Page 101 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

92 General principles of quantum theory

q 1 , q 2 , ... may be shown by comparing equations (3.46) and (3.54) for A equal
to the coordinate vector q

hqiˆhØjqjØiˆ Ø (q)Ø(q)q dô

hqiˆ r(q)q dô

For these two expressions to be mutually consistent, we must have

r(q) ˆ Ø (q)Ø(q)

Thus, this interpretation of Ø Ø follows from postulate 3 and for this reason
is not included in the statement of postulate 1.

Collapse of the state function
The measurement of a physical observable A gives one of the eigenvalues ë n of
^
the operator A. As stated by the fourth postulate, a consequence of this
measurement is the sudden change in the state function of the system from its
original form Ø to an eigenfunction or linear combination of eigenfunctions of
^
A corresponding to ë n .
At a ®xed time t just before the measurement takes place, the ket jØi may
^
be expanded in terms of the eigenkets jiái of A, as shown in equation (3.48). If
the measurement gives a non-degenerate eigenvalue ë n , then immediately after
the measurement the system is in state jni. The state function Ø is said to
collapse to the function jni. A second measurement of A on this same system,
if taken immediately after the ®rst, always yields the same result ë n . If the
eigenvalue ë n is degenerate, then right after the measurement the state function
is some linear combination of the eigenkets jnái, á ˆ 1, 2, ... , g n . A second,
immediate measurement of A still yields ë n as the result.
From postulates 4 and 5, we see that the state function Ø can change with
time for two different reasons. A discontinuous change in Ø occurs when some
property of the system is measured. The state of the system changes suddenly
from Ø to an eigenfunction or linear combination of eigenfunctions associated
with the observed eigenvalue. An isolated system, on the other hand, undergoes
a continuous change with time in accordance with the time-dependent SchroÈ-
dinger equation.

Time evolution of the state function
The ®fth postulate stipulates that the time evolution of the state function Ø is
determined by the time-dependent Schrodinger equation
È