<< Chapter < Page Chapter >> Page >
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} } } {} .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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} {} .

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

(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} {}

Got questions? Get instant answers now!

(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?

Got questions? Get instant answers now!

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} {}

Got questions? Get instant answers now!

(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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
Ali Reply
displacement in easy way.
Mubashir Reply
binding energy per nucleon
Poonam Reply
why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
why god made humenity
Ali
and he to multiply
Owofemi
stuff happens
Ju
God plays dice
Ju
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
Abrar
when we face electric shock these elements work as a conductor that's why we got this shock
Abrar
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
ximena Reply
why electromagnetic induction is not used in room heater ?
Gopi Reply
room?
Abrar
What is position?
Amoah Reply
What is law of gravition
sushil Reply
Practice Key Terms 7

Get the best College physics for ap... course in your pocket!





Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics for ap® courses' conversation and receive update notifications?

Ask