy Levels?

[unscientific]

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)
$$H = -\frac{e}{m^2c^2} \frac{1}{r} \frac{\partial \phi}{\partial r} \vec S . \vec L$$
$$-\mu . B = -\frac{e}{m^2c^2} \frac{1}{r} \frac{\partial \phi}{\partial r} \vec S . \vec L$$
$$\frac{e}{2m}\vec S . \vec B = -\frac{e}{m^2c^2} \frac{1}{r} \frac{\partial}{\partial r} \vec S . \vec L$$
$$|\vec B| = |\vec L| \frac{e}{m^2 c^2} \frac{m}{e} \frac{1}{r} \frac{\partial \phi}{\partial r}$$

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

$$|\vec B| = \frac{e\hbar}{mc^2 4\pi \epsilon_0 r^3}$$

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)

$$\Delta H = -\frac{e^2}{m^2c^24\pi \epsilon_0 r^3} (\vec S . \vec L)$$

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:
$$\frac{e^2}{m^2c^2 4\pi \epsilon_0} \frac{1}{(l+1)(2l+1)}\left(\frac{1}{na_0}\right)^3$$

For j = l - 1/2, the splitting becomes:
$$\frac{e^2}{m^2c^2 4\pi \epsilon_0} \frac{1}{l(2l+1)}\left(\frac{1}{na_0}\right)^3$$

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

 

[DrClaude]

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 $B = a_0$?

The electron is not at a fixed position, you have to integrate radially the wave function, which makes the $\langle r^{-3} \rangle$ appear.

 

[unscientific]

The electron is not at a fixed position, you have to integrate radially the wave function, which makes the $\langle r^{-3} \rangle$ appear.

Yes I get that, but for part (c), how does the magnetic field come into play?

 

[DrClaude]

You need to add an additional perturbation corresponding to the coupling with the magnetic field.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/zeeman.html

 

[unscientific]

You need to add an additional perturbation corresponding to the coupling with the magnetic field.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/zeeman.html

I've been reading binney's book and I'm extremely confused. He mentioned the 'Zeeman Spin Hamiltonian' and the 'Spin Orbit Hamiltonian' - what's the difference and what are they used for?

My understanding is that:

1. In the absence of an external electric field, the electron moving around the atom experiences a magnetic torque. This torque causes its spin to precess. This precession results in an interaction (energy) between the electron's orbital motion and spin.

2. Part (b) asks us to find the electron's own 'equivalent magnetic field' that's causing this effect. The spin interaction hamiltonian is simply $H_{ZS} = -\vec \mu_s \cdot \vec B = -\gamma \vec S \cdot \vec B = -\frac{g_sq}{2m}\vec S \cdot \vec B = -\frac{e}{m} \vec S \cdot \vec B$

3. But in the presence of an external magnetic field, two things happen. Firstly, the spin interacts with the external field, giving hamiltonian : $H_{ZS} = -\frac{e}{m} \vec S \cdot \vec B$.
Secondly, the electron's orbital angular momentum interacts with the external field, giving hamiltonian: $H_l = - \vec \mu_l \cdot \vec B = -\frac{g_l q}{2m} \vec L \cdot \vec B = -\frac{e}{2m} \vec L \cdot \vec B$. In this case, $g_l = 1$ because there is no degeneracy.

Together, the TOTAL external field interaction Hamiltonian is given by:
$$H = H_{ZS} + H_l = \frac{e}{2m} \left( \vec L + 2\vec S\right)\cdot \vec B$$

 

[unscientific]

The electron is not at a fixed position, you have to integrate radially the wave function, which makes the $\langle r^{-3} \rangle$ appear.

Also, for part (b), the expression for the magnetic flux density is this:

$$|\vec B| = \frac{e\hbar}{mc^2 4\pi \epsilon_0 r^3}$$

For a 2p configuration, when I use $r = a_0$ it doesn't give the right answer. Only $r = 4a_0$ gives the right answer. Why is that so?

 

[unscientific]

How do I evaluate $\frac{Be}{2m}\langle j,m | L_z + 2S_z|j,m\rangle$?

 

[unscientific]

I think I got it. In this question, l = 0, so $\frac{Be}{2m}\langle j,m | L_z + 2S_z|j,m\rangle = \frac{Be}{2m}\langle \pm |2S_z|\pm\rangle = \pm \frac{Be}{2m}(\hbar)$.

If $l \neq 0$, we have to add angular momenta.