Page 94 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.7 Postulates of quantum mechanics 85

3.7 Postulates of quantum mechanics
In this section we state the postulates of quantum mechanics in terms of the
properties of linear operators. By way of an introduction to quantum theory, the
basic principles have already been presented in Chapters 1 and 2. The purpose
of that introduction is to provide a rationale for the quantum concepts by
showing how the particle±wave duality leads to the postulate of a wave
function based on the properties of a wave packet. Although this approach,
based in part on historical development, helps to explain why certain quantum
concepts were proposed, the basic principles of quantum mechanics cannot be
obtained by any process of deduction. They must be stated as postulates to be
accepted because the conclusions drawn from them agree with experiment
without exception.
We ®rst state the postulates succinctly and then elaborate on each of them
with particular regard to the mathematical properties of linear operators. The
postulates are as follows.
1. The state of a physical system is de®ned by a normalized function Ø of the spatial
coordinates and the time. This function contains all the information that exists on
the state of the system.
^
2. Every physical observable A is represented by a linear hermitian operator A.
3. Every individual measurement of a physical observable A yields an eigenvalue of
^
the corresponding operator A. The average value or expectation value hAi from a
series of measurements of A for systems, each of which is in the exact same state
Ø, is given by hAiˆhØjAjØi.
4. If a measurement of a physical observable A for a system in state Ø gives the
^
eigenvalue ë n of A, then the state of the system immediately after the measurement
is the eigenfunction (if ë n is non-degenerate) or a linear combination of eigenfunc-
tions (if ë n is degenerate) corresponding to ë n .
5. The time dependence of the state function Ø is determined by the time-dependent
Schrodinger differential equation
È
@Ø
^
i" ˆ HØ
@t
^
where H is the Hamiltonian operator for the system.
This list of postulates is not complete in that two quantum concepts are not
covered, spin and identical particles. In Section 1.7 we mentioned in passing
that an electron has an intrinsic angular momentum called spin. Other particles
also possess spin. The quantum-mechanical treatment of spin is postponed until
Chapter 7. Moreover, the state function for a system of two or more identical
and therefore indistinguishable particles requires special consideration and is
discussed in Chapter 8.