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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

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, College physics -- hlca 1104. OpenStax CNX. May 18, 2013 Download for free at http://legacy.cnx.org/content/col11525/1.1
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