What is quantum numbers? in periodic classification topic!?

Question

Answer

Quantum numbers give information about the location of an electron or set of electrons. A full set of quantum numbers describes a unique electron for a particular atom.

Answer

Quantum no. is a set of four no. I.e.1)Principle quantum no. 2)Azimuthal quantum no. 3)Magnetic quantum no. 4)Spin quantum no. .. which gives complete information about electron revolving in orbits.

Answer

Quantum Numbers in Periodic Classification

Quantum numbers are a set of four numbers that describe the electron configuration of an atom. They are used to specify the energy levels, sub-levels, and the number of electrons in an atom. The quantum numbers help in explaining the periodic properties of elements. The four quantum numbers are:

Principal Quantum Number (n)
The principal quantum number determines the energy level of an electron. It is represented by the letter 'n' and can have positive integers (1, 2, 3, 4, ...) as its value. The higher the value of n, the higher the energy level of the electron.

Azimuthal Quantum Number (l)
The azimuthal quantum number determines the sub-level in which the electron resides. It is represented by the letter 'l' and can have integers from 0 to (n-1) as its value. The value of l determines the shape of the sub-level; for example, l=0 corresponds to an s sub-level, l=1 corresponds to a p sub-level, l=2 corresponds to a d sub-level, and l=3 corresponds to an f sub-level.

Magnetic Quantum Number (m)
The magnetic quantum number determines the orientation of the sub-level in space. It is represented by the letter 'm' and can have integers from -l to +l as its value. For example, if l=1, then m can have values of -1, 0, or +1.

Spin Quantum Number (s)
The spin quantum number determines the direction of the electron's spin. It is represented by the letter 's' and can have a value of +1/2 or -1/2.

Periodic Properties of Elements
The periodic properties of elements can be explained using the quantum numbers. For example, the atomic radius of an element increases as we move down a group because the number of energy levels increases, and the electrons are farther away from the nucleus. Similarly, the ionization energy and electronegativity of an element increase as we move across a period because the number of electrons in the outermost shell increases, making it harder to remove or attract them.

Conclusion
In conclusion, the quantum numbers are an important tool for understanding the electron configuration and the periodic properties of elements. They help in predicting the behavior of atoms and their chemical reactions.

Answer

A number which occurs in the theoretical expression for the value of some quantized property of a subatomic particle, atom, or molecule and can only have certain integral or half-integral values.

Answer

It is very important in chemistry, as it species the shape of the orbital, bond angle, chemical bonds.. ..

Answer

Azimuthal quantum number is the number which determines the sub shell in a principal energy shell to which an electron belongs. In the periodic table of the elements, a period indicates the value of the principal quantum number (n) for the outermost shells.

Answer

Quantum number

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Quantum mechanics
Part of a series of articles about
$i\hbar {\frac {\partial }{\partial t}}\|\psi (t)\rangle ={\hat {H}}\|\psi (t)\rangle$ Schr�dinger equation
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v t e

Quantum numbers describe values of

conserved quantities

in the dynamics of a

quantum system

. In the case of electrons, the quantum numbers can be defined as 'the sets of numerical values which give acceptable solutions to the

Schr�dinger wave equation

for the

hydrogen

atom

'. An important aspect of

quantum mechanics

is the

quantization

of observable quantities, since quantum numbers are

discrete sets of integers

or half-integers, although they could approach

infinity

in some cases. This distinguishes quantum mechanics from

classical mechanics

where the values that characterize the system such as mass, charge, or momentum, range continuously. Quantum numbers often describe specifically the

energy levels

of

electrons

in

atoms

, but other possibilities include

angular momentum

,

spin

, etc. An important family is

flavour quantum numbers

–

internal

quantum numbers which determine the type of a particle and its interactions with other particles through the

forces

. Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.

Contents

[hide]

1
How many quantum numbers exist?
2
Spatial and angular momentum numbers
- 2.1
  Traditional nomenclatures
  - 2.1.1
    Hund-Mulliken molecular orbital theory
3
Total angular momenta numbers
- 3.1
  Total momentum of a particle
- 3.2
  Nuclear angular momentum quantum numbers
4
Elementary particles
5
Multiplicative quantum numbers
6
See also
7
References and external links
- 7.1
  General principles
- 7.2
  Atomic physics
- 7.3
  Particle physics

How many quantum numbers exist?

[

edit

]

The question of how many quantum numbers are needed to describe any given system has no universal answer. Hence for each system one must find the answer for a full analysis of the system. A quantized system requires at least one quantum number. The dynamics of any quantum system are described by a quantum

Hamiltonian

, H. There is one quantum number of the system corresponding to the energy, i.e., the

eigenvalue

of the Hamiltonian. There is also one quantum number for each

operator

O that commutes with the Hamiltonian. These are all the quantum numbers that the system can have. Note that the operators O defining the quantum numbers should be

independent

of each other. Often, there is more than one way to choose a set of independent operators. Consequently, in different situations different sets of quantum numbers may be used for the description of the same system. we have four kinds of quantum number 1:principal 2:angular 3:magnetic 4:spin

Spatial and angular momentum numbers

[

edit

]

Single electron orbitals for hydrogen-like atoms with quantum numbers n=1,2,3 (blocks), ℓ (rows) and m (columns). The spin s is not visible, because it has no spatial dependence.

Four quantum numbers can describe an electron in an atom completely. As per the following model, these nearly-compatible quantum numbers are:

Principal quantum number
(n)
Azimuthal quantum number
(ℓ)
Magnetic quantum number
(m)
Spin quantum number
(s)

The spin-orbital interaction, however, relates these numbers. Thus, a complete description of the system can be given with fewer quantum numbers, if orthogonal choices are made for these basis vectors.

Traditional nomenclatures

[

edit

]

Hund-Mulliken molecular orbital theory

[

edit

]

Many different models have been proposed throughout the

history of quantum mechanics

, but the most prominent system of nomenclature spawned from the Hund-Mulliken

molecular orbital

theory of

Friedrich Hund

,

Robert S. Mulliken

, and contributions from

Schr�dinger

,

Slater

and

John Lennard-Jones

. This system of nomenclature incorporated

Bohr

energy levels, Hund-Mulliken orbital theory, and observations on electron spin based on

spectroscopy

and

Hund's rules

.^[1]

This model describes electrons using four quantum numbers, n, ℓ, m_ℓ, m_s, given below. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons). A quantum description of

molecular orbitals

require different quantum numbers, because the

Hamiltonian

and its symmetries are quite different.

The
principal quantum number
(n) describes the
electron shell
, or energy level, of an electron. The value of n ranges from 1 to the shell containing the outermost electron of that atom, i.e.^[2]
$n = 1, 2, ...$ .
For example, in
caesium (Cs)
, the outermost
valence
electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6.
For particles in a time-independent potential (see
Schr�dinger equation
), it also labels the nth eigenvalue of Hamiltonian (H), i.e. the energy, E with the contribution due to angular momentum (the term involving J²) left out. This number therefore has a dependence only on the distance between the electron and the nucleus (i.e., the radial coordinate, r). The average distance increases with n, and hence quantum states with different principal quantum numbers are said to belong to different shells.
The
azimuthal quantum number
(ℓ) (also known as the angular quantum number or orbital quantum number) describes the
subshell
, and gives the magnitude of the orbital
angular momentum
through the relation
$L 2 = ħ 2 ℓ (ℓ + 1)$ .
In chemistry and spectroscopy, 'ℓ = 0' is called an s orbital, 'ℓ = 1' a p orbital, 'ℓ = 2' a d orbital, and 'ℓ = 3' an f orbital.
The value of ℓ ranges from 0 to n − 1, so the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) appears in the third shell (n= 3), and so on:^[3]
$ℓ = 0, 1, 2,..., n - 1$ .
A quantum number beginning in 3, 0, … describes an electron in the s orbital of the third electron shell of an atom. In chemistry, this quantum number is very important, since it specifies the shape of an
atomic orbital
and strongly influences
chemical bonds
and
bond angles
.
The
magnetic quantum number
(m_ℓ) describes the specific
orbital
(or 'cloud') within that subshell, and yields the projection of the orbital
angular momentum
along a specified axis:
$L z = m ℓ ħ$ .
The values of m_ℓ range from −ℓ to ℓ, with integer intervals:^[4]
The s subshell (ℓ = 0) contains only one orbital, and therefore the m_ℓ of an electron in an s orbital will always be 0. The p subshell (ℓ = 1) contains three orbitals (in some systems, depicted as three 'dumbbell-shaped' clouds), so the m_ℓ of an electron in a p orbital will be −1, 0, or 1. The d subshell (ℓ = 2) contains five orbitals, with m_ℓ values of −2, −1, 0, 1, and 2.
The
spin projection quantum number
(m_s) describes the spin (intrinsic
angular momentum
) of the electron within that orbital, and gives the projection of the
spin
angular momentum S along the specified axis:
$S z = m s ħ$ .
In general, the values of m_s range from −s to s, where s is the
spin quantum number
, an intrinsic property of particles:^[5]
$m s = - s, - s + 1, - s + 2, ..., s - 2, s - 1, s$ .
An electron has spin number s = �, consequently m_s will be ��, referring to 'spin up' and 'spin down' states. Each electron in any individual orbital must have different quantum numbers because of the
Pauli exclusion principle
, therefore an orbital never contains more than two electrons.

Note that there is no universal fixed value for m_ℓ and m_s values. Rather, the m_ℓ and m_s values are random. The only requirement is that the naming schematic used within a particular set of calculations or descriptions must be consistent (e.g. the orbital occupied by the first electron in a p orbital could be described as m_ℓ = −1 or m_ℓ = 0 or m_ℓ = 1, but the m_ℓ value of the next unpaired electron in that orbital must be different; yet, the m_ℓ assigned to electrons in other orbitals again can be m_ℓ = −1 or m_ℓ= 0, or m_ℓ = 1 ).

These rules are summarized as follows:

Name	Symbol	Orbital meaning	Range of values	Value examples
principal quantum number	n	shell	1 ≤ n	n = 1, 2, 3, …
azimuthal quantum number ( angular momentum )	ℓ	subshell (s orbital is listed as 0, p orbital as 1 etc.)	0 ≤ ℓ ≤ n − 1	for n = 3: ℓ = 0, 1, 2 (s, p, d)
magnetic quantum number , (projection of angular momentum )	m_ℓ	energy shift (orientation of the subshell's shape)	−ℓ ≤ m_ℓ ≤ ℓ	for ℓ = 2: m_ℓ = −2, −1, 0, 1, 2
spin projection quantum number	m_s	spin of the electron (−� = 'spin down', � = 'spin up')	−s ≤ m_s ≤ s	for an electron s = �, so m_s = −�, �

Example: The quantum numbers used to refer to the outermost

valence

electrons

of the

Carbon

(C)

atom

, which are located in the 2p

atomic orbital

, are; n = 2 (2nd electron shell), ℓ = 1 (p orbital

subshell

), m_ℓ = 1, 0 or −1, m_s = � (parallel spins).

Results from

spectroscopy

indicated that up to two electrons can occupy a single orbital. However two electrons can never have the same exact quantum state nor the same set of quantum numbers according to

Hund's rules

, which addresses the

Pauli exclusion principle

. A fourth quantum number with two possible values was added as an ad hoc
assumption to resolve the conflict; this supposition could later be explained in detail by relativistic quantum mechanics and from the results of the renowned

Stern–Gerlach experiment

.

Total angular momenta numbers

[

edit

]

Total momentum of a particle

[

edit

]

When one takes the

spin–orbit interaction

into consideration, the L and S operators no longer

commute

with the

Hamiltonian

, and their eigenvalues therefore change over time. Thus another set of quantum numbers should be used. This set includes^[6]^[7]

The
total angular momentum quantum number
:
$j = | ℓ � s |,$
which gives the total
angular momentum
through the relation
$J 2 = ħ 2 j (j + 1)$ .
The
projection of the total angular momentum along a specified axis
:
$m j = - j, - j + 1, - j + 2, ..., j - 2, j - 1, j$
analogous to the above and satisfies
$m j = m ℓ + m s$ and $| m ℓ + m s | \leq j$ .
Parity
This is the
eigenvalue
under reflection: positive (+1) for states which came from even ℓ and negative (−1) for states which came from odd ℓ. The former is also known as even parity and the latter as odd parity, and is given by
$P = (-1) ℓ$ .

For example, consider the following 8 states, defined by their quantum numbers:

#	n	ℓ	m_ℓ	m_s	ℓ + s	ℓ − s	m_ℓ + m_s
1.	2	1	1	+1/2	3/2	~~1/2~~	3/2
2.	2	1	1	−1/2	3/2	1/2	1/2
3.	2	1	0	+1/2	3/2	1/2	1/2
4.	2	1	0	−1/2	3/2	1/2	−1/2
5.	2	1	−1	+1/2	3/2	1/2	−1/2
6.	2	1	−1	−1/2	3/2	~~1/2~~	−3/2
7.	2	0	0	+1/2	1/2	−1/2	1/2
8.	2	0	0	−1/2	1/2	−1/2	−1/2

The [quantum state]s in the system can be described as linear combination of these 8 states. However, in the presence of

spin–orbit interaction

, if one wants to describe the same system by 8 states that are

eigenvectors

of the

Hamiltonian

(i.e. each represents a state that does not mix with others over time), we should consider the following 8 states:

j	m_j	parity
3/2	3/2	odd	(coming from state (1) above)
3/2	1/2	odd	(coming from states (2) and (3) above)
3/2	−1/2	odd	(coming from states (4) and (5) above)
3/2	−3/2	odd	(coming from state (6) above)
1/2	1/2	odd	(coming from states (2) and (3) above)
1/2	−1/2	odd	(coming from states (4) and (5) above)
1/2	1/2	even	(coming from state (7) above)
1/2	−1/2	even	(coming from state (8) above)

Nuclear angular momentum quantum numbers

[

edit

]

In

nuclei

, the entire assembly of

protons

and

neutrons

(

nucleons

) has a resultant angular momentum due to the angular momenta of each nucleon, usually denoted '

I

'. If the total angular momentum of a neutron is j_n = ℓ + s and for a proton is j_p = ℓ + s (where s for protons and neutrons happens to be � again) then the nuclear angular momentum quantum numbers

I

are given by:

I = | j n - j p |, | j n - j p | + 1, | j n - j p | + 2, ..., (j n + j p) - 2, (j n + j p) - 1, (j n + j p)

Parity with the number

I

is used to label nuclear angular momentum states, examples for some isotopes of

Hydrogen

(H),

Carbon

(C), and

Sodium

(Na) are;^[8]

1 1 H	$I$ = (1/2)⁺	9 6 C	$I$ = (3/2)⁻	20 11 Na	$I$ = 2⁺
2 1 H	$I$ = 1⁺	10 6 C	$I$ = 0⁺	21 11 Na	$I$ = (3/2)⁺
3 1 H	$I$ = (1/2)⁺	11 6 C	$I$ = (3/2)⁻	22 11 Na	$I$ = 3⁺
		12 6 C	$I$ = 0⁺	23 11 Na	$I$ = (3/2)⁺
		13 6 C	$I$ = (1/2)⁻	24 11 Na	$I$ = 4⁺
		14 6 C	$I$ = 0⁺	25 11 Na	$I$ = (5/2)⁺
		15 6 C	$I$ = (1/2)⁺	26 11 Na	$I$ = 3⁺

The reason for the unusual fluctuations in

I

, even by differences of just one nucleon, are due to the odd/even numbers of protons and neutrons - pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd/even numbers of unpaired nucleons. The property of nuclear spin is an important factor for the operation of

NMR

spectroscopy in

organic chemistry

,^[7] and

MRI

in

nuclear medicine

,^[8] due to the

nuclear magnetic moment

interacting with an external

magnetic field

.

Elementary particles

[

edit

]

Elementary particles

contain many quantum numbers which are usually said to be intrinsic to them. However, it should be understood that the elementary particles are

quantum states

of the

standard model

of

particle physics

, and hence the quantum numbers of these particles bear the same relation to the

Hamiltonian

of this model as the quantum numbers of the

Bohr atom

does to its

Hamiltonian

. In other words, each quantum number denotes a symmetry of the problem. It is more useful in

quantum field theory

to distinguish between

spacetime

and

internal

symmetries.

Typical quantum numbers related to

spacetime symmetries

are

spin

(related to rotational symmetry), the

parity

,

C-parity

and

T-parity

(related to the

Poincar� symmetry

of

spacetime

). Typical internal symmetries^{[clarification needed]} are

lepton number

and

baryon number

or the

electric charge

. (For a full list of quantum numbers of this kind see the article on

flavour

.)

Multiplicative quantum numbers

[

edit

]

A minor but often confusing point is as follows: most conserved quantum numbers are additive, so in an elementary particle reaction, the sum of the quantum numbers should be the same before and after the reaction. However, some, usually called a parity
, are multiplicative; i.e., their product is conserved. All multiplicative quantum numbers belong to a symmetry (like parity) in which applying the symmetry transformation twice is equivalent to doing nothing (

involution

)

Answer

Magnetic Quantum no.

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