2.5. Particle in a Box

Course-wide Conventions & Notation

Overview and Learning Objectives¶

A particle in a box is the simplest quantized model of translation and provides a clean route to the translational partition function. We review the 1D and 3D energy levels, derive $q_{\mathrm{trans}}=V/\Lambda^3$ in the high-temperature/large-volume limit, and use $Q=q^N/N!$ to recover ideal-gas thermodynamic relations.

Learning objectives:

• Write the quantized energy levels for a particle in a 1D box and generalize to a 3D rectangular box and cube.

• Derive the single-particle partition function and its continuum approximation in the limit of small level spacing.

• Define the thermal de Broglie wavelength $\Lambda$ and express $q_{\mathrm{trans}}=V/\Lambda^3$.

• Use $Q=(V/\Lambda^3)^N/N!$ to obtain $U=\tfrac{3}{2}Nk_{\mathrm B}T$, $C_V=\tfrac{3}{2}Nk_{\mathrm B}$, and $P=Nk_{\mathrm B}T/V$.

Core Ideas and Derivations¶

Review of the Particle in a Box¶

Particle in a One-Dimensional Box¶

Source

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d.art3d import Poly3DCollection
from scipy.constants import k, eV
from labellines import labelLines
from myst_nb import glue

fig, axs = plt.subplot_mosaic([[0]], figsize=(4, 4))

# Plot a one-dimensional box
axs[0].plot([-0.5, -0.5], [18, 0], color='black', zorder=2.5, lw=4)
axs[0].plot([-0.5, 0.5], [0, 0], color='black', zorder=2.5, lw=4)
axs[0].plot([0.5, 0.5], [0, 18], color='black', zorder=2.5, lw=4)

# Plot the energy levels (blue lines)
for n in range(1, 5):
    energy = axs[0].plot([-0.5, 0.5], [n**2, n**2], color='blue', label=r'$E_{%d}$' % n)
    labelLines(energy, xvals=[-1/6], zorder=2.5)

# Plot the wavefunctions (red curves)
x = np.linspace(-0.5, 0.5, 100)
for n in range(1, 5):
    wavefunction = axs[0].plot(x, np.sin(n * np.pi * (x + 0.5)) + n**2, color='red', label=r'$\psi_{%d}$' % n)
    labelLines(wavefunction, xvals=[1/6], zorder=2.5)

axs[0].set_xlabel('Position')
axs[0].set_ylabel('Energy')
axs[0].set_xticks([-0.5, 0, 0.5])
axs[0].set_xticklabels([r'$-L/2$', r'$0$', r'$L/2$'])
axs[0].set_yticks([])
axs[0].spines['top'].set_visible(False)
axs[0].spines['right'].set_visible(False)
axs[0].spines['left'].set_visible(False)

plt.tight_layout()
plt.show()
plt.close(fig)

Energy levels $E_n$ (blue) and wavefunctions $\psi_n$ (red) for a particle in a one-dimensional box with boundaries at $x=\pm L/2$.

For a particle in a one-dimensional box of length $L$, the quantized energy levels are

where $h$ is Planck’s constant, $m$ is the particle mass, and $n$ is a positive integer.

Particle in a Three-Dimensional Box¶

In three dimensions, the energy levels are

where $n_x$, $n_y$, and $n_z$ are positive integers, and $L_x$, $L_y$, and $L_z$ are the respective side lengths of a rectangular box.

Particle in a Cube¶

For a cube of side $L$, $L_x = L_y = L_z = L$. The energy simplifies to

where $V = L^3$ is the cube volume.

Partition Function for a Particle in a Cube¶

The single-particle partition function for a three-dimensional cube is

Define $\alpha = \beta h^2 / (8 m V^{2/3})$. Then

Approximation of the Sum¶

When the level spacing is small (large $T$ or large $L$), the sum $\sum_{n=1}^{\infty} e^{-\alpha n^2}$ is well approximated by an integral:

Substituting $\alpha = \beta h^2/(8 m V^{2/3})$ gives

We often define the thermal de Broglie wavelength $\Lambda$ as

Hence,

So the single-particle partition function becomes

Partition Function for $N$ Particles in a Cube¶

For $N$ identical, non-interacting, indistinguishable particles, the total partition function $Q$ is

The factor of $N!$ corrects for particle indistinguishability: permutations of identical particles do not produce new states.

Ensemble Averages for $N$ Particles in a Cube¶

Natural Logarithm of the Partition Function¶

Taking the logarithm of $Q$ gives

Often, the Stirling approximation ($\ln N! \approx N \ln N - N$) is used for large $N$.

Internal Energy¶

The internal energy $U$ follows from

Heat Capacity at Constant Volume¶

From $U = \frac{3}{2}N k_{\mathrm B}T$, it follows that

Pressure¶

To find the pressure, we use

recovering the ideal gas law.

Translational Partition Function¶

Because the particle-in-a-box spectrum describes translational motion, the single-particle partition function

is called the translational partition function. It underpins the classical description of dilute gases when $\Lambda \ll$ (typical particle spacing), so quantum effects are negligible at ordinary densities and temperatures.

Computational Studio: Ideal Gas¶

Use the interactive studio below (or open it in a new tab) to vary particle count ($N$), volume ($V$), and temperature ($T$). The studio visualizes how these parameters affect the gas while showing $P$, $U$, and $S$ derived from the partition function.

You can open the studio in a new tab: Ideal Gas Computational Studio.

Worked Example¶

Estimating $\Lambda$ and $q_{\mathrm{trans}}$ (helium, 300 K)¶

For a particle of mass $m$, the thermal de Broglie wavelength is

Take helium: $m=4.0026\,u=6.65\times10^{-27}\ \mathrm{kg}$, $T=300\ \mathrm{K}$.

1. Compute $\Lambda$

   $$\Lambda=\frac{6.626\times10^{-34}}{\sqrt{2\pi(6.65\times10^{-27})(1.381\times10^{-23})(300)}} \approx 5.0\times10^{-11}\ \mathrm{m}.$$

   (18)

2. Compute the single-particle translational partition function in $V=1.00\ \mathrm{L}=10^{-3}\ \mathrm{m^3}$

   $$q_{\mathrm{trans}}=\frac{V}{\Lambda^3} =\frac{10^{-3}}{(5.0\times10^{-11})^3} \approx 7.8\times10^{27}.$$

   (19)

Result. $q_{\mathrm{trans}}\gg 1$ in macroscopic volumes, consistent with the continuum approximation used to derive $V/\Lambda^3$.

Concept Checks¶

1. Why does the sum over quantum numbers become well-approximated by an integral at high $T$ or large $V$?

2. What does the condition “interparticle spacing $\gg \Lambda$” mean physically?

3. Why does translation give $U\propto T$ but not $U\propto V$ for an ideal gas?

4. Where does the $1/N!$ factor enter when connecting microscopic states to macroscopic entropy?

Key Takeaways¶

• Quantized translation in a box leads, in the continuum limit, to $q_{\mathrm{trans}}=V/\Lambda^3$.

• $\Lambda$ sets the scale of quantum wavepacket “size”; classical behavior emerges when $\Lambda$ is small compared to typical spacings.

• For an ideal monatomic gas, $Q=(V/\Lambda^3)^N/N!$ reproduces $PV=Nk_{\mathrm B}T$ and $U=\tfrac{3}{2}Nk_{\mathrm B}T$.

• Partition functions connect microscopic spectra to thermodynamic equations of state via derivatives of $\ln Q$.