View the attached file for proper formatting of formulas.
Consider two hermitian operators A and B which satisfy the following commutation relation:
[A, B] = AB-BA=iC, where C is also a hermitian operator in general. Let us introduce a new operator Q defined by: Q=A+ iλB, with λ being a real number, and consider the following scalar product:
Where is any normalized wave function?
(a) Show that Eq. (1) leads to the following result:
(b) Define the uncertainties as follows: With U≡A - <A> and V≡B - < B>, respectively. Show that [U, V] = [A, B] = iC
(c) Thus, replacing A, B in (a) by U,V, we obtain: Regarding I(λ) as a quadratic function ofλ, show that I(λ) is minimum when
(d) Let A=X, B= px, show that the quantum condition [x, px]= i leads to the uncertainty principle:
The solution shows complete formulas together with explanations in solving the problem.