FOUNDATIONS OF QUANTUM MECHANICS

IV. 5. THE UNCERTAINTY RELATIONS 99
The Cauchy - Schwarz inequality (II. 12), p. 19, for the vectors A ψ |ψ⟩ and B ψ |ψ⟩ reads
⟨A ψ ψ | A ψ ψ⟩ ⟨B ψ ψ | B ψ ψ⟩ ∣ ∣ ⟨Aψ ψ | B ψ ψ⟩ ∣ ∣ 2 . (IV. 35)
Because A ψ and B ψ are self - adjoint, we can also write this inequality as follows,
⟨A 2 ψ ⟩ ψ ⟨B 2 ψ ⟩ ψ ∣ ∣⟨A ψ B ψ ⟩ ψ
∣ ∣
2 . (IV. 36)
Using both the commutator [· , ·] − and the anti - commutator [· , ·] + , we find for the right - hand side
of (IV. 36)
∣ ⟨Aψ B ψ ⟩ ψ
∣ ∣
2
where the cross - term disappears because of
Furthermore,
= ∣ 1
2 ⟨[A ψ, B ψ ] − ⟩ ψ + 1 2 ⟨[A ∣
ψ, B ψ ] + ⟩ ψ 2
= 1 ∣
∣
4 ⟨[Aψ , B ψ ] − ⟩ ψ 2 +
1
4 ⟨[A ψ, B ψ ] + ⟩ ψ 2 , (IV. 37)
⟨[A ψ , B ψ ] − ⟩ ∗ ψ = − ⟨[A ψ, B ψ ] − ⟩ ψ
⟨[A ψ , B ψ ] + ⟩ ∗ ψ = + ⟨[A ψ, B ψ ] + ⟩ ψ . (IV. 38)
[A ψ , B ψ ] − = [A, B] − , (IV. 39)
and we obtain the inequality
⟨A 2 ψ ⟩ ψ ⟨B 2 ψ ⟩ ψ 1 4
∣ ∣
∣⟨[A, B] − ⟩ ψ 2 +
1
4 ⟨[A ψ, B ψ ] + ⟩ ψ 2 . (IV. 40)
In view of the inequalities (IV. 26) and (IV. 40), we make a few remarks.
(i) Leaving out the last term on the right - hand side of inequality (IV. 40) gives the better known
but weaker inequality, derived by H.P. Robertson (1929),
⟨A 2 ψ ⟩ ψ ⟨B 2 ψ ⟩ ψ 1 4
∣
∣⟨[A, B] − ⟩ ψ
∣ ∣
2 . (IV. 41)
(ii) Notice that ⟨A 2 ψ ⟩ ψ is equal to the square of the standard deviation of the quantity A in the
state |ψ⟩,
⟨A 2 ψ ⟩ ψ = ⟨(A − ⟨A⟩ ψ ) 2 ⟩ = (∆ ψ A) 2 . (IV. 42)
(iii) For the special case A = Q and B = P , the Robertson inequality (IV. 41) transforms into the
Kennard inequality (IV. 26), and the expressions (IV. 29) and (IV. 31) correspond to ⟨Q 2 ψ ⟩ ψ in
the q - language and ⟨P 2
ψ ⟩ ψ in the p - language.
(iv) Notice that in deriving these uncertainty relations the interpretation of the uncertainties plays
no role.