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  • Define quantum number.
  • Calculate angle of angular momentum vector with an axis.
  • Define spin quantum number.

Physical characteristics that are quantized—such as energy, charge, and angular momentum—are of such importance that names and symbols are given to them. The values of quantized entities are expressed in terms of quantum numbers    , and the rules governing them are of the utmost importance in determining what nature is and does. This section covers some of the more important quantum numbers and rules—all of which apply in chemistry, material science, and far beyond the realm of atomic physics, where they were first discovered. Once again, we see how physics makes discoveries which enable other fields to grow.

The energy states of bound systems are quantized , because the particle wavelength can fit into the bounds of the system in only certain ways. This was elaborated for the hydrogen atom, for which the allowed energies are expressed as E n 1/ n 2 , where n = 1, 2, 3, ... . We define n to be the principal quantum number that labels the basic states of a system. The lowest-energy state has n = 1 , the first excited state has n = 2 , and so on. Thus the allowed values for the principal quantum number are

n = 1, 2, 3, ... . size 12{n=1, 2, 3, "." "." "." } {}

This is more than just a numbering scheme, since the energy of the system, such as the hydrogen atom, can be expressed as some function of n size 12{n} {} , as can other characteristics (such as the orbital radii of the hydrogen atom).

The fact that the magnitude of angular momentum is quantized was first recognized by Bohr in relation to the hydrogen atom; it is now known to be true in general. With the development of quantum mechanics, it was found that the magnitude of angular momentum L size 12{L} {} can have only the values

L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} l = 0, 1, 2, ... , n 1 , size 12{ left (l=0, 1, 2, "." "." "." ,n - 1 right )} {}

where l size 12{l} {} is defined to be the angular momentum quantum number    . The rule for l size 12{l} {} in atoms is given in the parentheses. Given n size 12{n} {} , the value of l size 12{l} {} can be any integer from zero up to n 1 size 12{n - 1} {} . For example, if n = 4 size 12{n=4} {} , then l size 12{l} {} can be 0, 1, 2, or 3.

Note that for n = 1 size 12{n=1} {} , l size 12{l} {} can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not h / 2 π as Bohr proposed. The picture of circular orbits is not valid, because there would be angular momentum for any circular orbit. A more valid picture is the cloud of probability shown for the ground state of hydrogen in [link] . The electron actually spends time in and near the nucleus. The reason the electron does not remain in the nucleus is related to Heisenberg’s uncertainty principle—the electron’s energy would have to be much too large to be confined to the small space of the nucleus. Now the first excited state of hydrogen has n = 2 size 12{n=2} {} , so that l size 12{l} {} can be either 0 or 1, according to the rule in L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} . Similarly, for n = 3 size 12{n=3} {} , l size 12{l} {} can be 0, 1, or 2. It is often most convenient to state the value of l size 12{l} {} , a simple integer, rather than calculating the value of L size 12{L} {} from L = l l + 1 h size 12{L= sqrt {l left (l+1 right )} { {h} over {2π} } } {} . For example, for l = 2 size 12{l=2} {} , we see that

L = 2 2 + 1 h = 6 h = 0 . 390 h = 2 . 58 × 10 34 J s . size 12{L= sqrt {2 left (2+1 right )} { {h} over {2π} } = sqrt {6} { {h} over {2π} } =0 "." "390"h=2 "." "58" times "10" rSup { size 8{ - "34"} } " J" cdot s} {}

It is much simpler to state l = 2 size 12{l=2} {} .

As recognized in the Zeeman effect, the direction of angular momentum is quantized . We now know this is true in all circumstances. It is found that the component of angular momentum along one direction in space, usually called the z size 12{z} {} -axis, can have only certain values of L z size 12{L rSub { size 8{z} } } {} . The direction in space must be related to something physical, such as the direction of the magnetic field at that location. This is an aspect of relativity. Direction has no meaning if there is nothing that varies with direction, as does magnetic force. The allowed values of L z size 12{L rSub { size 8{z} } } {} are

Questions & Answers

some questions on vector
Rose Reply
where are mcq?
wald Reply
Please what is the meaning of mcq
Appiah
what is static on its own
Aniedi Reply
where is the position of distance
AHMED Reply
what is physics
Mbaba Reply
calculate the angular displacement for an object which completes five revolutions
Bisalom Reply
I need someone to explain how white light disperses to form the "ROYGBIV".
Dera Reply
when it pass through a glass prism through a process called dispersion of light
Mahmud
What is an atom
Sulaimon Reply
An atom is the smallest indivisible particle of an element
Dera
When a toilet is flushed or a sink is drained, the water (and other material) begins to rotate about the drain on the way down. Assuming no initial rotation and a flow initially directly straight toward the drain, explain what causes the rotation and which direction it has in the northern hemisphere.
Collin Reply
find the change in entropy of a 2.00 kg block of gold at 1063^0C when it meets to become liquid gold at 1063^0C
precious Reply
if you are asked to make a very sensitive thermometer which of the following fluids would you choose
precious
between mercy and gasoline
precious
it good to use mercury because mercury does not wet glass and it does not evaporate easily
Desmond
0
firdaus
SFAR Sifar SIFAT -<SIFST
firdaus
how many particles are in 2 moles of chromium
Mario Reply
if so use the normal formula number of atom= number of particle/Avogadro's number
Aki
n= np/avogadtos constant. therefore n= 24/ 6.022×10²³
albert
24÷6.022×10²³
albert
@Albert is wrong
Aki
when you cross multiple it should give you Number of particles= mole*Avogadro's number X=2m*6.022*10^²³ X=1.20*10²⁴g
Aki
1.204×10^-22
Maame
please what is final velocity and initial velocity
Nonso Reply
don't know
Ekene
what do you want to become in future
Ekene
Y ar u asking pls
Nonso
I think initial velocity is the velocity that the mobile starts with at the start time (t=0s) but I don't think I heard abt final velocity
Malak
Malak where are you now I need to learn more from you
Ekene
initial velocity is the velocity an object possess at it intial position or is the starting velocity, while a final velocity is the velocity an object or body possess at it final stage or at the end of it motion
Mubarak
Bohr is kimia, of toksid, cloud, tree have cloud, tree, river but small from toksid fish or another.
firdaus Reply
Heavy, heavy kehidupan susah, kekayaan, berlambak, bergumpul. Dikenali.
firdaus Reply
Gravitional, Gravitional mean kehidup seseorang. Kehidupan bumi, kehidupan muka bumi, kehidupan dalam longitude, kehidupan dalam momentom, kehidupan dalam mongitude. Kehidupan dalam Pelajaran, mean Pelajar kolej.
firdaus
Nonconservative. Sains belajar
firdaus
hello
LFX
what are the types of kinetics
pawi Reply
Practice Key Terms 7

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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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