Quantum Numbers

An Introduction to Quantum Numbers

Quantum numbers are used to define the trajectory and movement of an electron within an atom. Additionally, the quantum numbers of every electron in an atom are combined; it should obey the Schrodinger equation.

Notably, this is a crucial topic in your syllabus. Not only do you need to learn about this topic for your syllabus, but also because it is vital for future curriculum in various examinations. Consequently, do learn the significance of quantum numbers in detail.


What are Quantum Numbers?

Quantum number is a value, which is used to describe the energy available in atoms and molecules. Moreover, an electron within an ion or an atom has four such numbers to portray its energy state. Additionally, it yields clarifications to the wave equation for hydrogen atoms by Schrodinger. These four quantum numbers are –

  • n is the principal quantum number, and it expresses the energy levels.

  • ℓ is azimuthal or angular momentum quantum number; it describes the subshell.

  • mℓ or m is a magnetic quantum number which expresses the orbital of a subshell.

  • ms or s is the spin quantum number expressing the spin.

Furthermore, every electron in an atom has a set of unique quantum numbers. According to the ‘Pauli exclusion principle’, these four quantum numbers and their combination cannot be the same in two electrons. 

Additionally, these quantum numbers are crucial, as they are used to comprehend the electron configuration of an atom. Besides, they also give an idea of the possible location of an atom’s electrons. Moreover, the significance of quantum number is that it is useful to gather information on characteristics of atoms like its radius, and ionisation energy.


Values of Quantum Numbers

As mentioned above, the ‘Pauli exclusion principle’ mentions that two electrons within an atom cannot have similar sets of quantum numbers. Hence, every number has either an integer or a half-integer value. Therefore, the values of these quantum number are –

  • The principal quantum number is an integer. Additionally, it is equivalent to the number of electron shells. Hence, its value is at least one and higher. Thus, it is never zero or negative.

  • Moreover, the angular momentum quantum number is also an integer, which represents the value of an electron’s orbital. Hence, ℓ is either greater than or equal to zero, and lower or equivalent to n-1.

  • The magnetic quantum number symbolises the orientation of the orbital. Here, the integer values are ranging from -ℓ to ℓ. Thus, for p orbital, where ℓ=1, m can have values of -1, 0, 1.

  • Lastly, the spin quantum number has a half-integer value, which is either -1/2, known as ‘spin down’ or 1/2 called ‘spin up’.


Principal Quantum Number

This principal quantum number portrays the electron shell or energy level of an atom. Here, the value on ‘n’ starts from one and gradually increases to the shell that contains the outermost electron of a particular atom. For instance, in caesium (Cs), the outmost valence electron within the shell has energy level 6. Hence, the ‘n’ value of an electron in caesium can range from 1 to 6. 

Moreover, particles that are in a time-independent potential have nth eigenvalue of Hamiltonian, as per the Schrodinger equation. Hamiltonian’s nth eigenvalue refers to the energy, i.e. E with contribution from angular momentum. However, the term that involves J2 is not considered here. 

Therefore, this number only depends on the distance between an electron and its nucleus, which is the radial coordinate ‘r’. Since the average number rises with ‘n’, quantum states with various principal quantum numbers are said to be a part of different shells.


Azimuthal Quantum Number

The azimuthal quantum number is commonly known as the angular or orbital quantum number. Moreover, it describes the subshell of an electron and its magnitude of the orbital angular momentum via relation. Additionally, in spectroscopy or chemistry where 

  • ℓ = 0, it is known as an s orbital,

  • ℓ = 1 is a p orbital,

  • ℓ = 2 represents a d orbital,

  • ℓ = 3 is an f orbital.

Therefore, the value of ℓ varies from 0 to n-1, because the first p orbital where ℓ=1 arrives in the second electron shell, i.e. n=2. Likewise, the first d orbital, i.e. ℓ=2, appears within the third shell, which is n=3, and so on. The azimuthal quantum number is very significant in chemistry, as it identifies the shape of an atomic orbital, and has a powerful effect on chemical bonds and bond angles. 


Magnetic Quantum Number

Magnetic quantum numbers articulate the energy available in a subshell and estimate the orbital angular momentum along a specific axis. Moreover, values associated with mℓ ranges between – to ℓ, but integer steps are associated. Additionally, the ‘s’ is a subshell where ℓ=0 has one orbital. Therefore, mℓ of an electron within a ‘s’ subshell will be zero always. 

Additionally, the ‘p’ subshell, i.e. ℓ=1 comprises three orbitals. It is also known as three ‘dumbbell-shaped’ clods. Hence, the mℓ of an electron in this ‘p’ subshell should be either -1, 0, 1.

Lastly, the ‘d’ subshell where ℓ=2 has five orbitals. Furthermore, here mℓ has values starting from -2 to +2. Additionally, the value of mℓ quantum number here is associated with orbital orientation.


Spin Projection Quantum Number

The fourth number on this list, quantum numbers spin describes intrinsic angular momentum or ‘spin’ of an electron within an orbital. Moreover, it provides a projection of the spin angular momentum (s) along a particular axis.

Additionally, the values of ms r start from –s to s. Here, ‘s’ defines the spin quantum number, an inherent property of particles. An electron that has a spin ‘s’ = 1/2, its ms will be ‘±’, confirming its spin and opposite spin. Moreover, every electron in any particular orbital should have different spins according to ‘Pauli Exclusion Principle’. Hence, an orbital cannot contain more than 2 electrons. 


Background of Quantum Numbers

The work of Broglie and Bohr have established how electrons have diverse discrete energy levels associated with their atomic radius. This model offered a comparatively, simpler spherical view. Moreover, this model by Bohr and Broglie indicated how the energy level of electrons is related to their principal quantum number. However, there are no numerical ways present in this model to classify additional behaviour of an electron in space. 

Furthermore, Schrodinger’s equation offered three additional quantum numbers to describe an electron’s behaviour in a more complicated multi-electron atom. This model was opposite to what Bohr and Broglie have done previously. Moreover, it opened new possibilities in the field of studying quantum numbers.

Additionally, based on these two model and further contributions from John Lennard-Jones and Slater, the Hund-Mulliken theory has been developed. Moreover, this theory is regarded as the most prominent system of nomenclature in the history of quantum mechanics.

Moreover, this nomenclature has incorporated Hund-Mulliken’s theory along with Bohr’s energy levels, and observations made on electron spin on spectroscopy and Hund’s rule.


Multiplicative Quantum Numbers

One negligible yet confusing point, which is related to the quantum numbers is that a large portion of these numbers is additive. Hence, in an elementary particle reaction, the sum value of such number must be similar before and after a reaction. 

However, some of these numbers, which are typically called parity, are multiplicative. It means their product is preserved. Moreover, these multiplicative quantum numbers are affiliated with a symmetry. Hence, applying it results in transformation twice as equal to that of not doing anything, i.e. involution.


Atomic Orbital

Solving Schrodinger equation results in obtaining a set of mathematical functions called wave functions. It indicates the probability of locating electrons at specific energy levels in an atom. Additionally, this wave function for an electron within an atom is called the atomic orbital. Moreover, it indicates a space where the probability of finding an electron is higher.

Quantum numbers class 11 chemistry is not a very difficult chapter to prepare if you get an interactive session with a subject expert. You will learn the concepts, real-life examples and how to solve equations with ease in such sessions. 

Therefore, if you are searching for such an interactive session on quantum numbers, then visit the official app of Vedantu. Subject experts from across the country conduct live and interactive classes which can be immensely helpful in clearing any doubt that you might have.

FAQ (Frequently Asked Questions)

1. What Is A Quantum Number?

Quantum number is a value that describes the energy available in atoms and molecules. Moreover, it defines the trajectory and movement of an electron in an atom. Furthermore, when every quantum number of electrons in an atom are combined, it must obey the Schrodinger equation.

2. What are The Types of Quantum Number?

There are four types of quantum numbers, principal, azimuthal or angular momentum, magnetic, and spin quantum number. Every number has its characteristics and have a different value system.

3. What is The Principal Quantum Number?

The principal quantum number is the one which portrays the number of electron shell or energy level within an atom. Moreover, the value of ‘n’ here begins from one and then gradually increases until it reaches the outermost electron of a particular atom. 

4. What is An Atomic Orbital?

An atomic orbital is the ‘wave function’ of an electron. Additionally, it indicates a space where the chances of locating an atom are higher. Moreover, it is obtained by solving the Schrodinger equation.