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## Homework Statement

Here is the entire problem set, but (obviously) you don't have to do it all, if you could just give me a few hints on where to even start, because I am completely lost.

Recall that we found the solutions of the Schrodinger equations

[itex] (x^2 - \partial_x ^2) V_n(x) = (2n+1) V_n(x) \quad \quad n = 0,1,2 [/itex]

[itex] V_0 (x) = e^{\frac{x^2}{2}} \quad \quad V_n (x) = (x - \partial x)^n V_0(x) [/itex]

1. Show that for each [itex] n [/itex] there is a polynomial [itex] P_n [/itex] such that [itex] V_n(x)=P_n(x)V_0(x) [/itex]

2. Show that for any smooth function [itex] v(x) [/itex]

[itex] (x - \partial_x) v(x) = -e^{\frac{x^2}{2}} \frac{d}{dx} \left ( e^{\frac{-x^2}{2}} v(x) \right ) [/itex]

## Homework Equations

During class we went over the commutator of [itex] [B,C] [/itex] and a lot its it's identities.

Also [itex] (x^2 + \partial_x ^2) V_n = (2n+1) V_n \rightarrow V_n := (A^*)V_0 \neq 0 [/itex]

## The Attempt at a Solution

I have no idea where to even start on this. None of this is in the book and our professor said that each of the problems can be solved using a few "tricks" and then after that they aren't that hard.