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# Quantum Numbers

Top
 Sub Topics When Schrodinger equation is applied to the motion of electron in an atom, it is found that the quantum state $\psi$ of electron is characterized by four numbers called quantum numbers. These quantum numbers are principal quantum number n, orbital quantum number l, magnetic quantum number m and spin quantum number m.The solution of Schrodinger wave equation called wave function gives all kind of information about the electron in the atom. Quantum states and the interactions between them are conveniently described by quantum numbers. In most cases, the quantum number of a system following a process will not change that is quantum number is conserved.

## Quantum Numbers Definition

"Quantum number is the number expressing the value of some property of a particle which occurs in quanta." The energy level to which each electron belongs is identified by the list of four quantum numbers.
The four quantum numbers are the
• Principle quantum number (n)
• Azimuthal quantum number or angular momentum quantum number (l)
• Magnetic quantum number (ml)
• Spin quantum number (mx)

## Four Quantum Numbers

Quantum number have definite rules. The Quantum Number Rules are mentioned below.
1. Principal quantum number - assigned values of n = 1,2,3... correspond to the principal energy level of the electron.
2. Orbital quantum number - assigned values of l = 0,1,2,3...n-1 correspond to the values of the angular momentum of the electron.
3. Magnetic quantum number - assigned values of whole numbers from ml = -l to +l correspond to the possible properties of an electron in a magnetic field.
4. Spin quantum number mx - +$\frac{1}{2}$ or -$\frac{1}{2}$ corresponds to the possible spin vectors or orientations of an electron in a magnetic field with spin vectors classified as spin up (+ or $\uparrow$ ) and spin down (- or $\downarrow$ ).

## Quantum Numbers Chart

When designating the quantum numbers for the electrons in an atom the lowest possible values of the first three quantum numbers are used first. The quantum numbers for the first 20 elements are listed below in the table.

 Element Principal quantum number (n) Azimuthal quantum number (l) Magnetic quantum number (ml) Spin quantum number (ms) H 1 0 0 +$\frac{1}{2}$ He 1 0 0 -$\frac{1}{2}$ Li 2 0 0 +$\frac{1}{2}$ Be 2 0 0 -$\frac{1}{2}$ B 2 1 -1 +$\frac{1}{2}$ C 2 1 0 +$\frac{1}{2}$ N 2 1 +1 +$\frac{1}{2}$ O 2 1 -1 -$\frac{1}{2}$ F 2 1 0 -$\frac{1}{2}$ Ne 2 1 +1 -$\frac{1}{2}$ Na 3 0 0 +$\frac{1}{2}$ Mg 3 0 0 -$\frac{1}{2}$ Al 3 1 -1 +$\frac{1}{2}$ Si 3 1 0 +$\frac{1}{2}$ P 3 1 +1 +$\frac{1}{2}$ S 3 1 -1 -$\frac{1}{2}$ Cl 3 1 0 -$\frac{1}{2}$ Ar 3 1 +1 -$\frac{1}{2}$ K 4 0 0 +$\frac{1}{2}$ Ca 4 0 0 -$\frac{1}{2}$

## Electron Quantum Numbers

Electrons in an atom are described by four quantum numbers.
• The four electron shells surrounding the nucleus are usually designated K, L, M and N and are assigned the principal quantum number n where n equals 1, 2, 3 and 4 respectively.
• These electron shells are split into sub shells, each with its own quantized energy level. These sub shells are labelled s, p, d and f. These sub shells are assigned the quantum number l which can have values l = 0, 1, 2...(n-1) where n is principal quantum number.
• Within each sub shell there are a number of possible orbitals. The number of orbitals in a sub shell equals 2l+1, where l is the sub shell quantum number. The energy values of the orbitals in a sub shell are normally degenerate. These energy levels are then described by the quantum number m which can have values from -l to +l.
• Each electron also has a spin quantum number s which can have one of two values +$\frac{1}{2}$ or -$\frac{1}{2}$. Thus every electron has its own set of quantum numbers describing its energy level. The four quantum numbers of all electrons in the first three shells and their energy levels are shown below.

## Periodic Table with Quantum Numbers

Each electron in an atom is described by a set of four quantum numbers. n represents the period of the periodic table while also noting that the d and f electrons have n quantum numbers that are 1 and 2 units less than the period in which they are found. Next the l quantum number represents various "blocks" within the periodic table similar to the periodic table shown below.

Finally the possible values of ml and mx result in the number of elements that can occupy each of the blocks mentioned. These blocks can also be designated with the letters s, p, d and f.

## Principal Quantum Number

This quantum number was introduced by Bohr and is denoted by n. The principal quantum number (n) can have any integer value from 1 to infinity. As the name implies it is the most important quantum number because the value of n determines the total energy of electron.

$E_{n} = \frac{2\pi^{2}Z^{2}me^{4}}{n^{2}h^{2}}$

This relation is similar to the expression given by Bohr.

Where
• m = Mass of electron
• e = Charge on electron
• h = Plancks constant
• En = Energy of electron in nth shell
• n = Principal quantum number
• Z = Atomic number

## Angular Momentum Quantum Number

Angular momentum quantum number or Azimuthal Quantum Number (l) = 0, 1, 2, ...(n-1) was first proposed by Sommerfield. This quantum number determines angular momentum of the electron as

$Angular\ momentum = \sqrt{l(l+1)}\times\frac{h}{2\pi}$

The principal quantum number gives total energy and position of an electron in general. A part of this energy however must be associated with the orbital motion of electron. This orbital motion is described by the orbital angular momentum of the electron. Therefore it is known as orbital angular momentum quantum number.

Various sub levels are designated as s, p, d and f depending upon the value of l as shown below.

 Value of l 0 1 2 3 4 Designation of sub shells s p d d f

## Magnetic Quantum Number

It explains the behavior of an electron in the external magnetic field or in other words it tells about orbitals of the electrons. It is denoted by the symbol ml. This quantum number refers to different orientations of electron cloud in a particular sub shell. These different orientations are called the orbitals.

The number of orbitals in a particular sub shell within a principal energy level is given by the number of values allowed to "m" which in turn depend upon the value of "l". The possible values of "m" are all integral values from +l through 0 to -l thus making a total of (2l+1) number of values.

 Energy level Principal quantum number Possible values of (l) Possible values of (m) K 1 0 0 L 2 0, 1 0 +1, 0, -1 M 3 0, 1, 2 0 +1, 0, -1+2, +1, 0, -1, -2

It can have the value 0, $\pm$1, $\pm$2, $\pm$3......$\pm$l. The value of n for a shell limits the number of sub shells that are possible within that shell. That is l can be no larger than n-1.

Magnetic quantum number defines orientation of electron in space which is given or explained on the basis of Zeeman effect.

## Spin Quantum Number

Spin quantum number was introduced by Uhlenbeck and Goudsmit in 1925. It id denoted by mx = +$\frac{1}{2}$ and -$\frac{1}{2}$. It does no follow wave mechanical treatment but arises from the spectral evidences That the electron in its motion around the nucleus also rotates or spins about its own axis. It may be either clockwise or anticlockwise.

The spin quantum number can have only two values +$\frac{1}{2}$ and -$\frac{1}{2}$ . The +$\frac{1}{2}$ value indicates the clockwise spin shown by an arrow pointing upward, and the -$\frac{1}{2}$ indicates the anticlockwise spin shown by an arrow pointing downward.

The spin angular momentum of an electron is represented as

$Spin\ Angular\ Momentum = \sqrt{S(S+1)}\times \frac{h}{2\pi}$

The spin of electrons in its own axis is shown below.

## Quantum Numbers Problems

Some of the solved Quantum Numbers Examples are given below.

### Solved Examples

Question 1: What sub shells are possible in n = 3 energy levels?
Solution:

Sub shells in n = 3 energy level
For n = 3, the possible values of l are 0,1 and 2.

The corresponding sub shells are
l = 0, s sub shell
l = 1, p sub shell
l = 2, d sub shell

Question 2: What are the possible values of l for an electron in
• Third energy level
• 3d sub shell

Solution:

• For third energy level n=3. Since l may have values from 0 to n-1, the possible values of l are 0, 1 and 2.
• For d sub shell l=2, hence in 3d sub shell the only possible value of l is 2.