- #1

- 45

- 0

## Homework Statement

I am to consider the Zeeman Effect. I need to calculate the energy level shifts for a given magnetic field corresponding to different quantum numbers. I'm having a hard time knowing when a quantum number [itex]Q[/itex] should be interpreted as just [itex]Q[/itex] or as [itex](Q, Q-1, ..., 0, ..., -Q)[/itex].

## Homework Equations

For low magnetic field values, the energy shift is given by:

[tex]\Delta E = \mu_{B} g_{F} B_{z} m_{F},[/tex]

where [itex] \mu_{B} [/itex] is the Bohr magneton, [itex] g_{F} [/itex] is the Lande g-factor, [itex] B_{z} [/itex] is the magnetic field, and [itex] m_{F} [/itex] is the projection of the total angular momentum quantum number [itex] F [/itex].

The Lande g-factor [itex] g_{F} [/itex] is[tex]

g_{F} = g_{J} \frac{ F(F+1) + J(J+1) - I(I+1) }{ 2F(F+1) }[/tex]

and [itex] g_{J} [/itex], is given by

[tex]g_{J} ≈ \frac{3}{2} + \frac{ 3/4 - L(L+1) }{ 2J(J+1) }[/tex]

The quantum numbers above are:

[itex]L[/itex], "orbital angular momentum quantum number",

*depends on state*

[itex]J[/itex], "total angular momentum quantum number", [itex]J = L + S[/itex]

([itex]S[/itex], "spin angular momentum quantum number", [itex]S = 1/2[/itex])

[itex]I[/itex], "nuclear spin quantum number",

*depends on isotope*

[itex]F[/itex], "(super?) total angular momentum quantum number", [itex]F = J + I[/itex]

**3. The Attempt at a Solution , pt. 1**

For concreteness, take potassium ( [itex]^{39} K[/itex] ). In this case, [itex] I = 3/2 [/itex]. The ground state is [itex] 4S_{1/2} [/itex], that is, [itex] n = 4 [/itex], [itex] L = 0 [/itex], and [itex] J = 1/2 [/itex].

(a) Determining [itex]F[/itex]:

Here is the first place that I run into trouble. The equation for [itex]F[/itex] simply states that [itex]F = J + I[/itex], from which follows that [itex]F = 1/2 + 3/2 = 2[/itex]. But I am looking at a diagram that shows

__two__[itex]F[/itex] states: [itex]F = 2[/itex] and [itex]F = 1[/itex]. (That is, the hyperfine splitting.)

So why are there two states and not one? My best guess is that I made a mistake earlier in interpreting the state label [itex] 4S_{1/2} [/itex]. The [itex]``1/2"[/itex] is not [itex]J[/itex], but actually [itex]|J|[/itex], and [itex]J[/itex] actually can take two values, [itex]J=+1/2,-1/2[/itex].

Of course, what is confusing about that is that [itex]J[/itex] was given as [itex]L + S[/itex]. But now I am supposing that I should interpret it as (what Wikipedia calls) the "main total angular momentum quantum number", whose values are determined by [itex]|L-S|≤J≤L+S[/itex].

Summary: Is it correct that the [itex]J[/itex] in spectroscopic notation is really telling you [itex]|J|[/itex], and that in general when you use [itex]J[/itex] to calculate [itex]F[/itex], the range of [itex]J[/itex] values should be used?

(b) Determining [itex] g_{J} [/itex]:

Here I run into trouble again. Should I interpret [itex] L [/itex] and [itex] J [/itex] to be [itex]0[/itex] and [itex] 1/2 [/itex], respectively? (Here [itex] L [/itex] is easy, since it's 0, but if it were nonzero, such as if it were 1, I'm

*assuming*that it would always be interpreted as 1 for this equation, and not its possible projections, right?)

My hunch is that the two possible values of [itex]J[/itex] here should be considered, and hence there will be

__two__different values of [itex] g_{J} [/itex] in this particular case, one for [itex]J=+1/2[/itex] and another for [itex]J=-1/2[/itex].

(c) Determining [itex] g_{F} [/itex]:

Again, should I expect to get several values of [itex] g_{F} [/itex] corresponding to the different possible values of [itex]F[/itex] (i.e. [itex]2[/itex] and [itex]1[/itex]) and [itex]J[/itex] (i.e. [itex]1/2[/itex] and [itex]-1/2[/itex])?

**4. The attempt at a solution, pt. 2**

Here's my guess at how to calculate [itex]g_{F} m_{F}[/itex] for the [itex] 4S_{1/2} [/itex] state of [itex]^{39} K[/itex]. (Note that ultimately I want to calculate [itex]\Delta E = \mu_{B} g_{F} B_{z} m_{F}[/itex], but I don't care about [itex]B_{z}[/itex] since it's the independent variable, and the Bohr magneton's just a number I can look up).

The two possible [itex]g_{J}[/itex] values, running on the assumption that I need to calculate it for the whole range of [itex]J[/itex] values:

[tex]g_{J=1/2} = \frac{3}{2} + \frac{\frac{3}{4} - 0(0+1)}{2(\frac{1}{2})(\frac{1}{2}+1)} = 2[/tex]

[tex]g_{J=-1/2} = \frac{3}{2} + \frac{\frac{3}{4} - 0(0+1)}{2(-\frac{1}{2})(-\frac{1}{2}+1)} = 0[/tex]

For [itex]F=2[/itex] (and [itex]J=1/2[/itex]):

[tex]g_{F=2} = 2 \frac{2(2+1) + \frac{1}{2}(\frac{1}{2}+1) - \frac{3}{2}(\frac{3}{2}+1)}{2(2)(2+1)} = \frac{1}{2}[/tex]

For [itex]F=2[/itex] (and [itex]J=-1/2[/itex]):

[tex]g_{F=2} = 0 \frac{2(2+1) + -\frac{1}{2}(-\frac{1}{2}+1) - \frac{3}{2}(\frac{3}{2}+1)}{2(2)(2+1)} = 0[/tex]

For [itex]F=1[/itex] (and [itex]J=1/2[/itex]):

[tex]g_{F=1} = 2 \frac{1(1+1) + \frac{1}{2}(\frac{1}{2}+1) - \frac{3}{2}(\frac{3}{2}+1)}{2(1)(1+1)} = -\frac{1}{2}[/tex]

For [itex]F=1[/itex] (and [itex]J=1/2[/itex]):

[tex]g_{F=1} = 0 \frac{1(1+1) + -\frac{1}{2}(-\frac{1}{2}+1) - \frac{3}{2}(\frac{3}{2}+1)}{2(1)(1+1)} = 0[/tex]

Finally, I multiply these (I think?) by all of their corresponding possible values of [itex]m_{F}[/itex] (the projection of [itex]F[/itex], which ranges from [itex]F[/itex] to [itex]-F[/itex] in integer steps.)

So for [itex]F=2[/itex] (and [itex]J=1/2[/itex]):

[itex]g_{F=2} m_{F=2} = (\frac{1}{2}) (2) = 1[/itex]

[itex]g_{F=2} m_{F=2} = (\frac{1}{2}) (1) = \frac{1}{2}[/itex]

[itex]g_{F=2} m_{F=2} = (\frac{1}{2}) (0) = 0[/itex]

[itex]g_{F=2} m_{F=2} = (\frac{1}{2}) (-1) = -\frac{1}{2}[/itex]

[itex]g_{F=2} m_{F=2} = (\frac{1}{2}) (-2) = -1[/itex]

For [itex]F=2[/itex] (and [itex]J=-1/2[/itex]):

[itex]g_{F=2} m_{F=2} = (0) (2) = 0[/itex]

[itex]g_{F=2} m_{F=2} = (0) (1) = 0[/itex]

[itex]g_{F=2} m_{F=2} = (0) (0) = 0[/itex]

[itex]g_{F=2} m_{F=2} = (0) (-1) = 0[/itex]

[itex]g_{F=2} m_{F=2} = (0) (-2) = 0[/itex]

So for [itex]F=1[/itex] (and [itex]J=1/2[/itex]):

[itex]g_{F=1} m_{F=1} = (-\frac{1}{2}) (1) = -\frac{1}{2}[/itex]

[itex]g_{F=1} m_{F=1} = (-\frac{1}{2}) (0) = 0[/itex]

[itex]g_{F=1} m_{F=1} = (-\frac{1}{2}) (-1) = \frac{1}{2}[/itex]

For [itex]F=1[/itex] (and [itex]J=-1/2[/itex]):

[itex]g_{F=1} m_{F=1} = (0) (1) = 0[/itex]

[itex]g_{F=1} m_{F=1} = (0) (0) = 0[/itex]

[itex]g_{F=1} m_{F=1} = (0) (-1) = 0[/itex]

So, in this case, if you applied a magnetic field to the atom and measured its ground state energy levels, you would find

__five__unique energy levels. If you had a way of counting the degeneracy of each level, you would find that the middle level had 10 degenerate states, while some of the others have 2 degenerate states (one for [itex]F=1[/itex] and one for [itex]F=2[/itex]).

Does that look correct? The alternative would be to have only

__one__value of [itex]g_{J}[/itex] (not two), and only

__two__values (or possibly one value) of [itex]g_{F}[/itex] (not four). That would be if I took [itex]J[/itex] in the above equations to mean [itex]|J_{max}|[/itex] (and possibly [itex]F[/itex] to be [itex]|F_{max}|[/itex]).

Last edited: