Quantum Mechanics: Evaluate the following using Ket-Bra notation

[Xyius]

This problem has 4 parts to it (that are unrelated to each other). I did the first two parts without too much trouble but these next two parts have me stumped!

Homework Statement

Evaluate the following using Ket-Bra notation
a.)$$exp[if(A)]=?$$
Where A is a Hermitian operator whose eigenvalues are known.

b.)$$\Sigma _{a'}\psi^{*}_{a'}(x') \psi_{a'}(x'')$$

Homework Equations

None that I can think of, I believe this is all mathematics.

The Attempt at a Solution

a.) Well I assume if(A) means a function of A multiplied by the imaginary number i. My gut instinct is to simply write this as a Taylor expansion.
$$exp[if(A)]=1+if(A)+\frac{(if(A))^2}{2!}+\frac{(if(A))^3}{3!}+...$$

I do not know where to go from here, or if this is the correct approach. It says to use ket-bra notation but I do not see where I could fit that in with this method.

b.) Honestly I have not tried this one yet so I do not wish to receive help on this just yet as I would like to try it for myself, I just put it up here because I am assuming I will have trouble with it when I finish part a.) :p

Can anyone help? :\

 

[Mike Pemulis]

I'm not sure about (a). I've taken some QM, and it seems like that expression is way too general to be simplified usefully. The only thing I can think of is that we can always write an operator as,

$A = \sum_{a'} a' | a' \rangle \langle a' |$

Where $a'$ is an eigenvalue and $|a' \rangle$ is the corresponding eigenket. That gets $A$ into bra-ket form, but it doesn't really get us closer to the answer.

But (b) is easy. Spoiler'd so you can view it if/when you need to.

Moderator note: Solution removed.

 

[Xyius]

A ha! That fact which I had forgotten, was enough for me to be able to figure it out. Thank you :]