The Schrödinger Equation

Quantum mechanics explains how entities like electrons have both particle-like and wave-like characteristics. The Schrödinger equation describes the wavefunction of a particle:

$\big[\frac{-\hbar}{2m}\nabla^{2}+\mathbf{V}\big]\mathbf{\Psi}(\vec{r},t)=i\hbar\frac{\partial \Psi(\vec{r},t)}{\partial t}$

where

$\nabla=\frac{\partial}{\partial x}\vec{i}+\frac{\partial}{\partial y}\vec{j}+\frac{\partial}{\partial z}\vec{k}$

In these equations, Ψ is the wavefunction, m is the mass of the particle, h is Planck’s constant, and V is the potential field in which the particle is moving. The product of Ψ with its complex conjugate—Ψ*Ψ, often written as |Ψ|2—is interpreted as the probability distribution of the particle.

The Schrödinger equation for a collection of particles like a molecule is very similar. In this case, Ψ would be a function of the coordinates of all the particles in the system as well as t.

The energy and many other properties of the particle can be obtained by solving the Schrödinger equation for Ψ, subject to the appropriate boundary conditions. Many different wavefunctions are solutions to it, corresponding to different stationary states of the system.

If V is not a function of time, the Schrödinger equation can be simplified using the mathematical technique known as separation of variables. If we write the wavefunction as the product of a spatial function and a time function:

$\Psi(\vec{r},t) = \psi(\vec{r})\tau(t)$

and then substitute these new functions into the first equation, we will obtain two equations, one of which depends on the position of the particle independent of time and the other of which is a function of time alone. For the problems in which we are interested, this separation is valid, and we focus entirely on the familiar time-independent Schrödinger equation:

$\textbf{H}\Psi(\vec{r}) = E\Psi(\vec{r})$

where E is the energy of the particle, and H is the Hamiltonian operator, equal to:

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

The various solutions to this equation correspond to different stationary states of the particle (molecule). The one with the lowest energy is called the ground state. Note that this equation is a non-relativistic description of the system which is not valid when the velocities of particles approach the speed of light. Thus, it does not give an accurate description of the core electrons in large nuclei.