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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} } } {} .

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

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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.

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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?

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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} {} .

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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?

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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.

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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?

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(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} {}

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(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?

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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} {}

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(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.

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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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Questions & Answers

what is angular velocity
Obaapa Reply
Why does earth exert only a tiny downward pull?
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Why is light bright?
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an 8.0 capacitor is connected by to the terminals of 60Hz whoes rms voltage is 150v. a.find the capacity reactance and rms to the circuit
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Valdes Reply
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is the study of matter in relation to energy
Kintu
a submersible pump is dropped a borehole and hits the level of water at the bottom of the borehole 5 seconds later.determine the level of water in the borehole
Obrian Reply
what is power?
aron Reply
power P = Work done per second W/ t. It means the more power, the stronger machine
Sphere
e.g. heart Uses 2 W per beat.
Rohit
A spherica, concave shaving mirror has a radius of curvature of 32 cm .what is the magnification of a persons face. when it is 12cm to the left of the vertex of the mirror
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Shii
1.75cm
Ridwan
my name is Abu m.konnek I am a student of a electrical engineer and I want you to help me
Abu
the magnification k = f/(f-d) with focus f = R/2 =16 cm; d =12 cm k = 16/4 =4
Sphere
what do we call velocity
Kings
A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows pa
Kavita Reply
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Godfred
what about the wind vane
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If a prism is fully imersed in water then the ray of light will normally dispersed or their is any difference?
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the same behavior thru the prism out or in water bud abbot
Ju
If this will experimented with a hollow(vaccum) prism in water then what will be result ?
Anurag
What was the previous far point of a patient who had laser correction that reduced the power of her eye by 7.00 D, producing a normal distant vision power of 50.0 D for her?
Jaydie Reply
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
A young woman with normal distant vision has a 10.0% ability to accommodate (that is, increase) the power of her eyes. What is the closest object she can see clearly?
Jaydie
29/20 ? maybes
Ju
In what ways does physics affect the society both positively or negatively
Princewill Reply
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rerry Reply
try to read several books on phy don't just rely one. some authors explain better than other.
Ju
And don't forget to check out YouTube videos on the subject. Videos offer a different visual way to learn easier.
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Practice Key Terms 7

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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