Quantum Mechanics: Evaluate the following using Ket-Bra notation

AI Thread Summary
The discussion revolves around evaluating two problems using Ket-Bra notation in quantum mechanics. The first part involves the expression exp[if(A)], where A is a Hermitian operator, and the user is unsure how to apply Ket-Bra notation after proposing a Taylor expansion approach. The second part, involving a summation of wave functions, has not been attempted yet, but the user anticipates difficulties. A participant suggests that while the first expression is too general for simplification, it can be represented in bra-ket form, although this does not lead to a solution. Ultimately, the user finds the necessary insight to solve the first part after some guidance.
Xyius
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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? :\
 
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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,

<br /> A = \sum_{a&#039;} a&#039; | a&#039; \rangle \langle a&#039; |<br />

Where a&#039; is an eigenvalue and |a&#039; \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.[/color]
 
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A ha! That fact which I had forgotten, was enough for me to be able to figure it out. Thank you :]
 
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