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L z = m l h size 12{L rSub { size 8{z} } =m rSub { size 8{l} } { {h} over {2π} } } {} m l = l , l + 1, ... , 1, 0, 1, ... l 1, l , size 12{ left (m rSub { size 8{l} } = - l, - l+1, "." "." "." , - 1, 0, 1, "." "." "." l - 1, l right )} {}

where L z size 12{L rSub { size 8{z} } } {} is the z size 12{z} {} -component of the angular momentum and m l size 12{m rSub { size 8{l} } } {} is the angular momentum projection quantum number. The rule in parentheses for the values of m l size 12{m rSub { size 8{l} } } {} is that it can range from l size 12{ - l} {} to l size 12{l} {} in steps of one. For example, if l = 2 size 12{l=2} {} , then m l size 12{m rSub { size 8{l} } } {} can have the five values –2, –1, 0, 1, and 2. Each m l size 12{m rSub { size 8{l} } } {} corresponds to a different energy in the presence of a magnetic field, so that they are related to the splitting of spectral lines into discrete parts, as discussed in the preceding section. If the z size 12{z} {} -component of angular momentum can have only certain values, then the angular momentum can have only certain directions, as illustrated in [link] .

The image shows two possible values of component of a given angular momentum along z-axis. One circular orbit above the original circular orbit is shown for m sub l value of plus one. Another circular orbit below the original circular orbit is shown for m sub l value of minus one. The angular momentum vector for the top circular orbit makes an angle of theta sub one with the vertical axis. The horizontal angular momentum vector at original circular orbit makes an angle of theta sub two with the vertical axis. The angular momentum vector for the bottom circular orbit makes an angle of theta sub three with the vertical axis.
The component of a given angular momentum along the z -axis (defined by the direction of a magnetic field) can have only certain values; these are shown here for l = 1 , for which m l = 1, 0, and +1 . The direction of L is quantized in the sense that it can have only certain angles relative to the z -axis.

What are the allowed directions?

Calculate the angles that the angular momentum vector L size 12{L} {} can make with the z size 12{z} {} -axis for l = 1 size 12{l=1} {} , as illustrated in [link] .

Strategy

[link] represents the vectors L size 12{L} {} and L z size 12{L rSub { size 8{z} } } {} as usual, with arrows proportional to their magnitudes and pointing in the correct directions. L size 12{L} {} and L z size 12{L rSub { size 8{z} } } {} form a right triangle, with L size 12{L} {} being the hypotenuse and L z size 12{L rSub { size 8{z} } } {} the adjacent side. This means that the ratio of L z size 12{L rSub { size 8{z} } } {} to L size 12{L} {} is the cosine of the angle of interest. We can find L size 12{L} {} and L z size 12{L rSub { size 8{z} } } {} using L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} and L z = m h size 12{L rSub { size 8{z} } =m { {h} over {2π} } } {} .

Solution

We are given l = 1 size 12{l=1} {} , so that m l size 12{m rSub { size 8{l} } } {} can be +1, 0, or −1. Thus L size 12{L} {} has the value given by L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} .

L = l l + 1 h = 2 h size 12{L= { { sqrt {l left (l+1 right )} h} over {2π} } = { { sqrt {2} h} over {2π} } } {}

L z size 12{L rSub { size 8{z} } } {} can have three values, given by L z = m l h size 12{L rSub { size 8{z} } =m rSub { size 8{l} } { {h} over {2π} } } {} .

L z = m l h = { h , m l = + 1 0, m l = 0 h , m l = 1

As can be seen in [link] , cos θ = L z /L, and so for m l =+ 1 size 12{m rSub { size 8{l} } "=+"1} {} , we have

cos θ 1 = L Z L = h 2 h = 1 2 = 0 . 707. size 12{"cos"θ rSub { size 8{1} } = { {L rSub { size 8{Z} } } over {L} } = { { { {h} over {2π} } } over { { { sqrt {2} h} over {2π} } } } = { {1} over { sqrt {2} } } =0 "." "707"} {}

Thus,

θ 1 = cos 1 0.707 = 45 . 0º.

Similarly, for m l = 0 size 12{m rSub { size 8{l} } =0} {} , we find cos θ 2 = 0 size 12{"cos"θ rSub { size 8{2} } =0} {} ; thus,

θ 2 = cos 1 0 = 90 . 0º. size 12{θ rSub { size 8{2} } ="cos" rSup { size 8{ - 1} } 0="90" "." 0°} {}

And for m l = 1 size 12{m rSub { size 8{l} } = - 1} {} ,

cos θ 3 = L Z L = h 2 h = 1 2 = 0 . 707, size 12{"cos"θ rSub { size 8{3} } = { {L rSub { size 8{Z} } } over {L} } = { { - { {h} over {2π} } } over { { { sqrt {2} h} over {2π} } } } = - { {1} over { sqrt {2} } } = - 0 "." "707"} {}

so that

θ 3 = cos 1 0 . 707 = 135 . 0º. size 12{θ rSub { size 8{3} } ="cos" rSup { size 8{ - 1} } left ( - 0 "." "707" right )="135" "." 0°} {}

Discussion

The angles are consistent with the figure. Only the angle relative to the z size 12{z} {} -axis is quantized. L size 12{L} {} can point in any direction as long as it makes the proper angle with the z size 12{z} {} -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 of m l = 0 , namely m l = l . For that smallest angle,

cos θ = L z L = l l l + 1 , size 12{"cos"θ= { {L rSub { size 8{z} } } over {L} } = { {l} over { sqrt {l left (l+1 right )} } } } {}

which approaches 1 as l size 12{l} {} becomes very large. If cos θ = 1 size 12{"cos"θ=1} {} , then θ = . Furthermore, for large l , there are many values of m l , so that all angles become possible as l gets very large.

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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 have intrinsic 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 been found that the magnitude of the intrinsic (internal) spin angular momentum    , S size 12{S} {} , of an electron is given by

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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