Question

The angular wave function depends upon which quantum numbers?
(A) n and l
(B) l and m
(C) l only
(D) m only

Answer 1

Hint: The wave function is used in the Schrodinger’s equation which plays the role of Newton’s laws and conservation of energy. In quantum mechanics, Newton’s laws are inapplicable because the particles are so small in a quantum level that it doesn’t obey Newton’s laws of mechanics.

Complete step by step solution:
In order to know about the dependency of the angular wave function of quantum numbers, let’s get a basic idea about quantum numbers first.
Quantum number is used to give the address to each specific electron in an atomic system. To completely describe an electron in an atom, four quantum numbers are needed. They are as follows:
-The Principal Quantum Number (n): The first quantum number describes the energy level of an atom. The value of n ranges from 1 to the shell containing the outermost electron of that particular atom. For example, in caesium, the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have the value of n ranges from 1 to 6.
-The Azimuthal Quantum Number (l) : It is the second quantum number that describes the subshell and gives the magnitude of the orbital angular momentum. In chemistry and spectroscopy, ℓ = 0 is called an s orbital, ℓ = 1 a p orbital, ℓ = 2 a d orbital and ℓ = 3 is known as f orbital. The value of ℓ ranges from 0 to n − 1 where n is the principal quantum number.
-The magnetic quantum number (m): The magnetic quantum number describes the energy levels available within a subshell of an atom and yields the projection of the orbital angular momentum along a specified axis. The values of m range from −l to ℓ (where l is in integer).
-The spin quantum number: The fourth quantum number describes the spin (ingrained angular momentum) of the electron within that particular orbital and gives the projection of the spin angular momentum along the specified axis. Its values range from −s to s, where s is the spin quantum number, an intrinsic property of particles.
From the above discussion, it is clear that the quantum numbers l and m decide the shape of the electron cloud and its spatial orientation. Therefore, they decide the angular wave function of the orbital.

So, the answer of the given question is option (B) l and m.

Note: Remember that the question of how many quantum numbers are needed to completely describe any given system has no universal answer; for each system, one must find the answer by performing a full analysis of the system separately.