QUANTUM OPERATORS
19.2.2 Uncertainty principlesThe next topic we explore is the quantitative consequences of a non-zero com-
mutator for two quantum (Hermitian) operators that correspond to physical
variables.
As previously noted, the expectation value in a state|ψ〉of the physical quantityAcorresponding to the operatorAisE[A]=〈ψ|A|ψ〉. Any one measurement of
Acan only yield one of the eigenvalues ofA. But if repeated measurements could
be made on a large number of identical systems, a discrete or continuous range
of values would be obtained. It is a natural extension of normal data analysis
to measure the uncertainty in the value ofAby the observed variance in the
measured values ofA, denoted by (∆A)^2 and calculated as the average value of
(A−E[A])^2. The expected value of this variance for the state|ψ〉is given by
〈ψ|(A−E[A])^2 |ψ〉.
We now give a mathematical proof that there is a theoretical lower limit forthe product of the uncertainties in any two physical quantities, and we start by
proving a result similar to the Schwarz inequality. Let|u〉and|v〉be any two state
vectors and letλbe anyrealscalar. Then consider the vector|w〉=|u〉+λ|v〉and,
in particular, note that
0 ≤〈w|w〉=〈u|u〉+λ(〈u|v〉+〈v|u〉)+λ^2 〈v|v〉.This is a quadratic inequality inλand therefore the quadratic equation formed
by equating the RHS to zero must have no real roots. The coefficient ofλis
(〈u|v〉+〈v|u〉)=2Re〈u|v〉and its square is thus≥0. The condition for no real
roots of the quadratic is therefore
0 ≤(〈u|v〉+〈v|u〉)^2 ≤ 4 〈u|u〉〈v|v〉. (19.36)This result will now be applied to state vectors constructed from|ψ〉, the statevector of the particular system for which we wish to establish a relationship be-
tween the uncertainties in the two physical variables corresponding to (Hermitian)
operatorsAandB.Wetake
|u〉=(A−E[A])|ψ〉 and |v〉=i(B−E[B])|ψ〉. (19.37)Then
〈u|u〉=〈ψ|(A−E[A])^2 |ψ〉=(∆A)^2 ,
〈v|v〉=〈ψ|(B−E[B])^2 |ψ〉=(∆B)^2.Further,
〈u|v〉=〈ψ|(A−E[A])i(B−E[B])|ψ〉=i〈ψ|AB|ψ〉−iE[A]〈ψ|B|ψ〉−iE[B]〈ψ|A|ψ〉+iE[A]E[B]〈ψ|ψ〉=i〈ψ|AB|ψ〉−iE[A]E[B].