19.2 PHYSICAL EXAMPLES OF OPERATORS
In the second line, we have moved expectation values, which are purely numbers,
out of the inner products and used the normalisation condition〈ψ|ψ〉=1.
Similarly
〈v|u〉=−i〈ψ|BA|ψ〉+iE[A]E[B].Adding these two results gives
〈u|v〉+〈v|u〉=i〈ψ|AB−BA|ψ〉,and substitution into (19.36) yields
0 ≤(i〈ψ|AB−BA|ψ〉)^2 ≤4(∆A)^2 (∆B)^2At first sight, the middle term of this inequality might appear to be negative, but
this is not so. SinceAandBare Hermitian,AB−BAis anti-Hermitian, as is easily
demonstrated. Sinceiis also anti-Hermitian, the quantity in the parentheses in
the middle term is real and its square non-negative. Rearranging the equation and
expressing it in terms of the commutator ofAandBgives the generalised form
of theUncertainty Principle. For any particular state|ψ〉of a system, this provides
the quantitative relationship between the minimum value that the product of the
uncertainties inAandBcan have and the expectation value, in that state, of
their commutator,
(∆A)^2 (∆B)^2 ≥^14 |〈ψ|[A, B]|ψ〉|^2. (19.38)Immediate observations include the following:
(i) IfAandBcommute there is no absolute restriction on the accuracy with
which the corresponding physical quantities may be known. That is not to
say that ∆Aand ∆Bwill always be zero, only that they may be.
(ii) If the commutator ofAandBis a constant,k= 0, then the RHS of
equation (19.38) is necessarily equal to^14 |k|^2 , whatever the form of|ψ〉,
and it is not possible to have ∆A=∆B=0.
(iii) Since the RHS depends upon|ψ〉, it is possible, even for two operators
that do not commute, for the lower limit of (∆A)^2 (∆B)^2 to be zero. This
will occur if the commutator[A, B]is itself an operator whose expectation
value in the particular state|ψ〉happens to be zero.To illustrate the third of these, we might consider the components of angularmomentum discussed in the previous subsection. There, in equation (19.27), we
found that the commutator of the operators corresponding to thex-andy-
components of angular momentum is non-zero; in fact, it has the valueiLz.
This means that if the state|ψ〉of a system happened to be such that〈ψ|Lz|ψ〉=0,
as it would if, for example, it were the eigenstate ofLz,|ψ〉=|, 0 〉, then there
would be no fundamental reason why the physical values of bothLxandLy
should not be known exactly. Indeed, if the state were spherically symmetric, and