Example 1: Ideal Gas of Structureless Quantum Particles

An ideal gas of N non-interacting structureless paticles of mass $m$ is described by the N-particle Hamiltonian

$$\hat{H} = \sum_{j=1}^N \hat{h}_j,$$ (161)

where $\hat{h}_j$ is the one-particle Hamiltonian

$$\hat{h}_j = -\frac{\hbar^2}{2 m} \nabla^2_{{\bf R}_j} =-\frac{\hb... ...2}+\frac{\partial^2}{\partial y_j^2}+\frac{\partial^2}{\partial z_j^2} \Bigg ),$$ (162)

with ${\bf R}_j = (x_j,y_j,z_j)$. The eigenstates of $\hat{h}_j$ are the free-particle states

$$\phi_j(x,y,z) = A e^{{\bf k}_j \cdot {\bf R}},$$ (163)

where ${\bf k}_j = (kx_j,ky_j,kz_j)$, and $A$ is a normalization constant determined by the volume of the box that contains the gas. The one-particle eigenstates satisfy the eigenvalue problem

$$\hat{h}_j \vert\phi_j > = \epsilon_j \vert\phi_j >,$$ (164)

with $\epsilon_j = (\hbar {\bf k}_j)^2/(2 m_j)$. Note that since the volume of the box is V=Lx$\times$Ly$\times$Lz, and $\vert \phi_j >$ are stationary states, then Kx Lx = nx $\pi, \hspace{.1cm}$ Ky Ly = ny $\pi \hspace{.1cm}$ and Kz Lz = nz $\pi, \hspace{.1cm}$ with nx,ny,nz=0,1,2,... Therefore,

$$\sum_{nx,ny,nz}= \sum_{Kx,Ky,Kz} \frac{V}{\pi^3}...$$

and

$$Z=\frac{1}{N!} \Bigg ( \sum_j e^{-\beta \epsilon_j } \Bigg )^N =\... ...int_0^{\infty} dK_z e^{-\frac{\beta \hbar^2}{2m}(K_x^2+K_y^2+K_z^2) } \Bigg)^N.$$ (165)

Computing the Gaussian integrals analytically, we obtain

$$Z = \frac{1}{N! \pi^{3N}} \Bigg ( \frac{V}{2^3} \Bigg ( \frac{\pi... ...ac{V^N}{2^{3N} N! \pi^{3N}} \Bigg ( \frac{2\pi m}{\beta \hbar^2} \Bigg )^{3/2},$$ (166)

since $\int_0^{\infty} e^{-\alpha x^2 }dx =\frac{1}{2} \sqrt{\frac{\pi}{\alpha}}$. Therefore,

$$Z = \frac{V^N}{N! h^{3N}} \Bigg ( \frac{2\pi m}{\beta} \Bigg )^{3... ...- \frac{\partial \text{ln} Z}{\partial \beta} \Bigg )_{V,N} =\frac{3}{2} N k T.$$ (167)

In addition, defining the pressure $p$ according to

$$\beta p \equiv \frac{\partial \text{ln} Z}{\partial V} \Bigg )_{T,N}$$ (168)

we obtain

$$\beta p =\frac{N}{V} \Rightarrow \boxed{p V = N k T},$$ (169)

which is the equation of state for an ideal gas of structureless particles.