Mathematical Methods for Physics and Engineering - Matematica.NET

19.2 PHYSICAL EXAMPLES OF OPERATORSIn 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 givesand substitution into (19.36) yields〈u | v〉 + 〈v | u〉 = i〈ψ | AB − BA|ψ〉,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, butthis is not so. Since A and B are Hermitian, AB −BA is anti-Hermitian, as is easilydemonstrated. Since i is also anti-Hermitian, the quantity in the parentheses inthe middle term is real and its square non-negative. Rearranging the equation andexpressing it in terms of the commutator of A and B gives the generalised formof the Uncertainty Principle. For any particular state |ψ〉 of a system, this providesthe quantitative relationship between the minimum value that the product of theuncertainties in A and B can have and the expectation value, in that state, oftheir commutator,Immediate observations include the following:(∆A) 2 (∆B) 2 ≥ 1 4 |〈ψ | [ A, B ] | ψ〉|2 . (19.38)(i) If A and B commute there is no absolute restriction on the accuracy withwhich the corresponding physical quantities may be known. That is not tosay that ∆A and ∆B will always be zero, only that they may be.(ii) If the commutator of A and B is a constant, k ≠ 0, then the RHS ofequation (19.38) is necessarily equal to 1 4 |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 operatorsthat do not commute, for the lower limit of (∆A) 2 (∆B) 2 to be zero. Thiswill occur if the commutator [ A, B ] is itself an operator whose expectationvalue 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), wefound that the commutator of the operators corresponding to the x- and y-components of angular momentum is non-zero; in fact, it has the value iL z .This means that if the state |ψ〉 of a system happened to be such that 〈ψ|L z |ψ〉 =0,as it would if, for example, it were the eigenstate of L z , |ψ〉 = |l, 0〉, then therewould be no fundamental reason why the physical values of both L x and L yshould not be known exactly. Indeed, if the state were spherically symmetric, and665