10.1: Principles of quantum statistical mechanics
- Page ID
- 5260
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 \(\langle q_1...q_{3N}\vert\):
\[ \langle q_1...q_{3N}\vert H\vert\Psi(t)\rangle \nonumber \] |
\(=\) |
\( {\partial \over \partial t}\langle q_1...q_{3N}\vert\Psi(t)\rangle\) |
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\[ \left[-\sum_{i=1}^{3N}{\hbar^2 \over 2m_i}{\partial^2 \over \partial q_i^2} +U(q_1,...,q_{3N})\right]\Psi(q_1,...,q_{3N},t) \nonumber \] |
\(=\) |
\( i\hbar {\partial \over \partial t}\Psi(q_1,...,q_{3N},t) \) |
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
\[\langle A \rangle = \int dq_1\cdots dq_{3N}\Psi^*(q_1,...,q_{3N}) A \left (q_1, \cdots, q_{3N}, {\hbar \over i}{\partial \over \partial q_1}, \cdots , {\hbar \over i}{\partial \over\partial q_{3N}}\right)\Psi(q_1,...,q_{3N}) \nonumber \]