# Hermitian conjugate of operators

1. Nov 17, 2011

### v_pino

1. The problem statement, all variables and given/known data

Find the hermitian conjugates, where A and B are operators.

a.) AB-BA
b.) AB+BA
c.) i(AB+BA)
d.) $A^\dagger A$

2. Relevant equations

$(AB)^\dagger =B^\dagger A^\dagger$

3. The attempt at a solution

Are they correct and can I simplify them more?

a.) $(AB-BA)^\dagger = (AB)^\dagger - (BA)^\dagger= B^\dagger A^\dagger - A^\dagger B^\dagger = BA-AB = -(AB-BA)$

b.) $(AB+BA)^\dagger = (AB)^\dagger + (BA)^\dagger= B^\dagger A^\dagger + A^\dagger B^\dagger = BA+AB$

c.) $[i(AB+BA)]^\dagger = -i[(AB)^\dagger + (BA)^\dagger]= -i(B^\dagger A^\dagger + A^\dagger B^\dagger) = BA+AB$

d.) $[A^\dagger A]^\dagger = A^\dagger (A^\dagger)^\dagger=A^\dagger A$

2. Nov 17, 2011

### dextercioby

You might review point c.)

3. Nov 17, 2011

### v_pino

For point C.) $-i(B^\dagger A^\dagger + A^\dagger B^\dagger)$

4. Nov 17, 2011

### dextercioby

Alright. The last equality is not true. That's what I meant.

5. Nov 17, 2011

### v_pino

Is it ok until $-i(B^\dagger A^\dagger + A^\dagger B^\dagger)$?

6. Nov 17, 2011

Yes, it is.