# 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

What does mean ohms law imply
what is matter
Anything that occupies space
Kevin
Any thing that has weight and occupies space
Victoria
Anything which we can feel by any of our 5 sense organs
Suraj
the time rate of increase in velocity is called
acceleration
Emma
What is uniform velocity
Victoria
Greetings,users of that wonderful app.
how to solve pressure?
how do we calculate weight and eara eg an elefant that weight 2000kg has four fits or legs search of surface eara is 0.1m2(1metre square) incontact with the ground=10m2(g =10m2)
Cruz
P=F/A
Mira
can someone derive the formula a little bit deeper?
Bern
what is coplanar force?
what is accuracy and precision
How does a current follow?
follow?
akif
which one dc or ac current.
akif
how does a current following?
Vineeta
?
akif
AC current
Vineeta
AC current follows due to changing electric field and magnetic field.
akif
Abubakar
ok bro thanks
akif
flows
Abubakar
but i wanted to understand him/her in his own language
akif
but I think the statement is written in English not any other language
Abubakar
my mean that in which form he/she written this,will understand better in this form, i write.
akif
ok
Abubakar
ok thanks bro. my mistake
Vineeta
u are welcome
Abubakar
what is a semiconductor
substances having lower forbidden gap between valence band and conduction band
akif
what is a conductor?
Vineeta
replace lower by higher only
akif
convert 56°c to kelvin
Abubakar
How does a current follow?
Vineeta
A semiconductor is any material whose conduction lies between that of a conductor and an insulator.
AKOWUAH
what is Atom? what is molecules? what is ions?
What is a molecule
Is a unit of a compound that has two or more atoms either of the same or different atoms
Justice
A molecule is the smallest indivisible unit of a compound, Just like the atom is the smallest indivisible unit of an element.
Rachel
what is a molecule?
Vineeta
what is a vector
A quantity that has both a magnitude AND a direction. E.g velocity, acceleration, force are all vector quantities. Hope this helps :)
deage
what is the difference between velocity and relative velocity?
Mackson
Velocity is the rate of change of displacement with time. Relative velocity on the other hand is the velocity observed by an observer with respect to a reference point.
Chuks
what do u understand by Ultraviolet catastrophe?
Rufai
A certain freely falling object, released from rest, requires 1.5seconds to travel the last 30metres before it hits the ground. (a) Find the velocity of the object when it is 30metres above the ground.
Mackson
A vector is a quantity that has both magnitude and direction
Rufus
the velocity Is 20m/s-2
Rufus
derivation of electric potential
V = Er = (kq/r^2)×r V = kq/r Where V: electric potential.
Chuks
what is the difference between simple motion and simple harmonic motion ?
syed
hi
Peace
hi
Rufus
hi
Chip
simple harmonic motion is a motion of tro and fro of simple pendulum and the likes while simple motion is a linear motion on a straight line.
Muinat
a body acceleration uniform from rest a 6m/s -2 for 8sec and decelerate uniformly to rest in the next 5sec,the magnitude of the deceleration is ?
The wording not very clear kindly
Moses
6
Leo
9.6m/s2
Jolly
the magnitude of deceleration =-9.8ms-2. first find the final velocity using the known acceleration and time. next use the calculated velocity to find the size of deceleration.
Mackson
wrong
Peace
-3.4m/s-2
Justice
Hi
Abj
Firstly, calculate final velocity of the body and then the deceleration. The final ans is,-9.6ms-2
Muinat
8x6= 48m/-2 use v=u + at 48÷5=9.6
Lawrence
can i define motion like this motion can be define as the continuous change of an object or position
Any object in motion will come to rest after a time duration. Different objects may cover equal distance in different time duration. Therefore, motion is defined as a change in position depending on time.
Chuks