Postulates of quantum mechanics 367that lead from the initiation point to the detection point. This sum over paths is re-
ferred to as theFeynman path integral. As we will see in Chapter 12, the classical path,
i.e. the path predicted by extremizing the classical action, is the most probable path,
thereby indicating that classical mechanics naturally emerges as anapproximation to
quantum mechanics.
Proceeding as we did for classical statistical mechanics, this chapter will review
the basic principles of quantum mechanics. In the next chapter, wewill lay out the
statistical mechanical rules for connecting the quantum description of the microscopic
world to macroscopic observables. These chapters are by no means meant to be an
overview of the entire field of quantum mechanics, which could (and does) fill entire
books. Here, we seek only to develop the quantum-mechanical concepts that we will
use in our treatment of quantum statistical mechanics.
9.2 Review of the fundamental postulates of quantum mechanics
The fundamental postulates and definitions of quantum mechanicsaddress the follow-
ing questions:
- How is the physical state of a system described?
- How are physical observables represented?
- What are the possible outcomes of a given experiment?
- What is the expected result when an average over a very large number of obser-
vations is performed? - How does the physical state of a system evolve in time?
- What types of measurements are compatible with each other?
Let us begin by detailing how we describe the physical state of a system.
9.2.1 The state vector
In quantum theory, it is not possible to determine the precise outcome of a given single
experimental measurement. Thus, unlike in classical mechanics, where the microscopic
state of a system is specified by providing a definite set of coordinates and velocities of
the particles at any timet, the microscopic state of a system in quantum mechanics is
specified in terms of the probability amplitudes for the possible outcomes of different
measurements made on the system. Since we must be able to describe any type of
measurement, the specification of the amplitudes remains abstract until a particular
measurement is explicitly considered. The procedure for converting a set of abstract
amplitudes to probabilities associated with the outcomes of particular measurements
will be given shortly. For now, let us choose a mathematically useful construct for
listing these amplitudes. Such a list is conveniently represented as a vector of complex
numbers, which we can specify as a column vector:
|Ψ〉=
α 1
α 2
α 3
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