Particle in a Box Wave Function Plotter

Last Updated on September 14, 2024 by Max

Theory of the Particle in an Infinite Potential Well

The wave functions in the above code correspond to a quantum mechanical system known as the particle in a one-dimensional infinite potential well, also referred to as a particle in a box. This is a fundamental problem in quantum mechanics that demonstrates the quantization of energy levels and the behavior of particles at the quantum scale.

Description of the System

Consider a particle of mass \( m \) confined in a one-dimensional box of length \( L \). The potential inside the box is zero (free space), and it is infinitely large outside the box. This means the particle cannot exist outside the box, and its wave function must vanish at the boundaries \( x = 0 \) and \( x = L \).

The potential \( V(x) \) is defined as:

\[V(x) =\begin{cases} 0, & 0 \leq x \leq L \\\infty, & \text{otherwise}\end{cases}\]

Schrödinger’s Equation

The time-independent Schrödinger equation for a particle of mass \( m \) in a potential \( V(x) \) is given by:

\[-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x)\]

For the region inside the box (where \( V(x) = 0 \)), this simplifies to:

\[-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} = E \psi(x)\]

or,

\[\frac{d^2 \psi(x)}{dx^2} + k^2 \psi(x) = 0\]

where:

\[k = \sqrt{\frac{2mE}{\hbar^2}}\]

Solution of the Schrödinger Equation

The general solution to this differential equation is:

\[\psi(x) = A \sin(kx) + B \cos(kx)\]

where \( A \) and \( B \) are constants determined by the boundary conditions.

Boundary Conditions:

1. At \( x = 0 \), \( \psi(0) = 0 \):
   \[\psi(0) = A \sin(0) + B \cos(0) = B = 0\]
   Thus, \( B = 0 \), so the solution simplifies to:
   \[\psi(x) = A \sin(kx)\]
2. At \( x = L \), \( \psi(L) = 0 \):
   \[\psi(L) = A \sin(kL) = 0\]
   For a non-trivial solution (\( A \neq 0 \)), we require:
   \[\sin(kL) = 0\]
   This gives:
   \[kL = n\pi \quad \text{where } n = 1, 2, 3, \ldots\]
   Thus:
   \[k = \frac{n\pi}{L}\]

Wave Functions

Substituting for \( k \), the wave functions become:

\[\psi_n(x) = A \sin\left(\frac{n \pi x}{L}\right)\]

The normalization condition requires that the total probability of finding the particle within the box is 1:

\[\int_0^L |\psi_n(x)|^2 \, dx = 1\]

This gives the normalization constant \( A \):

\[A = \sqrt{\frac{2}{L}}\]

Therefore, the normalized wave functions are:

\[\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right), \quad n = 1, 2, 3, \ldots\]

Quantized Energy Levels

The corresponding energy levels for these wave functions are given by:

\[E_n = \frac{\hbar^2 k^2}{2m} = \frac{\hbar^2}{2m} \left(\frac{n \pi}{L}\right)^2 = \frac{n^2 \pi^2 \hbar^2}{2mL^2}, \quad n = 1, 2, 3, \ldots\]

These energy levels are discrete (quantized) and increase with \( n^2 \).

Physical Interpretation

Nodes: Each wave function \( \psi_n(x) \) has \( (n-1) \) nodes (points where the wave function passes through zero), and the number of nodes increases with increasing \( n \), indicating higher energy states.

Wave Function \(( \psi_n(x) )\): Describes the probability amplitude of finding the particle at position \( x \) within the box. The square of the wave function, \( |\psi_n(x)|^2 \), gives the probability density.

Quantization: The condition that the wave function must be zero at the boundaries (due to the infinite potential outside) leads to discrete energy levels. This is a direct result of the wave nature of particles at the quantum scale.