Partition Function of a Particle in 3D Box

Separation of Partition Function

Z = Z_x Z_y Z_z

Z_x = Σ exp[-β (ħ²π² / 2m) (n_x² / L_x²)]

Similarly, Z_y and Z_z are defined. To evaluate these sums, we convert summation into integration.

Z_x = ∫ exp[-β (ħ²π² / 2m)(n_x² / L_x²)] dn_x

Let:

n_x² (β / 2m)(ħπ / L_x)² = u²

For lower limit n_x = 1, u ≈ 0, so limits become 0 to ∞.

Z_x = ∫ (2m/β)^1/2 (L_x / ħπ) e^-u² du

Z_x = (m / 2πβħ²)^1/2 L_x

∫₀^∞ e^-u² du = √π / 2

Z_y = (m / 2πβħ²)^1/2 L_y

Z_z = (m / 2πβħ²)^1/2 L_z

Z = Z_x Z_y Z_z

Z = (m / 2πβħ²)^3/2 V

Where V = L_x L_y L_z is the volume of the container.

If probability of i-th state is P_i, then average energy:

Ē = Σ P_i E_i