- #1

fifthrapiers

- 3

- 0

1.) [tex]x,\hat{H}[/tex]

My work:

[tex][x,\hat{H}]\psi &= x\hat{H}\psi - \hat{H}x\psi[/tex]

[tex]= x \left ( \frac{p^2}{2m} + V(x) \right )\psi - \left ( \frac{p^2}{2m} + V(x) \right )x\psi[/tex]

[tex]= \frac{xp^2\psi}{2m} + x V(x)\psi - \frac{p^2x\psi}{2m} - V(x) x\psi[/tex]

2.) [tex][\hat{p}, \hat{H} + x][/tex]

[tex][\hat{p}, \hat{H} + x]\psi &= \hat{p}(\hat{H}+x)\psi + (\hat{H}+x)\hat{p}\psi[/tex]

[tex]= i\hbar\frac{\partial}{\partial p} \left(\left( \frac{p^2}{2m} + V(x) \right ) + x\right)\psi + \left(\left( \frac{p^2}{2m} + V(x) \right ) + x\right)i\hbar\frac{\partial}{\partial p}\psi[/tex]

Yikes.