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Analytic function, creation and annihilation operators proof

  1. Nov 24, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that

    f(aa)a = af(aa + 1)

    Where f is any analytic function and a and a† satisfy commutation relation [a, a] = 1.

    3. The attempt at a solution
    I have used [a, a†] = aa†-a†a=1 to write the expression like

    f(a†a)a†= a†f(aa†)

    but I don't know what to do next.

    I know that analytic function can be written like f(x)=Ʃ kn(x-x0)2 and that it is infinitely differentiable, but I don't see how can I sucesssfuly
    apply this, or there some other trick here.
     
  2. jcsd
  3. Nov 24, 2011 #2

    dextercioby

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    Science Advisor
    Homework Helper

    So f is essentially a power series and you can then use induction for an arbitrary power of (a+ a)
     
  4. Nov 24, 2011 #3
    I think now I understand,
    so on the left hand side for f(a†a) I will have powers of a†a
    like a†a + a†aa†a ...
    and on the right hand side for f(aa†) i will have powers of aa†
    like aa† + aa†aa† ...

    but since I have to multiply these series by a† from the opposite sides
    (a†a + a†aa†a ... ) /a† = a†aa† + a†aa†aa† ...
    a†\ ( aa† + aa†aa† ... ) = a†aa† + a†aa†aa† ...

    they turn out to be the same, right ?
     
  5. Nov 24, 2011 #4
    also I have a question which builds on this one (so I will stay in this thread),
    A bosonic one level system can be described by the Hamiltonian
    H= εa†a,

    The expectation value of n = a†a is defined as n(ε) = <n> = tr(ρa†a) where

    ρ=(1/ZG)*Exp[-β(ε-μ)a†a] , tr(ρ)=1

    is the grand canonical density matrix.
    Use f(a†a)a† = a†f(a†a + 1) to show that the form

    n(ε) = 1/(Exp[β(ε-μ)]

    of the Bose-Einstein distribution directly follows from the bosonic commutation
    relation for a and a†.

    I don't see where to apply my result from the first part, to trace function maybe ?
    But that does not have appropriate form.
     
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