In quantum mechanics, the one-dimensional infinite potential well is a common model used to study the behavior of a particle that is confined to moving in one dimension within a finite region of space.

This model is useful for understanding many physical phenomena, including the properties of atoms, molecules, and solid-state materials.

## The Model

The one-dimensional infinite potential well model consists of a particle of mass $m$ that is confined to move within a region of length $L$ along a single axis, typically the $x$-axis.

The potential energy of the particle is infinite outside of the region, and zero within the region, so that the particle is effectively trapped within the well.

The potential energy function $V(x)$ for this system is given by:

V(x) = 0 if 0 < x L

$\infty$ otherwise

## Wavefunction Solutions

The Schrödinger equation for the one-dimensional infinite potential well can be solved analytically. The time-independent Schrödinger equation is given by: \begin{equation} -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \end{equation}

Within the region of the well, where the potential energy is zero, the Schrödinger equation reduces to: \begin{equation} -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} = E\psi(x) \end{equation}

The general solution to this differential equation is a linear combination of sine and cosine functions: \begin{equation} \psi_n(x) = A\sin\left(\frac{n\pi x}{L}\right) + B\cos\left(\frac{n\pi x}{L}\right) \end{equation} where $n$ is a positive integer that specifies the energy level of the particle.

The constants $A$ and $B$ are determined by the boundary conditions, which require that the wavefunction be continuous and finite everywhere within the region of the well.

## Energy Levels

The energy levels of the particle in the one-dimensional infinite potential well are quantized, meaning that the particle can only have certain discrete values of energy.

The energy of the particle in the nth energy level is given by: \begin{equation} E_n = \frac{n^2\pi^2\hbar^2}{2mL^2} \end{equation}

The ground state of the particle, corresponding to the lowest energy level, is given by $n=1$ and has an energy of $E_1 = \frac{\pi^2\hbar^2}{2mL^2}$.

The energy levels increase as $n^2$, so that the energy spacing between adjacent levels becomes smaller as $n$ increases.