Page 108 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.11 Heisenberg uncertainty principle 99

dhxi dhyi dhzi
m ˆhp x i, m ˆhp y i, m ˆhp z i
dt dt dt
or, in vector notation
dhri
m ˆhpi
dt
which is one of the Ehrenfest theorems discussed in Section 2.3. The other
Ehrenfest theorem,

dhpi
ˆÿh=V(r)i
dt
^
may be obtained from equation (3.72) by setting A successively equal to ^ p x ,
^ p y , and ^ p z .

3.11 Heisenberg uncertainty principle
We have shown in Section 3.5 that commuting hermitian operators have
simultaneous eigenfunctions and, therefore, that the physical quantities asso-
ciated with those operators can be observed simultaneously. On the other hand,
^
^
if the hermitian operators A and B do not commute, then the physical
observables A and B cannot both be precisely determined at the same time. We
begin by demonstrating this conclusion.
^
^
Suppose that A and B do not commute. Let á i and â i be the eigenvalues of A ^
^
and B, respectively, with corresponding eigenstates já i i and jâ i i
^
Ajá i iˆ á i já i i (3:73a)
^
Bjâ i iˆ â i jâ i i (3:73b)
Some or all of the eigenvalues may be degenerate, but each eigenfunction has a
unique index i. Suppose further that the system is in state já j i, one of the
^
eigenstates of A. If we measure the physical observable A, we obtain the result
á j . What happens if we simultaneously measure the physical observable B?To
answer this question we need to calculate the expectation value hBi for this
system
^
hBiˆ há j jBjá j i (3:74)
If we expand the state function já j i in terms of the complete, orthonormal set
jâ i i
X
já j iˆ c i j â i i
i
where c i are the expansion coef®cients, and substitute the expansion into
equation (3.74), we obtain