Page 100 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.7 Postulates of quantum mechanics 91

dP ë ˆ r(ë)dë ˆ 1

As an example, we consider a particle in a one-dimensional box as discussed
in Section 2.5. Suppose that the state function Ø(x) for this particle is time-
independent and is given by

ðx
Ø(x) ˆ C sin 5 , 0 < x < a
a
where C is a constant which normalizes Ø(x). The eigenfunctions jni and
^
eigenvalues E n of the Hamiltonian operator H are
r 
2 2
2 nðx n h
jniˆ sin , E n ˆ , n ˆ 1, 2, ...
a a 8ma 2
^
Obviously, the state function Ø(x) is not an eigenfunction of H. Following the
general procedure described above, we expand Ø(x) in terms of the eigenfunc-
tions jni. This expansion is the same as an expansion in a Fourier series, as
described in Appendix B. As a shortcut we may use equations (A.39) and
(A.40) to obtain the identity
1
5
sin è ˆ (10 sin è ÿ 5 sin 3è ‡ sin 5è)
16
so that the expansion of Ø(x)is
" #

C ðx 3ðx 5ðx
Ø(x) ˆ 10 sin ÿ 5 sin ‡ sin
16 a a a
r 
C a
ˆ (10j1iÿ 5j3i‡j5i)
16 2
A measurement of the energy of a particle in state Ø(x) yields one of three
values and no other value. The values and their probabilities are
h 2 10 2 100
E 1 ˆ , P 1 ˆ ˆ ˆ 0:794
2
2
8ma 2 10 ‡ 5 ‡ 1 2 126
9h 2 5 2
E 3 ˆ , P 3 ˆ ˆ 0:198
8ma 2 126
25h 2 1 2
E 5 ˆ , P 5 ˆ ˆ 0:008
8ma 2 126
The sum of the probabilities is unity,
P 1 ‡ P 3 ‡ P 5 ˆ 0:794 ‡ 0:198 ‡ 0:008 ˆ 1

The interpretation that the quantity Ø (q 1 , q 2 , ... , t)Ø(q 1 , q 2 , ... , t)is
the probability density that the coordinates of the system at time t are