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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

S = s s + 1 h ( s = 1 / 2 for electrons), size 12{s=1/2} {}

where s size 12{s} {} is defined to be the spin quantum number    . This is very similar to the quantization of L size 12{L} {} given in L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} , except that the only value allowed for s size 12{s} {} for electrons is 1/2.

The direction of intrinsic spin is quantized , just as is the direction of orbital angular momentum. The direction of spin angular momentum along one direction in space, again called the z size 12{z} {} -axis, can have only the values

S z = m s h size 12{S rSub { size 8{z} } =m rSub { size 8{s} } { {h} over {2π} } } {} m s = 1 2 , + 1 2 size 12{ left (m rSub { size 8{s} } = - { {1} over {2} } , + { {1} over {2} } right )} {}

for electrons. S z size 12{S rSub { size 8{z} } } {} is the z size 12{z} {} -component of spin angular momentum and m s size 12{S rSub { size 8{z} } } {} is the spin projection quantum number    . For electrons, s size 12{s} {} can only be 1/2, and m s size 12{m rSub { size 8{s} } } {} can be either +1/2 or –1/2. Spin projection m s =+ 1 / 2 size 12{m rSub { size 8{s} } "=+"1/2} {} is referred to as spin up , whereas m s = 1 / 2 size 12{m rSub { size 8{s} } = - 1/2} {} is called spin down . These are illustrated in [link] .

Intrinsic spin

In later chapters, we will see that intrinsic spin is a characteristic of all subatomic particles. For some particles s size 12{s} {} is half-integral, whereas for others s size 12{s} {} is integral—there are crucial differences between half-integral spin particles and integral spin particles. Protons and neutrons, like electrons, have s = 1 / 2 size 12{s=1/2} {} , whereas photons have s = 1 size 12{s=1} {} , and other particles called pions have s = 0 size 12{s=0} {} , and so on.

To summarize, the state of a system, such as the precise nature of an electron in an atom, is determined by its particular quantum numbers. These are expressed in the form n, l, m l , m s —see [link] For electrons in atoms , the principal quantum number can have the values n = 1, 2, 3, ... . Once n is known, the values of the angular momentum quantum number are limited to l = 1, 2, 3, ... , n 1 . For a given value of l , the angular momentum projection quantum number can have only the values m l = l , l + 1, ... , 1, 0, 1, ... , l 1, l . Electron spin is independent of n, l, and m l , always having s = 1 / 2 . The spin projection quantum number can have two values, m s = 1 / 2 or 1 / 2 .

Atomic quantum numbers
Name Symbol Allowed values
Principal quantum number n 1, 2, 3, ...
Angular momentum l 0, 1, 2, ... n 1
Angular momentum projection m l l , l + 1, ... , 1, 0, 1, ... , l 1, l ( or 0, ±1, ±2, ... , ± l )
Spin The spin quantum number s is usually not stated, since it is always 1/2 for electrons s 1/2 ( electrons )
Spin projection m s 1/2, + 1/2

[link] shows several hydrogen states corresponding to different sets of quantum numbers. Note that these clouds of probability are the locations of electrons as determined by making repeated measurements—each measurement finds the electron in a definite location, with a greater chance of finding the electron in some places rather than others. With repeated measurements, the pattern of probability shown in the figure emerges. The clouds of probability do not look like nor do they correspond to classical orbits. The uncertainty principle actually prevents us and nature from knowing how the electron gets from one place to another, and so an orbit really does not exist as such. Nature on a small scale is again much different from that on the large scale.

Questions & Answers

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
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what happens to the size of charge if the dielectric is changed?
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omega= omega not +alpha t derivation
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u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
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Because HE needs someone to dominate the earth (Gen. 1:26)
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Practice Key Terms 7

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