Exponential of an operator into bra-ket notation

[Bravus]

Homework Statement

The question is to evaluate the expression e^-iA, where A is a Hermitian operator whose eigenvalues are known (but not given) using bra-ket algebra.

Homework Equations

See above.

The Attempt at a Solution

I have been looking around, reading the textbook and course notes, checking the web and so on. I have learned that an expression in this form is a 'unitary operator', and what that means.

I'm pretty comfortable with the bra-ket notation, but I'm struggling just to *get into it*. ;-)

If you can help me find a 'way in' to get started, that'd be great: at this point I'm still just kinda staring blankly at it, despite all my efforts so far.

Oh, yeah, I overhead another student mention using a Taylor series. I get how to convert the exponential to a Taylor series, but I'm not so sure how it helps... and the other student might be wrong.

 

[tiny-tim]

Welcome to PF!

Hi Bravus! Welcome to PF!

(try using the X² button just above the Reply box )

The question is to evaluate the expression e^-iA, where A is a Hermitian operator whose eigenvalues are known (but not given) using bra-ket algebra.

I think they mean if A|x_n> = λ_n |x_n>,

then what is e^-iA|∑a_nx_n> ?

 

[Bravus]

Thanks for the tip on the nicenesses allowed by the forum software - so nice after spending time answering math and physics questions on 'Yahoo Answers' and having to struggle with all-text!

Not sure the answer is giving me a way in, though: maybe I'm just thick. The question definitely just says 'evaluate e^-iA'.

 

[tiny-tim]

Hi Bravus!

(just got up :zzz:)

The question definitely just says 'evaluate e^-iA'.

but anyway, what is e^-iA|∑a_nx_n> ?

 

[Bravus]

Thanks, and the hint is definitely a handy one. It's Mr Taylor and his Series that really gets the job done in this instance, though...