# 30.8 Quantum numbers and rules

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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 ${E}_{n}\propto 1/{n}^{2}$ , 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

$n=1, 2, 3, ....$

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 $n$ , 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 $L$ can have only the values

$L=\sqrt{l\left(l+1\right)}\frac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}\left(l=0, 1, 2, ...,\phantom{\rule{0.25em}{0ex}}n-1\right)\text{,}$

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

Note that for $n=1$ , $l$ can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not $h/2\pi$ 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 $n=2$ , so that $l$ can be either 0 or 1, according to the rule in $L=\sqrt{l\left(l+1\right)}\frac{h}{2\pi }$ . Similarly, for $n=3$ , $l$ can be 0, 1, or 2. It is often most convenient to state the value of $l$ , a simple integer, rather than calculating the value of $L$ from $L=\sqrt{l\left(l+1\right)}\frac{h}{2\pi }$ . For example, for $l=2$ , we see that

$L=\sqrt{2\left(2+1\right)}\frac{h}{2\pi }=\sqrt{6}\frac{h}{2\pi }=0\text{.}\text{390}h=2\text{.}\text{58}×{\text{10}}^{-\text{34}}\phantom{\rule{0.25em}{0ex}}\text{J}\cdot s.$

It is much simpler to state $l=2$ .

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 $z$ -axis, can have only certain values of ${L}_{z}$ . 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 ${L}_{z}$ are

#### Questions & Answers

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
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why uniform circular motion is called a periodic motion?.
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there is enough information to calculate an AVERAGE acceleration
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what are the types of radioactivity
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It is the opposite of kinetic friction
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static fiction is friction between two surfaces in contact an none of sliding over on another, while Kinetic friction is friction between sliding surfaces in contact.
MINDERIUM
I don't get it,if it's static then there will be no friction.
author
It means that static friction is that friction that most be overcome before a body can move
kingsley
static friction is a force that keeps an object from moving, and it's the opposite of kinetic friction.
author
It is a force a body must overcome in order for the body to move.
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what is force
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force is something which is responsible for the object to change its position
MINDERIUM
more technically it is the product of mass of an object and Acceleration produced in it
MINDERIUM
wave is disturbance in any medium
iqra
energy is distributed in any medium through particles of medium.
iqra
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Eboh Reply
we have to first figure out .... wats a particle accelerator first
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The resistive force of surface.
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on both a string and peldulum?
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spring*
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Simple harmonic motion
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how.an.equipotential.line is two dimension and equipotential surface is three dimension ?
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definition of mass of conversion
umezurike Reply
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
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Adeshina Reply
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
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Praise what level are you
yusuf
If u are doing a levels in your first year you do AS topics therefore you do 5 big topic i.e particles radiation, waves and optics, mechanics,materials, electricity. After that you do A level topics like Specific Harmonic motion circular motion astrophysics depends really
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Heat nd Co for a level
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the range of objects and phenomena studied in physics is
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