# Commutation relation for Hermitian operators

1. Nov 27, 2014

### jimmycricket

1. The problem statement, all variables and given/known data
The Hermitian operators $\hat{A},\hat{B},\hat{C}$ satisfy the commutation relation$[\hat{A},\hat{B}]=c\hat{C}$.
Show that c is a purely imaginary number.

3. The attempt at a solution

I don't usually post questions without some attempt at an answer but I am at a loss here.

2. Nov 27, 2014

### Staff: Mentor

What happens if you take the Hermitian conjugate of that equality?

P.-S.: Don't forget that the homework template has three sections. You should've written down the relevant equations.

3. Nov 27, 2014

### jimmycricket

$[\hat{A},\hat{B}]^{\dagger}=[\hat{B},\hat{A}]=-[\hat{A},\hat{B}]$
How does this help?

4. Nov 27, 2014

### Staff: Mentor

You forgot the right-hand-side of the original equality.

5. Nov 27, 2014

### jimmycricket

$[\hat{A},\hat{B}]^{\dagger}=[\hat{B},\hat{A}]=-[\hat{A},\hat{B}]=(c\hat{C})^{\dagger}$

6. Nov 27, 2014

### Staff: Mentor

Continue...

7. Nov 27, 2014

### jimmycricket

$[\hat{A},\hat{B}]^{\dagger}=[\hat{B},\hat{A}]=-[\hat{A},\hat{B}]=(c\hat{C})^{\dagger}=c^{*}\hat{C}^{\dagger}$

8. Nov 27, 2014

### Staff: Mentor

You're almost there. To help you, I'll rewrite what you have there so that it may be more obvious what the last steps are.
\begin{align*} [\hat{A},\hat{B}]^{\dagger} &=(c\hat{C})^{\dagger} \\ [\hat{B},\hat{A}] &=c^{*}\hat{C}^{\dagger} \\ -[\hat{A},\hat{B}] &=c^{*}\hat{C}^{\dagger} \end{align*}

9. Nov 27, 2014

### jimmycricket

Im still having some trouble seeing where to go from here

10. Nov 27, 2014

### Staff: Mentor

What properties does $\hat{C}$ have? Can you replace the left-hand-side by something else?

11. Nov 27, 2014

### jimmycricket

Ahh ok I think Im there $-[\hat{A},\hat{B}]=(c\hat{C})^{\dagger}=c^{*}\hat{C}^{\dagger}=c^*\hat{C}$ (since C is hermitian)
So we now have $-[\hat{A},\hat{B}]=-c\hat{C}=c^*\hat{C}$
$$\Longrightarrow \frac{c^*}{c}=-1$$
Only pure imaginary numbers satisfy this last condition.

12. Nov 28, 2014

### Staff: Mentor

You got it!

13. Nov 28, 2014

### jimmycricket

thanks for your patience and for not giving it away too easily