10.1: Principles of quantum statistical mechanics

The problem of quantum statistical mechanics is the quantum mechanical treatment of an \(N\)-particle system. Suppose the corresponding \(N\)-particle classical system has Cartesian coordinates

\[q_1,...,q_{3N} \nonumber \]

and momenta

\[p_1, \cdots , p_{3N} \nonumber \]

and Hamiltonian

\[H = \sum_{i=1}^{3N} {p_i^2 \over 2m_i} + U(q_1,...,q_{3N}) \nonumber \]

Then, as we have seen, the quantum mechanical problem consists of determining the state vector \(\vert \Psi (t) \rangle \) from the Schrödinger equation

\[H\vert\Psi(t)\rangle = i\hbar{\partial \over \partial t}\vert\Psi(t)\rangle \nonumber \]

Denoting the corresponding operators, \(Q_1, \cdots , Q_{3N} \) and \(P_1, \cdots , P_{3N} \), we note that these operators satisfy the commutation relations:

\[ \left[Q_i,Q_j\right] \nonumber \]	\(=\)	\( \left[P_i,P_j\right] = 0\)
\[ \left[Q_i,P_j\right] \nonumber \]	\(=\)	\(\displaystyle i\hbar I \delta_{ij}\)

and the many-particle coordinate eigenstate \(\vert q_1...q_{3N}\rangle \) is a tensor product of the individual eigenstate \(\vert q_1\rangle ,...,\vert q_{3N}\rangle \):

\[\vert q_1...q_{3N}\rangle = \vert q_1\rangle \cdots \vert q_{3N}\rangle \nonumber \]

The Schrödinger equation can be cast as a partial differential equation by multiplying both sides by \(\langle q_1...q_{3N}\vert\):

\[ \langle q_1...q_{3N}\vert H\vert\Psi(t)\rangle \nonumber \]	\(=\)	\( {\partial \over \partial t}\langle q_1...q_{3N}\vert\Psi(t)\rangle\)
\[ \left[-\sum_{i=1}^{3N}{\hbar^2 \over 2m_i}{\partial^2 \over \partial q_i^2} +U(q_1,...,q_{3N})\right]\Psi(q_1,...,q_{3N},t) \nonumber \]	\(=\)	\( i\hbar {\partial \over \partial t}\Psi(q_1,...,q_{3N},t) \)

where the many-particle wave function is \(\Psi(q_1,....,q_{3N},t) =\langle q_1...q_{3N}\vert\Psi(t)\rangle \). Similarly, the expectation value of an operator \(A=A(Q_1,...,Q_{3N},P_1,...,P_{3N})\) is given by

\[\langle A \rangle = \int dq_1\cdots dq_{3N}\Psi^*(q_1,...,q_{3N}) A \left (q_1, \cdots, q_{3N}, {\hbar \over i}{\partial \over \partial q_1}, \cdots , {\hbar \over i}{\partial \over\partial q_{3N}}\right)\Psi(q_1,...,q_{3N}) \nonumber \]