Theoretical Background

Abstract

Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation,

$$ \left( { - \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + v} \right)\Psi = {\rm E}\Psi,\hbar = \frac{h}{{2\pi }}, $$

or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression

$$ H = - \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + V; $$

h is Planck’s constant; ∇² is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and $\frac{{ - {\hbar ^2}}}{{2m}}{\nabla ^2}$ the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.