Question

Angle between z axis and angular momentum

Answer 1

What state has a minimum value of the angle between the angular momentum and the z-axis; the electron spin or the atomic n=3 state? [with l not equal to 0]

2. Relevant equations
L
=
√
l
(
l
+
1
)
ℏ
L=l(l+1)ℏ

Lz = m
ℏ
ℏ

S
=
√
s
(
s
+
1
)
ℏ
S=s(s+1)ℏ

Sz = ms
ℏ
ℏ


3. The attempt at a solution
For the atominc n=3 state..
Since n = 3 (and l isn't 0), l = 1, 2 ... and ml = -l ... l = -2,-1,0,1,2
So then when you draw it.. (I don't seem to be able to get the picture in here]
The Lz vector is pointing up along the z-axis and the L vector is some degrees to the right of it.

The angle between the z-axis and angular momentum can be written using cosine...
c
o
s
(
θ
)
=
L
z
L
cos(θ)=LzL

I understand up to there.

But then the next part the professor did is add in numbers and I'm not sure where the numbers came from. He wrote this:
c
o
s
(
θ
)
=
2
ℏ
√
6
ℏ
=
2
√
6
cos(θ)=2ℏ6ℏ=26

The
√
6
6
comes from
L
=
√
l
(
l
+
1
)
ℏ
L=l(l+1)ℏ
using l = 2. But why l = 2 and not l = 1?
And I don't know where the 2 on top came from.

Then he wrote another one..
c
o
s
(
θ
)
=
1
/
2
√
3
/
4
cos(θ)=1/23/4