THE CANONICAL ENSEMBLE4.2 THE UNIFORM QUANTUM ENSEMBLE64Chapter 4The Canonical Ensemble4.1QUANTUM ENSEMBLESA statistical ensemble for a dynamical system is a collection of system points in the system phase space.Each system point represents a full system in a particular dynamical state. Therefore, a statistical ensemblemay be pictured as a vast collection of exact replicas of a particular system. If the system under study is abottle of beer, then the ensemble can be thought of as a huge warehouse filled with individual, noninteracting,bottles of beer. It cannot be pictured as a very large barrel of beer, which would be a single, different system.Each system in the ensemble has exactly the same Hamiltonian function, but, in general, different systems inthe ensemble are in different dynamical states. Since the systems all have the same Hamiltonian function, itmakes sense to represent the dynamical states of all the systems by points in a single phase space. Thus, thestate of the complete ensemble is represented by a “dust cloud” in the phase space, containing a vast numberof discrete points. The danger inherent in using this picture of an ensemble is that it is very easy to confuseit with the similar picture of a gas containing a vast number of point particles. This is unfortunate becausethe two things behave in quite different ways. The motion of a single gas particle is affected by the positionsand velocities of its neighbors, because the gas particles interact. In contrast, each point in the ensemblerepresentation is a system point for an isolated system that moves through the phase space according toHamilton’s equations with no regard for the states of the other isolated systems in the ensemble. For theuniform ensemble the dust cloud is of uniform density inside the energy surface and of zero density outsideit. For the microcanonical ensemble the system points of the ensemble are confined to the energy surfaceitself.A quantum system is defined by a Hermitian Hamiltonian operator rather than a Hamiltonian function.The possible energy states of the system are the solutions of Schr¨odinger’s equationHψn(r1, . . . ,rN) =Enψn(r1, . . . ,rN)(4.1)A statistical ensemble for a quantum system will be pictured as a vast collection of replicas of the system.Each will have the same Hamiltonian operator, but, in general, different members of the ensemble willbe in different quantum states.The quantum states are assumed to be energy eigenstates with energieswithin some particular energy interval.This is not the only reasonable way to choose the set of allowedmicrostates.A more general way is to allow any microstate that is a normalized linear combination ofthe energy eigenstates within the above-mentioned energy interval.That would include states that werenot themselves exact energy eigenstates. In the problems, the reader will be asked to show that this moregeneral procedure actually leads to exactly the same results as the assumption made here.
