# Info on wave mechanics prob in one dimension need

1. Dec 1, 2005

### 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 dunno where to go from there tho....

2. Dec 1, 2005

### 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?