Page 110 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.11 Heisenberg uncertainty principle 101
^ ^ 2 ^ ^
h(A ÿhAi)Øj(A ÿhAi)Øi‡ ë h(B ÿhBi)Øj(B ÿhBi)Øi
^
^
^
^
‡ iëh(A ÿhAi)Øj(B ÿhBi)Øiÿ iëh(B ÿhBi)Øj(A ÿhAi)Øi > 0
^
^
or, since A and B are hermitian
^ 2 2 ^ 2
hØj(A ÿhAi) jØi‡ ë hØj(B ÿhBi) jØi
^
^
‡ iëhØj[A ÿhAi, B ÿhBi]jØi > 0
Applying equations (3.75) and (3.78), we have
2
2
2
(ÄA) ‡ ë (ÄB) ÿ ëhCi > 0
If we complete the square of the terms involving ë, we obtain
2 2
hCi hCi
2
2
(ÄA) ‡ (ÄB) ë ÿ ÿ > 0
2(ÄB) 2 4(ÄB) 2
Since ë is arbitrary, we select its value so as to eliminate the second term
hCi
ë ˆ (3:80)
2(ÄB) 2
thereby giving
2 2 1 2
(ÄA) (ÄB) > hCi
4
or, upon taking the positive square root,
1
ÄAÄB > jhCij
2
Substituting equation (3.77) into this result yields
1 ^ ^
ÄAÄB > jh[A, B]ij (3:81)
2
This general expression relates the uncertainties in the simultaneous measure-
^
^
ments of A and B to the commutator of the corresponding operators A and B
and is a general statement of the Heisenberg uncertainty principle.
Position±momentum uncertainty principle
^
We now consider the special case for which A is the variable x (A ˆ x) and B
^ ^
^
is the momentum p x (B ˆÿi" d=dx). The commutator [A, B] may be evaluated
by letting it operate on Ø

dØ dxØ
^ ^
[A, B]Ø ˆÿi" x ÿ ˆ i"Ø
dx dx
^ ^
so that jh[A, B]ij ˆ " and equation (3.81) gives
"
ÄxÄp x > (3:82)
2
The Heisenberg position±momentum uncertainty principle (3.82) agrees
with equation (2.26), which was derived by a different, but mathematically