Show that: Translator operator can be expressed in terms of

[Diomarte]

Homework Statement

The translation operator T(a) is defined to be such that: T(a)ψ(x) = ψ(x+a)
Show that:
T(a) can be expressed in terms of the operator p = -iħ d/dx
and T(a) is unitary.

Homework Equations

T(a)ψ(x) = ψ(x+a)
p = -iħ d/dx

The Attempt at a Solution

I honestly have no idea how to start this expression, I figured there may be some way of doing it in a Taylor's expansion maybe, but I'm not sure how that would work, and that's the only guess I have on how to start this. Could anybody possibly give me some direction on where to start, please? Thank you very much.

 

[diazona]

Enough experience in physics will teach you that Taylor expansions can do anything ;-) In particular, whenever you have a parametrized operator like this, it's usually useful to consider small values of the parameter. So try assuming that a is very small, write out the series expansion for whatever needs to be expanded, and see where that takes you.

 

[Diomarte]

Ok, I seem to have figured that part out, now my second part is to show that T(a) is Unitary.
Thank you very much for your help, and hopefully showing that T(a) is unitary shouldn't be terrible difficult since if I'm correct I've got I think the correct results came out to be T(a)ψ(x) = exp(iap/ħ)ψ(x) = f(x+a)