Mathematical Methods for Physics and Engineering - Matematica.NET

QUANTUM OPERATORSlater algebraic convenience:[ ] [∞∑A, eλB= A,==∞∑n=0∞∑n=1n=0(λB) nn!]∞∑=n=0λ nn! [ A, Bn ]λ nn! nBn−1 [ A, B ] , using the earlier result,λ nn! nBn−1 [ A, B ]∞∑ λ m B m= λ [ A, B ] ,writingm = n − 1,m!m=0= λe λB [ A, B ] .Now consider the derivative with respect to λ of the functionf(λ) =e λA e λB e −λ(A+B) .In the following calculation we use the fact that the derivative of e λC is Ce λC ; this is thesame as e λC C, since any two functions of the same operator commute. Differentiating thethree-factor product givesdfdλ = eλA Ae λB e −λ(A+B) + e λA e λB Be −λ(A+B) + e λA e λB (−A − B)e −λ(A+B)= e λA (e λB A + λe λB [ A, B ] )e −λ(A+B) + e λA e λB Be −λ(A+B)− e λA e λB Ae −λ(A+B) − e λA e λB Be −λ(A+B)= e λA λe λB [ A, B ] e −λ(A+B)= λ [ A, B ] f(λ).In the second line we have used the result obtained above to replace Ae λB ,andinthelastline have used the fact that [ A, B ] commutes with each of A and B, and hence with anyfunction of them.Integrating this scalar differential equation with respect to λ and noting that f(0) = 1,we obtainln f = 1 2 λ2 [ A, B ] ⇒ e λA e λB e −λ(A+B) = f(λ) =e 1 2 λ2 [ A,B ] .Finally, post-multiplying both sides of the equation by e λ(A+B) and setting λ =1yieldse A e B = e 1 2 [ A,B ]+A+B . ◭19.2 Physical examples of operatorsWe now turn to considering some of the specific linear operators that play apart in the description of physical systems. In particular, we will examine theproperties of some of those that appear in the quantum-mechanical descriptionof the physical world.As stated earlier, the operators corresponding to physical observables are restrictedto Hermitian operators (which have real eigenvalues) as this ensures thereality of predicted values for experimentally measured quantities. The two basic656