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Operator in exponential function and generating function

  1. Jun 18, 2009 #1
    J.J. Sakurai Modern Quantum Mechanics p. 74
    It says,
    [A,H] = 0;
    H|a'> = Ea' |a'>
    where H is the Hamiltonian A is any observable |a'> is eigenket of A

    then,

    exp ( -iHt/h)|a'> = exp (-iEa't/h)|a'>

    where h is the reduced planck's constant.
    I want to know WHY ?
    and besides, I would like to ask what is generating function means?
    I have encountered
    xnew= x + dx
    pnew = p
    then generating function for infinitesimal translation is
    F(x,P) = x · P + p · dx
    How can I understand it ?

    Thank You ! I wish this time is readily to read !
     
  2. jcsd
  3. Jun 18, 2009 #2

    CompuChip

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    Science Advisor
    Homework Helper

    You can use the definition:

    [tex]e^{-iHt/h} = \sum_{n = 0}^\infty \frac{1}{n!} \left( -\frac{i}{h} H t \right)^n[/tex]
    Now t and i/h are numbers, so
    [tex]\left( -\frac{i}{h} H t \right)^n = \left( -\frac{i}{h} t \right)^n H^n[/tex]
    and you can easily show that
    [tex]H^n |a'\rangle = E_{a'}^n |a'\rangle[/tex]
    to get
    [tex]\sum_{n = 0}^\infty \frac{1}{n!} \left( -\frac{i}{h} E_{a'} t \right)^n = e^{-i E_{a'} t / h}[/tex]
     
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