Plug in thatλ.
(∆A)^2 −
1
4
〈ψ|[U,V]|ψ〉^2
(∆B)^2+
〈ψ|[U,V]|ψ〉^2
2(∆B)^2≥ 0
(∆A)^2 (∆B)^2 ≥−
1
4
〈ψ|[U,V]|ψ〉^2 =〈ψ|i
2[U,V]|ψ〉^2This result is theuncertainty for non-commuting operators.
(∆A)(∆B)≥
i
2〈[A,B]〉
If the commutator is a constant, as in the case of [p,x], the expectation values can be removed.
(∆A)(∆B)≥
i
2[A,B]
For momentum and position, this agrees with the uncertainty principle we know.
(∆p)(∆x)≥
i
2〈[p,x]〉=
̄h
2(Note that we could have simplified the proof by just stating that wechoose to dot (U+i2(∆〈[U,VB)] 2 〉V)ψ
into itself and require that its positive. It would not have been clear that this was the strongest
condition we could get.)
11.4 Time Derivative of Expectation Values*.
We wish to compute the time derivative of the expectation value of anoperatorAin the stateψ.
Thinking about the integral, this has three terms.
d
dt
〈ψ|A|ψ〉 =〈
dψ
dt
|A|ψ〉
+
〈
ψ∣
∣
∣
∣
∂A
∂t∣
∣
∣
∣ψ〉
+
〈
ψ|A|dψ
dt〉
=
− 1
i ̄h〈Hψ|A|ψ〉+1
i ̄h〈ψ|A|Hψ〉+〈
ψ∣
∣
∣
∣
∂A
∂t∣
∣
∣
∣ψ〉
=
i
̄h〈ψ|[H,A]|ψ〉+〈
ψ∣
∣
∣
∣
∂A
∂t∣
∣
∣
∣ψ〉
This is an important general result for thetime derivative of expectation values.
d
dt〈ψ|A|ψ〉=i
̄h〈ψ|[H,A]|ψ〉+〈
ψ∣
∣
∣
∣
∂A
∂t∣
∣
∣
∣ψ