Tru Physics

Particle in a Box

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The particle in a box, also known as the particle in a cubic box, is a fundamental quantum mechanical model that describes a particle confined to a three-dimensional (3D) box with infinite potential energy at and beyond the walls of the box. This model provides insights into energy quantization, wavefunctions, and quantum states for particles in confined systems, and serves as a starting point for understanding more complex quantum systems.

For the 1D case, view the page on Infinite Square Well.

Potential and System Description

The 3D box potential is defined as:

V(x, y, z) = \begin{cases}0 & 0 < x < L_x, 0 < y < L_y, 0 < z < L_z \\\infty & \text{otherwise}\end{cases}

where L_x, L_y, and L_z are the dimensions of the box. A particle of mass m is confined to this potential and can only exist within the box.

Note that the solutions to this problem do not actually require the box to be cubic. Rectangular boxes are also allowed.

Time-Independent Schrödinger Equation

The time-independent Schrödinger equation for the 3D box potential is given by:

-\dfrac{\hbar^2}{2m} \left(\dfrac{\partial^2\psi(x, y, z)}{\partial x^2} + \dfrac{\partial^2\psi(x, y, z)}{\partial y^2} + \dfrac{\partial^2\psi(x, y, z)}{\partial z^2}\right) = E\psi(x, y, z)

where:

  • \hbar is the reduced Planck constant,
  • m is the mass of the particle,
  • E is the energy of the particle,
  • \psi(x, y, z) is the particle’s wavefunction.

Wavefunction and Energy Quantization

Solving the time-independent Schrödinger equation for the 3D box yields the following wavefunction and energy quantization:

\psi_{n_x, n_y, n_z}(x, y, z) = \sqrt{\dfrac{8}{L_x L_y L_z}}\sin\left(\dfrac{n_x\pi x}{L_x}\right)\sin\left(\dfrac{n_y\pi y}{L_y}\right)\sin\left(\dfrac{n_z\pi z}{L_z}\right)

E_{n_x, n_y, n_z} = \dfrac{\pi^2 \hbar^2}{2m} \left(\dfrac{n_x^2}{L_x^2} + \dfrac{n_y^2}{L_y^2} + \dfrac{n_z^2}{L_z^2}\right)

where n_x, n_y, and n_z are positive integers called the quantum numbers.

Note the the above equation can be rewritten for a cubic box as:

E_{n_x, n_y, n_z} = \dfrac{\pi^2 \hbar^2}{2mL^2} \left(n_x^2 + n_y^2 + n_z^2 \right)

The wavefunctions represent the stationary states of the particle, and the quantized energy levels demonstrate that the particle can only have discrete energy values.

Probability Distribution and Expectation Values

The probability distribution for a particle in a 3D box is given by the square of the wavefunction:

P_{n_x, n_y, n_z}(x, y, z) = \left|\psi_{n_x, n_y, n_z}(x, y, z)\right|^2 = \dfrac{8}{L_x L_y L_z}\sin^2\left(\dfrac{n_x\pi x}{L_x}\right)\sin^2\left(\dfrac{n_y\pi y}{L_y}\right)\sin^2\left(\dfrac{n_z\pi z}{L_z}\right)

This distribution describes the likelihood of finding the particle at a particular location within the box. Expectation values for position and momentum can be calculated using the probability distribution and wavefunctions, providing further insight into the particle’s behavior within the box.

Applications and Limitations

The 3D particle in a box model serves as a fundamental starting point for understanding quantum mechanics in confined systems. Applications of this model can be found in areas such as quantum chemistry, solid-state physics, and nanotechnology. However, this model is based on idealized assumptions, such as infinitely high walls and a particle with no interactions other than confinement. Consequently, more complex models are often required to accurately describe real-world systems.

Chapter Summary

In this page, we have discussed the particle in a 3D box, a fundamental quantum mechanical model that describes a particle confined to a cubic box with infinitely high walls. We covered the potential and system description, the time-independent Schrödinger equation, wavefunctions, energy quantization, probability distribution, expectation values, and the applications and limitations of this model. The particle in a 3D box serves as an important foundation for understanding more complex quantum systems and has applications in various fields of science and technology.

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  1. Infinite Square Well – Tru Physics

    […] For the 3D case, view the page for Particle in a Box. […]

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