Physical Chemistry Third Edition

684 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

16.1 The First Two Postulates of Quantum Mechanics

The Schrödinger equation does not provide a complete theory of quantum mechanics.

Schrödinger, Heisenberg, and others devised severalpostulates(unproved fundamental

assumptions) that form a consistent logical foundation for nonrelativistic quantum

mechanics. In any theory based on postulates, the validity of the postulates is tested by

comparing the consequences of the postulates with experimental fact. The postulates of

quantum mechanics do pass this test. These postulates can be stated in slightly different

ways. We will state five postulates in a form similar to that of Mandl^1 and Levine.^2 The

first two postulates were introduced in Chapter 15 without calling them postulates. We

now state them explicitly:

Werner Karl Heisenberg, 1901–1976,
was a German physicist who
invented matrix mechanics, a form of
quantum mechanics equivalent to the
Schrödinger formulation. He discovered
the uncertainty principle, for which he
received the 1932 Nobel Prize in
physics.

Postulate 1. All information that can be obtained about the state of a mechanical system

is contained in a wave functionΨ, which is a continuous, finite, and single-valued function

of time and of the coordinates of the particles of the system.

This postulate implies that there is a one-to-one relationship between the state of

the system and a wave function. That is, each possible state corresponds to one wave

function, and each possible wave function corresponds to one state. The terms “state

function” and “wave function”are often used interchangeably. Information about values

of mechanical variables such as energy and momentum must be obtained from the wave

function, instead of from values of coordinates and velocities as in classical mechanics.

The fourth postulate will provide the method for obtaining this information.

Postulate 2. The wave functionΨobeys the time-dependent Schrödinger equation

̂HΨih ̄∂Ψ

∂t

(16.1-1)

whereĤis the Hamiltonian operator of the system.

The time-independent Schrödinger equation can be derived from the time-dependent

equation, as was shown in Chapter 15, by assuming that the wave function is a product

of a coordinate factor and a time factor:

Ψ(q,t)ψ(q)η(t) (16.1-2)

whereqstands for all of the coordinates of the particles in the system and where the

coordinate wave functionψsatisfies the time-independent Schrödinger equation. Not

all wave functions consist of the two factors in Eq. (16.1-2), but all wave functions

must obey the time-dependent Schrödinger equation.

16.2 The Third Postulate. Mathematical Operators

and Mechanical Variables

The time-independent Schrödinger equation determines the energy eigenvalues for a

given system. These eigenvalues are the possible values that the energy of the system

(^1) F. Mandl,Quantum Mechanics, Butterworths Scientific Publications, London, 1957, p. 60ff.
(^2) I. N. Levine,Quantum Chemistry, 5th ed., Prentice-Hall, Englewood Cliffs, NJ, 2000.