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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.

The image shows probability clouds for the electron in the ground state and several excited states of hydrogen. Sets of quantum numbers given as n l m subscript l are shown for each state. The ground state is zero zero zero. The probability of finding the electron is indicated by the shade of color.
Probability clouds for the electron in the ground state and several excited states of hydrogen. The nature of these states is determined by their sets of quantum numbers, here given as n , l , m l size 12{ left (n, l, m rSub { size 8{l} } right )} {} . The ground state is (0, 0, 0); one of the possibilities for the second excited state is (3, 2, 1). The probability of finding the electron is indicated by the shade of color; the darker the coloring the greater the chance of finding the electron.

We will see that the quantum numbers discussed in this section are valid for a broad range of particles and other systems, such as nuclei. Some quantum numbers, such as intrinsic spin, are related to fundamental classifications of subatomic particles, and they obey laws that will give us further insight into the substructure of matter and its interactions.

Phet explorations: stern-gerlach experiment

The classic Stern-Gerlach Experiment shows that atoms have a property called spin. Spin is a kind of intrinsic angular momentum, which has no classical counterpart. When the z-component of the spin is measured, one always gets one of two values: spin up or spin down.

Stern-Gerlach Experiment

Section summary

  • Quantum numbers are used to express the allowed values of quantized entities. The principal quantum number n size 12{n} {} labels the basic states of a system and is given by
    n = 1, 2, 3, . . . . size 12{n=1, 2, 3, "." "." "." } {}
  • The magnitude of angular momentum is given by
    L = l l + 1 h l = 0, 1, 2, ... , n 1 ,
    where l size 12{l} {} is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the z size 12{z} {} -axis is given by
    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 ,
    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. Similarly, the electron’s intrinsic spin angular momentum S size 12{S} {} is given by
    S = s s + 1 h ( size 12{S= sqrt {s left (s+1 right )} { {h} over {2π} } } {} s = 1 / 2 for electrons), size 12{s=1/2} {}
    s size 12{s} {} is defined to be the spin quantum number. Finally, the direction of the electron’s spin along the z size 12{z} {} -axis is given by
    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 )} {}
    where 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{m rSub { size 8{s} } } {} is the spin projection quantum number. 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. [link] summarizes the atomic quantum numbers and their allowed values.

Conceptual questions

Define the quantum numbers n, l, m l , s , and m s size 12{m rSub { size 8{s} } } {} .

For a given value of n size 12{n} {} , what are the allowed values of l size 12{l} {} ?

For a given value of l size 12{l} {} , what are the allowed values of m l size 12{m rSub { size 8{l} } } {} ? What are the allowed values of m l size 12{m rSub { size 8{l} } } {} for a given value of n size 12{n} {} ? Give an example in each case.

List all the possible values of s size 12{s} {} and m s size 12{m rSub { size 8{s} } } {} for an electron. Are there particles for which these values are different? The same?

Problem exercises

If an atom has an electron in the n = 5 size 12{n=5} {} state with m l = 3 size 12{m rSub { size 8{l} } =3} {} , what are the possible values of l size 12{l} {} ?

l = 4, 3 are possible since l < n size 12{l<n} {} and m l l size 12{ lline m rSub { size 8{l} } rline {underline {<}} l} {} .

An atom has an electron with m l = 2 size 12{m rSub { size 8{l} } =2} {} . What is the smallest value of n size 12{n} {} for this electron?

What are the possible values of m l size 12{m rSub { size 8{l} } } {} for an electron in the n = 4 size 12{n=4} {} state?

n = 4 l = 3, 2, 1, 0 m l = ± 3, ± 2, ± 1, 0 are possible.

What, if any, constraints does a value of m l = 1 size 12{m rSub { size 8{l} } =1} {} place on the other quantum numbers for an electron in an atom?

(a) Calculate the magnitude of the angular momentum for an l = 1 size 12{l=1} {} electron. (b) Compare your answer to the value Bohr proposed for the n = 1 size 12{n=1} {} state.

(a) 1 . 49 × 10 34 J s size 12{1 "." "49" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}

(b) 1 . 06 × 10 34 J s size 12{1 "." "06" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}

(a) What is the magnitude of the angular momentum for an l = 1 size 12{l=1} {} electron? (b) Calculate the magnitude of the electron’s spin angular momentum. (c) What is the ratio of these angular momenta?

Repeat [link] for l = 3 size 12{l=3} {} .

(a) 3 . 66 × 10 34 J s size 12{3 "." "66" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}

(b) s = 9 . 13 × 10 35 J s size 12{s=9 "." "14" times "10" rSup { size 8{ - "35"} } " J" cdot s} {}

(c) L S = 12 3 / 4 = 4 size 12{ { {L} over {S} } = { { sqrt {"12"} } over { sqrt {3/4} } } =4} {}

(a) How many angles can L size 12{L} {} make with the z size 12{z} {} -axis for an l = 2 size 12{l=2} {} electron? (b) Calculate the value of the smallest angle.

What angles can the spin S size 12{S} {} of an electron make with the z size 12{z} {} -axis?

θ = 54.7º, 125.3º

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Source:  OpenStax, Basic physics for medical imaging. OpenStax CNX. Feb 17, 2014 Download for free at http://legacy.cnx.org/content/col11630/1.1
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