Info on wave mechanics prob in one dimension need

[belleamie]

Hi there, i need help in a couple of questions that I'm just stumped
one of them :
A) use induction to show that
[ x (hat)^n, p(hat) sub "x" ] = i (hbar)n x(hat)^(n-1)

- so far I've figured out this equation is in relation to solve the above eq, but I'm not entirely sure how to connect the two
[ f (x (hat)), p(hat) sub "x"] = i h(bar) (partial F/ partial x) * (x (hat))

B) I'm not sure how to show the symbol "pitch fork" but i will refer to it as "tsi"

Show for infinitesimal translation for
|tsi> --> |tsi'> =T(hat) (dirac delta x)|tsi>
that <x> ---> <x> + dirac delta x and
< P (as in momentum) sub x > ----> < P subx >

SO far I have gotten {T is for hte translator)
<tsi| T(hat with dagger) (diarc delta x) x T hat (diarc delta x)| tsi>
= <tsi| (1+(idelta sub x P hat sub x/ hbar) x hat (1- i diarc delta x P sub x /hbar) |tsi>

I don't know where to go from there tho...

 

[Physics Monkey]

For part A, have you written out your induction hypothesis and verified that the first case is satisfied? Once you've done this, you might try multiplying both sides by x (from the left) and seeing what you can do. As for the formula you have given, I wouldn't think you could use that since its basically what you're trying to prove. If you can use it, you might want to look at the special case where f is a power of x.

For part B, you're post seems somewhat confusing. Why do you have dirac delta functions in your equation $$|\psi'\rangle = \hat{T} |\psi \rangle$$? As I read it now, it looks somewhat nonsensical. Could you check the problem again and then try to be a little more clear about what you're trying to do?