A Linear algebra problem
I have this linear algebra problem in the context of quantum mechanics. Let be a family of linear operators so to each we have a linear operator where is a complex vector space if one is unfamiliar with functional analysis (like I am) or is a Hilbert space if one is. Let’s suppose that this family is differentiable.
Suppose further that is always a Hermitian operator. Suppose that has a discrete spectrum of eigenvalues . I need to show the following:
Now here is a “proof,” it is not quite rigorous since there are probably a lot of technical details regarding functional analysis that I’m missing out on but:
Proof We begin by differentiating the eigenvalue equation with respect to using the product rule:
After multiplying by and rearranging terms we have the following:
Now we can take the adjoint of both sides of the eigenvalue equation to get that since because the eigenvalues of a normal operator are real.…