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Quantum Mechanics: Evaluate the following using Ket-Bra notation

  1. Feb 3, 2012 #1
    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!

    1. The problem statement, all variables and given/known data
    Evaluate the following using Ket-Bra notation
    a.)[tex]exp[if(A)]=?[/tex]
    Where A is a Hermitian operator whose eigenvalues are known.

    b.)[tex]\Sigma _{a'}\psi^{*}_{a'}(x') \psi_{a'}(x'')[/tex]

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


    3. 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.
    [tex]exp[if(A)]=1+if(A)+\frac{(if(A))^2}{2!}+\frac{(if(A))^3}{3!}+...[/tex]

    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? :\
     
  2. jcsd
  3. Feb 3, 2012 #2
    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,

    [itex]
    A = \sum_{a'} a' | a' \rangle \langle a' |
    [/itex]

    Where [itex]a'[/itex] is an eigenvalue and [itex]|a' \rangle[/itex] is the corresponding eigenket. That gets [itex]A[/itex] 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.
     
    Last edited by a moderator: Feb 3, 2012
  4. Feb 3, 2012 #3
    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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