# Hermitian conjugate of operators

## Homework Statement

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

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

## Homework Equations

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

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

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dextercioby
Homework Helper
You might review point c.)

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

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

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

dextercioby