Quantum Numbers

The route and mobility of an electron in an atom can be described using quantum numbers. The Schrodinger equation must be met when the quantum numbers of all the electrons in an atom are summed together.

Quantum numbers are the values of the conserved quantities in a quantum system. Electronic quantum numbers (quantum numbers that describe electrons) are numerical quantities that provide solutions to the Schrodinger wave equation for hydrogen atoms.

Brief Introduction of 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.

The following four quantum numbers can be used to fully characterise all of the characteristics of an atom's electrons:

• n is the principal quantum number.

• The quantum number of orbital angular momentum (also known as the azimuthal quantum number) is indicated by the letter l.

• ml stands for a magnetic quantum number.

• ms stands for the electron spin quantum number.

What are Quantum Numbers?

The position and energy of an electron in an atom are described by quantum numbers, which are a collection of numbers. An atom is made up of a vast number of orbitals that are distinguishable from one another by their shape, size, and spatial orientation. The orbital properties are utilised to thoroughly define an electron's state and are expressed in terms of three numbers: 

• Principal quantum number

• Azimuthal quantum number

• Magnetic quantum number

• Spin quantum number

The numbers that designate and distinguish various atomic orbitals and electrons present in an atom are known as quantum numbers. Quantum Numbers are a collection of four numbers that may be used to obtain all of the information about all of the electrons in an atom, including their energy, location, space, kind of orbital occupied, and even the direction of that orbital.

Quantum Number Values

• No two electrons in an atom may have the same set of quantum numbers, according to the Pauli exclusion principle. A half-integer or integer value is used to represent each quantum number.

• The number of the electron's shell is the primary quantum number, which is an integer. The value is one or more (never 0 or negative).

• The value of the electron's orbital is represented by the angular momentum quantum number (s=0, p=1). l is less than or equal to n-1 and bigger than or equal to zero.

• With integer values ranging from -l to l, the magnetic quantum number is the orbital's orientation. As a result, for the p orbital, where l=1, m might be -1, 0, or 1.

• The spin quantum number is a half-integer value that is either -1/2 (referred to as "spin down") or 1/2 (referred to as "spin up") (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 outermost 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 the nth given value 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’ clouds. 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 models 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 a 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 the 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.