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P operator

  1. Nov 21, 2008 #1
    1. The problem statement, all variables and given/known data
    given that X(operator) and P (operator) operate on functions,and the relation [X,P]=ih/2π,show that if X(operator)=x ,and P (operator) has the representation P=-ih/2π*∂/∂x +f(x)
    where f(x) is an arbitrary function of x

    2. Relevant equationsquantum mechanic by Liboff

    3. The attempt at a solutionI wrote the commutator relation of P and x on an arbitrary function like g(x) ,[x,p]g(x) so XP(g(x))-PX(g(x))=ih/2pi(g(x)) because of X=x so I wrote
    xP(g(x))-P(xg(x))=xP(g(x))-xP(g(x))-g(x)Px=-g(x)Px=ih/2pi(g(x)) so I can derive just this part of equation-ih/2π*∂/∂x , what can I do for the part of f(x)?
  2. jcsd
  3. Nov 21, 2008 #2


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    If I'm interpreting the question correctly, your not supposed to derive the commutation relation; it's actually a given. You are also told that [itex]X=x[/itex].What you don't know is what [itex]P[/itex] is. You're supposed to show that given [tex] [X,P]=i \hbar[/tex] and [itex]X=x[/itex] that [itex]P[/itex] must take the form [tex]-i \hbar \frac{d}{dx} +f(x)[/tex]....to do this, just operate on a function [itex]g(x)[/itex] by [itex] [x,P][/itex] while leaving [itex]P[/itex] as an unknown operator....what do you get when you do that?
  4. Nov 21, 2008 #3
    thanx now I get it!!
  5. Nov 21, 2008 #4
    I think I must derive that P is this form P=-ih/2π*∂/∂x +f(x)
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