Physics 309

Commutators and Hermitian Operators

1. What is the commutator of the momentum $\hat p$ and energy $\hat H$ operators? Can you draw any conclusions about the eigenfunctions of $\hat p$ and $\hat H$?

2. If the operator $\hat A$ is hermitian show
   
   $$\langle \phi \vert \hat A \phi \rangle = \langle \hat A \phi \vert \phi \rangle$$
   
   

3. Show that for two hermitian operators, $\hat {A~}$ and $\hat {B~}$, if
   
   $$[\hat {A~}, \hat {B~}]= i \hat {C~}$$
   
   
   then
   
   $$[\delta \hat {A~}, \delta \hat {B~}]= i \hat {C~}$$
   
   
   as well, where
   
   $$\delta \hat {A~} = \hat {A~} - <A>, \qquad \delta \hat {B~} = \hat {B~} - <B> \quad .$$
   
   

4. For the definition of the inner product in one dimension that we have been using in class, show
   
   $$\langle \gamma \vert \gamma \rangle \geq 0$$
   
   
   where $\vert\gamma\rangle$ is an element/eigenfunction of a Hilbert space.

5. Show that for any two elements $\phi$ and $\psi$ in a Hilbert space with lengths/norms $\vert\vert\phi \vert\vert$ and $\vert\vert\psi \vert\vert$ the Cauchy-Schwart Inequality is true
   
   $$\vert\langle \phi \vert \psi \rangle \vert \leq \vert\vert\phi \vert\vert ~ \vert\vert\psi \vert\vert$$
   
   
   where $\vert\vert\phi \vert\vert^2 = \langle\phi \vert \phi \rangle$. This is the same problem as 4.20 in Liboff, but with the hint below added.

   Hint: Let $\vert\gamma\rangle = \vert\psi\rangle - \frac{\langle\phi\vert\psi\rangle}{\langle\phi \vert \phi \rangle}\vert\phi\rangle$ and use the result in the previous problem.