Question

The probability of finding out an electron at a point within an atom is proportional to the:
(A) square of the orbital wave function, i.e., ${\psi ^2}$.
(B) orbital wave function, i.e., $\psi$.
(C) Hamiltonian operator, i.e, H.
(D) Principal quantum number, i.e, n.

Answer 1

A wave function describes the wave like behavior of material particles such as electrons. A Hamiltonian operator represents the sum of potential energy and the kinetic energy of an electron.

Complete answer:
Matter can also exhibit wave-like behavior. This is known as wave-matter dualism. Hence, matter is also associated with a wave function. The wave function gives the amplitude of the matter-wave.
An atomic orbital is represented by the mathematical wave function. This mathematical wave function is the orbital wave function $\psi$. When this mathematical wave function for the atomic orbital is squared, the probability density is obtained. The probability density has always a positive sign. In the term probability density, the term probability refers to the probability of finding an electron in an orbital of the atom.
A Hamiltonian operator represents total energy of the system. The Schrödinger equation is $H\psi = E\psi$ .
The principal quantum number represents the atomic orbital/shell. It represents the size and energy of the atomic orbital. When the Schrödinger equation is solved, the principal quantum number is obtained as a solution.

Hence, the option (A) is the correct option.

Note: The probability of finding an electron in the nucleus is zero. An orbital represents the region around the nucleus, where the probability of finding the electron is maximum. Thus, an orbital is the three dimensional description of the most likely location of the electron in an atom.