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Chapter Summary:

4. States and Observables

This chapter describes the basic structure of any quantum mechanical theory. The first step in the mathematical formulation is to associate a suitable Hilbert space with a physical system. The vectors in that Hilbert space (or rather the one-dimensional subspaces spanned by these vectors) describe the possible states of the system. Certain linear operators correspond to the physically observable quantities. The expectation value of an observable defines the link between the mathematical quantities and physical experiments. It is interpreted as the mean value of many measurements of that observable.

For the single particle systems described in this book, the Hilbert space is always represented as a set of square-integrable functions. You will learn how to find the operators corresponding to the classical observables, and you will learn how to predict the probability for measuring certain values of an observable in a given state.

The transition from classical to quantum mechanics can be formally achieved by replacing the classical observables with linear operators according to the following substitution rule. Always replace the position variable x with the operator of multiplication by x and the momentum p by the differential operator -i \hbar \nabla. Unlike the classical quantities, the quantum mechanical operators of position and momentum do not commute. Instead, they satisfy "canonical commutation relations" where the commutator replaces the classical Poisson brackets. A pair of non-commuting operators leads to a generalized uncertainty relation and limits the accuracy with which certain "incompatible" properties of a physical system can be predicted simultaneously.

The quantum mechanical observable corresponding to the kinetic energy of a particle is the Laplace operator in the Schrödinger equation. We can obtain the Schrödinger equation in an electromagnetic field by replacing the Laplace operator with the operator describing the energy of a particle in an electromagnetic field. The classical energy is described by the Hamiltonian function which expresses the energy in terms of the position and momentum variables. An application of the substitution rule leads to the "Hamiltonian operator" of quantum mechanics. We describe this process for electric and magnetic fields and discuss the "gauge freedom" related to the non-uniqueness of the electromagnetic potentials. Finally, projection operators are introduced to describe "properties" of a physical system. They are used to determine the probability that the measured value of an observable is found within a given set of possible values.