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

$q_1,...,q_{3N}$

and momenta

$p_1, \cdots , p_{3N}$

and Hamiltonian

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

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

$H\vert\Psi(t)\rangle = i\hbar{\partial \over \partial t}\vert\Psi(t)\rangle$

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

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

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

$\vert q_1...q_{3N}\rangle = \vert q_1\rangle \cdots \vert q_{3N}\rangle$

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

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

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

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