7.4. THEGENERALIZEDUNCERTAINTYPRINCIPLE 117
andsquaringbothsides,wehave
|<ψ 1 |ψ 1 >||<ψ 2 |ψ 2 >| ≥ |<ψ 1 |ψ 2 >|^2(∆A)^2 (∆B)^2 ≥1
2
[
|<ψ 1 |ψ 2 >|^2 +|<ψ 2 |ψ 1 >|^2]≥
1
2
[
|<DAψ|DBψ>|^2 +|<DBψ|DAψ>|^2]≥
1
2
[
|<ψ|DADBψ>|^2 +|<ψ|DBDAψ>|^2](7.123)
Nextwrite
DADB = F+G
DBDA = F−G (7.124)where
F =1
2
(DADB+DBDA)
G =
1
2
[DA,DB]=
1
2
[A,B] (7.125)
Substitutingtheseexpressionsinto(7.123)weget
(∆A)^2 (∆B)^2 ≥1
2
[(+)(+)∗
+(−)(−)∗]
≥ ||^2 +||^2
≥ ||^2
≥
1
4
|<ψ|[A,B]|ψ>|^2 (7.126)TakingthesquarerootofbothsidesestablishesthegeneralizedUncertaintyPrinciple.
Inparticular, theHeisenbergUncertaintyprinciplefollowsbychoosingA=xand
B=p,inwhichcase
∆x∆p ≥1
2
|<ψ|i ̄h|ψ>|≥̄h
2(7.127)
asseeninLecture6.
Problem:Byminimizingthenormofthevector
|u>+q|v> (7.128)withrespecttoq,provetheCauchy-Schwarzinequality
|u||v|≥|<u|v>| (7.129)