Quantum Number: Definition, Types, Uses, Principle

Quantum Number: We all have a specific address where people can find us. Similarly, electrons inside an atom also have an address where we can find an electron. So, quantum numbers are the way to represent the address of an electron as we have the name of the country, state, city, etc. There are four quantum numbers viz. principal, azimuthal, magnetic and spin quantum numbers. With the help of quantum numbers, we can derive the approximate position and energy of the electron in an atom. Please note that it is not possible to exactly calculate the position of an electron with 100% accuracy.

In this article, we will be discussing all quantum numbers, their types, and their uses.

What are Quantum Numbers?

Atoms have many orbitals around the nucleus differing in their shape, size, and orientation in space. The different characteristics of orbitals are represented primarily by the principal quantum number, Azimuthal quantum number, and magnetic quantum number, and these numbers are derived from Schrodinger’s wave equations. The spin quantum number on the other side shows the spin or rotation of the electron along its axis only and does not denote any characteristic of the orbital.

Quantum number may be defined as a set of four numbers with the help of which we can get complete information about all the electrons in an atom, i.e., location, energy, the type of occupied, shape, and orientation of the orbital, etc. Quantum numbers distinguish different orbitals based on size, shape, and orientation in space.

Quantum Mechanics

After the failure of Bohr’s model of the atom due to its inability of explaining Heisenberg’s uncertainty principle and dual nature of particles and the failure of classical physics for microscopic particles, there was a need for a new branch of science to be introduced called “Quantum Mechanics”.

Quantum mechanics was developed by Erwin Schrodinger in \(1926,\) keeping account of wave nature associated with particles like electrons. For the wave motion of electrons in a three-dimensional space around the nucleus, he put forward the wave equation known as Schrodinger’s wave equation, Quantum mechanics.

\({\text{-}}\frac{{{{\text{h}}^{\text{2}}}}}{{{\text{2}}{{\text{m}}_{\text{e}}}}}\left({\frac{{{\partial ^{\text{2}}}{\text{? }}}}{{\partial {{\text{x}}^{\text{2}}}}}{\text{ + }}\frac{{{\partial ^{\text{2}}}{\text{? }}}}{{\partial {{\text{y}}^{\text{2}}}}}{\text{ + }}\frac{{{\partial ^{\text{2}}}{\text{? }}}}{{\partial {{\text{z}}^{\text{2}}}}}} \right){\text{ + V(x,y,z)? =E? }}\).

Or

\(\rm{\hat H{\text{?}}} = E{\text{?}}\) (General form of Equation)

In wave equation, \( {\text{?}}\) is the magnitude of amplitude of the wave, \(\text{E}\) is the electron’s energy, and \(\text{H}\) is the mathematical operator called Hamiltanion Operator. After solving the equation, we get \(\text{E}\) and \( {\text{?}}\), where \(\text{E}\) is quantized, and \( {\text{?}}\) is the wave functions.

The wave function of the electron does not have any physical significance. Still, the \( { {\text{?}}^2}\) denotes the intensity of electron at a particular position or, as in Heisenberg’s uncertainty principle, it represents the probability of electron at a position and the position where the probability of finding an electron is high \(90\% \) is known as the orbital of an atom.

Types of Quantum Number

The four-quantum number completely specifies an electron in an atom or gives a complete address of an electron in an atom. The four quantum numbers are:

• Principal quantum number \(\text{(n)}\)
• Azimuthal or secondary or angular quantum number \(\text{(l)}\)
• Magnetic quantum number \(\left({{{\text{m}}_{\text{1}}}}  \right)\)
• Spin quantum number \(\text{(s)}\)

1. Principal Quantum Number (n)

Principle quantum number is one of the most critical quantum numbers that tells about an electron’s principal energy level, orbit, or shell. It is denoted by the letter “\(\text{n}\)” and can only have an integral value except zero, i.e., \(\text{n} = 1,2,3,4…\) The different principal quantum numbers are also represented by the letters \(\text{K, L, M, N}\), etc., starting from the nucleus. The number also helps understand the lines of the spectrum based on electron jumps within the shells.

The principal quantum number gives the following information:

1. It determines the size of the electron cloud i.e., it gives the average distance of the electron from the nucleus.
2. It determines the energy of the electron in the hydrogen atom and hydrogen-like particles (i.e., which contain only one electron) by the following equation:
   \({{\rm{E}}_{\rm{n}}}{\rm{ = }}\,{\rm{ – }}\frac{{{\rm{2}}{{\rm{\pi }}^{\rm{2}}}{\rm{m}}{{\rm{e}}^{\rm{4}}}{{\rm{Z}}^{\rm{2}}}}}{{{{\rm{n}}^{\rm{2}}}{{\rm{h}}^{\rm{2}}}}}\)
   Where \(\text{m}\) is the mass, \(\text{e}\) is the charge of the electron, \(\text{Z}\) is the atomic number, n is the principal shell number, and \(\text{h}\) is Planck’s constant.
   The energies of various shells follow the sequence: \(\text{K < L < M < N < O}\,…..\).
3. The maximum number of electrons present in any principal shell is given by \(\text{2n}^2\), where \(\text{n}\) is the number of the principal shell.

2. Azimuthal Quantum Number or Angular Momentum or Subsidiary (l) Quantum Number

From the line spectrum, it was well observed that the spectrum has main lines and some fine lines. To explain the fine lines obtained in the spectrum, and was suggested that electrons in any shell of multiple electron atoms do not have the same energy as they move on a different path and have different angular momentum. Hence within the same shell, subshells or sub-energy levels are present.
The azimuthal quantum number gives us various other information as well, like

1. The number of subshells present in the main shell of an atom.
2. The angular momentum of electrons present in the subshell.
3. The energies or relative energies of subshells.
4. The shape of the various subshells in the main shell.

The azimuthal number is denoted by the letter “\(\text{l}\)”, and the value of the subshell is an integral value that can range from \(0\) to \(\text{n}-1\).

For example, for the first shell \(\text{(K)}\), \(\text{n} = 1\), \(\text{l}\) can have only one value, i.e., \(\text{l} = 0\) and for the second shell \(\text{(L)}\), \(\text{n} = 2\), \(\text{l}\) can have two value, i.e., \(\text{l} = 0\) and \(1\).

Depending upon the value of \(\text{l}\), i.e., \(\text{l} = 0, 1, 2\), and \(3\), the different subshells are designated as \(\text{s, p, d}\), and \(\text{f}\), respectively. These notations are the initial letters of the words, sharp, principal, diffused, and fundamental, formerly used to describe different spectral lines.

The number of subshells present in any principal shell is equal to the number of the principal shell or the principal quantum number. The possible subshell in the first four shells and their designation is summed up in the following table.

Due to the motion of the electron in an orbital, it possesses angular momentum. Angular momentum of the electron in an orbital \({\text{=}}\sqrt {{\text{l(l + 1)}}} \frac{{\text{h}}}{{{{2\pi }}}}\), where \(\text{l}\) is the azimuthal quantum number and \(\text{h}\) is Plank’s constant.

3. Magnetic Quantum Number (m or m₁)

This quantum number is present to explain the Zeeman effect, where the spectral lines split up into more lines whenever present in the influence of the magnetic field. The magnetic quantum number determines the number of preferred orientations of the electrons present in a subshell. The magnetic quantum number is denoted by the letter “\(\text{m}\)” or “\(\text{m}_\text{l}\)”, and the value for a given value of “\(\text{l}\)” is in the range of \(\text{-l}\) to \(\text{l}\), including zero. Hence the number of values of m is \(\text{2l}+1\).

So, for \(\text{l}=0\), the number of values of \(\text{m}\) is \(= {\rm{2l}} + 1{\rm{ }} = {\rm{ }}2\left( 0 \right){\rm{ + }}1{\rm{ }} = {\rm{ }}1\), i.e., \(\text{m}=0\)
for \(\text{l}=1\), the number of values of \(\text{m}\) is \(= {\rm{2l}}\,{\rm{ + }}\,1{\rm{ }} = {\rm{ }}2\left( 1 \right) + 1{\rm{ }} = {\rm{ }}3\) i.e., \({\rm{m}} = \, – 1,0,1\)
for \(\text{l}=2\), the number of values of \(\text{m}\) is \( = {\rm{2l}}\, + 1 = {\rm{ }}2\left( 2 \right) + 1 = 5\) i.e., \({\rm{m}} = \, – 2,{\rm{ }} – 1,{\rm{ }}0,{\rm{ }}1,{\rm{ }}2\) and so on

The values of \(\text{m}_\text{l}\) can be summarised as follows:

4. Spin Quantum Number (s)

This quantum number is to know that electrons rotate on their own axis and revolve around the nucleus. This number gives information about the spin direction of an electron. It is denoted by the letter “\(\text{s}\)”. As the electron spins clockwise or anti-clockwise only, spin, the quantum number has two values only, i.e. \( + 1/2\) and \( – 1/2\) and can be represented by arrows pointing upwards \(↑\) and downwards \(↓\).

This quantum number helps in determining the magnetic nature of a substance. The spinning electron behaves like a micro magnet, and two electrons with opposite spins cancel each other’s magnetism. If an atom has all fully-filled orbitals, the net magnetic moment is zero, and the atom is diamagnetic. If an atom has half-filled orbitals, the magnetic moment will be maximum, and the atom is paramagnetic.

The electron is associated with the magnetic field, hence the electron has a magnetic moment. The spin magnetic moment can be calculated using the formula, \({\rm{\mu }_{\text{s}}}{\text{=}}\sqrt {{\text{n(n + 2)}}} {\text{BM}},\) number of unpaired electrons and \(\text{BM} =\) Bohr magneton, a unit of the magnetic moment.

The values of \(\text{n, m}\), and \(\text{m}_\text{l}\) can be summed up as follows:

Limitations of Quantum Numbers

There are some restrictions and limitations to quantum numbers. The accuracy with which they can predict the position and energy of electrons is governed by the following principles:

1. Pauli’s Exclusion Principle: This is a theory of quantum mechanics that was propounded by Wolfgang Pauli in 1925. (Exclusion means to leave, apply a different rule, etc.) According to this principle:

(i) No two identical fermions can be in the same quantum state at the same time. For electrons in the same atom, this law states that “no two electrons can have the same four (ie all) quantum numbers”.
(ii) According to this principle, no two particles (whose spin, colour charge, angular momentum, etc. are in the same state or share similar properties) can stay in the same place at the same time.
(iii) The particles which follow this principle are called fermions, such as electrons, atoms, neutrons etc.; and the particles which do not obey this principle are called bosons, such as photons, gluons, gauge bosons.

2. Hund’s Rule: According to this rule “First, one electron is filled in all the sub-orbitals of any orbital and after that pairing starts, that is, pairing is made later, first one electron is filled in all the sub-orbitals.” When an orbital is half full or completely full, then this orbital is comparatively more stable. Therefore, according to Hund’s law, all the sub-orbitals of any orbital are filled with one electron first and their pairing starts.

This Hund’s law is also called the law of maximum modalities.

3. Heisenberg’s Uncertainty Principle: According to this principle, it is not possible to find the value of both the position and the momentum of microscopic particles like electrons, protons, etc. simultaneously at any given moment.

Electron Configuration of Elements

Here we have provided the electronic configuration of the 1st, 2nd, 3rd, and 4th-row elements.

Solved Examples on Quantum Numbers

Q.1. An electron is present in \(\text{4f}\) sub-shell. Write the possible values for the quantum numbers.
Ans: For \(\text{4f}\)-electron, principal quantum number \(\text{(n)} = 4\)
Azimuthal quantum number \(\text{(l)} = 3\)
Magnetic quantum number \(\left( {{{\rm{m}}_{\rm{l}}}} \right) = – 3, – 2, – 1,0, + 1, + 2, + 3\)
Spin quantum number \(\text{(s)}= + \frac{1}{2}\) or \(\frac{1}{2}\) for any of these orbitals.

Q.2. An atomic orbital has \(\text{n}=3\), what are the possible values of \(\text{l}\)?
Ans: For \(\text{n} = 3\), \(\text{l}\) can have three possible values. These are: \(\text{l}=0\), \(\text{l}=1\), \(\text{l}=2\).

Q.3. Using \(\text{s, p}\) and \(\text{d}\) notations, describe the orbitals with following quantum numbers:
a) \(\text{n} = 1\), \(\text{l} = 0\)
b) \(\text{n} = 3\), \(\text{l} = 1\)
c) \(\text{n} = 4\), \(\text{l} = 2\)
Ans: a) \(\text{n} = 1\), \(\text{l} = 0\) is \(\text{1s}\) orbital.
b) \(\text{n} = 3\), \(\text{l} = 1\) is \(\text{3p}\) orbital.
c) \(\text{n} = 4\), \(\text{l} = 2\) is \(\text{4d}\) orbital.

Q.4. What are the possible values of \(\text{m}_\text{l}\), for \(\text{l} = 3\)?
Ans: For \(\text{l} = 3\), \(\text{m}_\text{l}\) can have seven possible values.
These are: \(-3, -2, -1, 0, +1, +2, +3\).

Q5. If an electron jumped from energy level n = 5 to energy level n = 3, did absorption or emission of a photon occur?
Ans: Emission, because energy is lost by the release of a photon.

Q6. List the possible combinations of all four quantum numbers when n=2, l=1, and m_l=0.
Ans: The fourth quantum number is independent of the first three, allowing the first three quantum numbers of two electrons to be the same. Since the spin can be +1/2 or =1/2, there are two combinations:

• n=2, l=1, m=0, m_l=0, ms=+1/2
• n=2, l=1, m=0, m_l=0, ms=−1/2

Summary

In this article, you have understood four types of quantum numbers i.e., principal quantum number \(\rm{(n)}\), azimuthal or secondary or angular quantum number \(\rm{(l)}\), magnetic quantum number \((\rm{m}_\rm{l})\), and spin quantum number \(\rm{(s)}\) and their applications.

FAQs on Quantum Numbers

Q.1. Who proposed the principal quantum number?
Ans: Schrodinger proposes the concept of quantum numbers and quantum mechanics after the failure of Bohr’s model with the help of wave equations and wave functions.

Q.2. What is the spin of an electron?
Ans: This number gives information about the spin direction of an electron. It is denoted by the letter “\(\rm{s}\)”. As the electron spins clockwise or anti-clockwise only, spin, the quantum number has two values only, i.e. \( + 1/2\) and \( – 1/2\) and can be represented by arrows pointing upwards \(↑\) and downwards \(↓\).

Q.3. Which energy level has the least energy?
Ans: The 1s orbital, which is the closest to the nucleus of an atom, has the least energy, and the energy keeps on increasing as the electrons keep moving away from the nucleus.

Q.4. What are the four quantum numbers?
Ans: Different quantum numbers represent the different characteristics of orbitals like shape, size, and orientation.
1. Principle quantum number- It talks about the electron’s principal energy level, orbit, or shell.
2. Azimuthal quantum number-It talks about the number of subshells present in the main shell of an atom.
3. Magnetic quantum number- It indicates the number of preferred orientations of the electrons present in a subshell.
4. Spin quantum number- It gives information about the spin direction of an electron.

Q.5. How are quantum numbers calculated?
Ans: The principal quantum is calculated as the number of the shell, \(\rm{n}.\)
The azimuthal quantum number is calculated by \(\rm{l = n-1}.\)
The magnetic quantum number is calculated by \(\rm{m = -l}\) to \(\rm{l}\)
The spin quantum number has two values, only \( + 1/2\) and \( – 1/2.\)

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