Solution for a Single Free Particle

Claimed by Carlos M. Silva (Spring 2022)

The Schrödinger Equation is a linear partial differential equation that governs the wave function of a quantum mechanical system. Similar to Newton's Laws, the Schrödinger Equation is an equation of motion, meaning that it's capable of describing the time-evolution of a position analog of a system.

The free particle is the name given to the system consisting of a single particle subject to a null or constant potential everywhere in space. It's the simplest system to which the Schrödinger Equation has a solution with physical meaning.

Although the free-particle solution does not have ample practical use in the field of Physics, the methods and conclusions that come from the solution of this system are of great use in a plethora of other quantum systems.

The General Schrödinger Equation

The general formulation of the Schrödinger Equation for a time-dependent system of non-relativistic particles in Bra-Ket notation is:

[math]\displaystyle{ i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H\vert\Psi(t)\rangle }[/math]

Here [math]\displaystyle{ i = \sqrt{-1} }[/math] is the imaginary unit. [math]\displaystyle{ \hbar }[/math] is the reduced Planck's constant. [math]\displaystyle{ \vert\Psi(t)\rangle }[/math] is the state vector of the quantum system at time [math]\displaystyle{ t }[/math], and [math]\displaystyle{ \hat H }[/math] is the Hamiltonian operator. For a single particle, the general Schrödinger Equation reduces to a single linear partial differential equation:

[math]\displaystyle{ i\hbar\Psi(\vec{r},t) = - \frac{\hbar^2}{2m} \nabla ^2 \Psi(\vec{r},t) + V(\vec{r},t)\Psi(\vec{r},t) }[/math]

[math]\displaystyle{ \Psi(\vec{r},t) }[/math] becomes the wave function of the particle at position [math]\displaystyle{ \vec{r} }[/math] and time [math]\displaystyle{ t }[/math]. [math]\displaystyle{ V(\vec{r},t) }[/math] is the scalar potential energy of the particle at position [math]\displaystyle{ \vec{r} }[/math] and time [math]\displaystyle{ t }[/math].