Figure 30.55represents the vectorsLandLzas usual, with arrows proportional to their magnitudes and pointing in the correct directions.L
andLzform a right triangle, withLbeing the hypotenuse andLzthe adjacent side. This means that the ratio ofLztoLis the cosine of
the angle of interest. We can findLandLzusingL= l(l+ 1)h
2π
andLz=mh
2π
.
SolutionWe are givenl= 1, so thatmlcan be +1, 0, or −1. ThusLhas the value given byL= l(l+ 1)h
2π
.
(30.45)
L=
l(l+ 1)h
2π
=^2 h
2π
Lzcan have three values, given byLz=mlh
2π
.
(30.46)
Lz=mlh
2π
=
⎧
⎩
⎨
⎪
⎪
h
2π
, ml = +1
0, ml = 0
−h
2π
, ml = −1
As can be seen inFigure 30.55,cosθ= Lz/L,and so forml=+1, we have
(30.47)
cosθ 1 =
LZ
L
=
h
2π
2 h
2π=^1
2
= 0.707.
Thus,θ (30.48)
1 = cos
−10.707 = 45.0º.
Similarly, forml= 0, we findcosθ 2 = 0; thus,
θ (30.49)
2 = cos
−10 = 90.0º.
And forml= −1,
(30.50)
cosθ 3 =
LZ
L
=
−2πh
2 h
2π= −^1
2
= −0.707,
so thatθ (30.51)
3 = cos
−1(−0.707)= 135.0º.
DiscussionThe angles are consistent with the figure. Only the angle relative to thez-axis is quantized.Lcan point in any direction as long as it makes the
proper angle with thez-axis. Thus the angular momentum vectors lie on cones as illustrated. This behavior is not observed on the large scale.
To see how the correspondence principle holds here, consider that the smallest angle (θ 1 in the example) is for the maximum value ofml= 0,
namelyml=l. For that smallest angle,
(30.52)
cosθ=
Lz
L
= l
l(l+ 1)
,
which approaches 1 aslbecomes very large. Ifcosθ= 1, thenθ= 0º. Furthermore, for largel, there are many values ofml, so that all
angles become possible aslgets very large.
Intrinsic Spin Angular Momentum Is Quantized in Magnitude and Direction
There are two more quantum numbers of immediate concern. Both were first discovered for electrons in conjunction with fine structure in atomic
spectra. It is now well established that electrons and other fundamental particles haveintrinsic spin, roughly analogous to a planet spinning on its
axis. This spin is a fundamental characteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic
angular momentum is quantized independently of orbital angular momentum. Additionally, the direction of the spin is also quantized. It has beenfound that themagnitude of the intrinsic (internal) spin angular momentum,S, of an electron is given by
S= s(s+ 1)h (30.53)
2π
(s= 1 / 2 for electrons),
1094 CHAPTER 30 | ATOMIC PHYSICS
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