Solution to Show Relation of X, P with Representation of P=-ih/2π*∂/∂x +f(x)

[najima]

Homework Statement

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

Homework Equations

quantum mechanic by Liboff

The Attempt at a Solution

I 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)?

 

[gabbagabbahey]

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 X=x.What you don't know is what P is. You're supposed to show that given [X,P]=i \hbar and X=x that P must take the form -i \hbar \frac{d}{dx} +f(x)...to do this, just operate on a function g(x) by [x,P] while leaving P as an unknown operator...what do you get when you do that?

 

[najima]

thanx now I get it!

 

[najima]

I think I must derive that P is this form P=-ih/2π*∂/∂x +f(x)