Quiz 1, 1/24/03

(10 pts) (A) What is a statistical ensemble and why is it necessary to to describe macroscopic systems in terms of ensembles ?

(20 pts) (B) How do you describe an ensemble and its time evolution ?
(B.1) In classical statistical mechanics (CSM).
(B.2) In quantum statistical mechanics (QSM).

(10 pts) (C) How do you describe a microscopic state ?
(C.1) In classical statistical mechanics (CSM).
(C.2) In quantum statistical mechanics (QSM).

(20 pts) (D) How do you compute an ensemble average ?
(D.1) In classical statistical mechanics (CSM).
(D.2) In quantum statistical mechanics (QSM).

(20 pts) (E) How do you describe the maximum entropy ensemble of a system with a fixed number of particles and fixed volume ?

(20 pts) (F) Prove that

$$E= \frac{\partial (\beta A)}{\partial \beta} \Bigg )_{V,N},$$ (99)

when $A=E-TS$.

Solution

(A) The collection of a large number of independently prepared replicas of the system is called an ensemble. An ensemble of N replicas of systems is in a statistical mixture of states $\vert\phi _k>$, with probabilities $p_k$, when $n_k$ members of the ensemble are in state $\vert\phi _k>$, with $p_k=n_k/N$. (See page 6 of the lecture notes) Measurements on macroscopic systems must be described in terms of ensemble averages simply because, in practice, macroscopic systems can only be prepared in thermodynamic states (i.e., in a statistical mixtures of quantum states) characterized by a few physical quantities (e.g., the temperature, the pressure, the volume and the number of particles). (See pages 3 and 4 of the lecture notes)

(B.1) An ensemble is described in classical statistical mechanics by the density of states $\rho = \rho(x_1, ...x_f, p_1, ..., p_f)$, where $(x_1, ...x_f)$ and $(p_1, ..., p_f)$ are the coordinates and momenta. The density of states evolves in time according to the following equation of motion:

$$\frac{\partial \rho}{\partial t} = -\left( \rho , H \right),$$ (100)

where $H = \sum_{j=1}^f p_j^2/(2 m_j) + V(x_1, ... x_f)$ is the classical Hamiltonian and $(\rho , H)$ represents the Poisson bracket of $\rho$ and $H$. (See page 10 of the lecture notes)

(B.2) An ensemble is described in quantum statistical mechanics by the density operator

$$\hat{\rho} = \sum_k p_k \vert\phi_k><\phi_k\vert,$$ (101)

where $\vert\phi _k>$ are the possible quantum states that the system can populate and $p_k$ is the probability of populating state $\vert\phi _k>$. (See page 10 of the lecture notes) The density operator evolves in time according to the following equation:

$$\frac{\partial \hat{\rho}}{\partial t} = -\frac{[\hat{\rho},\hat{H}]}{i\hbar},$$ (102)

where $[\hat{\rho},\hat{H}]$ is the commutator of $\hat{\rho}$ and $\hat{H}$.(See page 9 of the lecture notes)

(C.1) A microscopic state is described in classical statistical mechanics by a point in phase-space defined by a set of coordinates $(x_1, ...x_f)$ and momenta $(p_1, ..., p_f)$. (See page 9 of the lecture notes)

(C.2) A microscopic state is described in quantum statistical mechanics by a quantum state in Hilbert space, defined in terms of a set quantum numbers associated with a ket-vector. (See page 9 of the lecture notes)

(D.1) An ensemble average is computed in classical statistical mechanics according to the following equation:

$$\bar{A} = \frac{\int d{\bf x} \int d{\bf p} \rho(x_1, ..., x_f, p_1, ..., p_f) A}{\int d{\bf x} \int d{\bf p} \rho(x_1, ..., x_f, p_1, ..., p_f)},$$ (103)

where $d{\bf x} d{\bf p}$ stands for a volume element in phase space and $A$ is the quantity of interest. (See page 10 of the lecture notes)

(D.2) An ensemble average is computed in quantum statistical mechanics according to the following equation:

$$A = Tr\{\hat{\rho} \hat{A}\},$$ (104)

where $\hat{\rho}$ is the density operator and $A$ is the quantity of interest. (See page 7 of the lecture notes)

(E) The maximum entropy ensemble of a system with a fixed number of particles and fixed volume is described by the density operator

$$\hat{\rho} = Z^{-1} (-\beta \hat{H}),$$ (105)

where $Z$ is the partition function

$$Z \equiv Tr \{ exp(-\beta \hat{H}) \},$$ (106)

and $\beta=(k T)^{-1}$. (See page 12 of the lecture notes) (F)

$$E \equiv Tr\{ \hat{\rho} \hat{H}\} = -\frac{\partial \text{ln} Z}{\partial \beta} \Bigg )_X.$$ (107)

$$S \equiv -k Tr\{\hat{\rho} \hat{\rho} \}= k \beta E + k Z.$$ (108)

Therefore,

$$A \equiv E - T S = -k T Z,$$ (109)

and

$$E = \frac{\partial(\beta A)}{\partial \beta} \Bigg )_X.$$ (110)