# Hermitian Conjugate

pretty simple question. have to prove $\hat{O} \hat{O}\dagger$ is a Hermitian operator.

i found that

$\left( \int \int \int \psi^{\star}(\vec{r}) \hat{O} \hat{O}^{\dagger} \phi(\vec{r}) d \tau \right)^{\star} = \int \int \int \phi^{\star}(\vec{r}) \hat{O}^{\dagger} \hat{O} \phi(\vec{r}) d \tau$

so all we need to get the result is to establish if the operator commutes with it's hermitian conjugate. i'm guessing it does but don't know why - can someone explain?

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malawi_glenn
Homework Helper
do you know how to use bra-ket notation? it is a bit easier then.

ok.

$\langle \psi|\hat{O} \hat{O}^{\dagger}|\phi \rangle^{\star} = \langle \phi|\hat{O}^{\dagger} \hat{O}| \psi \rangle$

but then i again encounter the problem of showing that
$\hat{O} \hat{O}^{\dagger}=\hat{O}^{\dagger} \hat{O}$
????

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malawi_glenn
Homework Helper
you are not using bra-ket correctly.

an operator is hermitian if <b|R|a> = <a|R|b>*

remeber to use dual-correspondence

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sorry, i edited the bra-ket. now from the definition of hermition conjugate, O is hermitian iff

$\hat{O}\hat{O}^{\dagger}=\hat{O}^{\dagger}\hat{O}$

malawi_glenn
Homework Helper
have you used the dual correspondance and the fundamental property of inner product?

i don't know what those are. sorry...

malawi_glenn
Homework Helper
i don't know what those are. sorry...
ok, if you did not know what bra-ket is and how it works you should have said so....

yeah, well we use bra-kets in our course so i guess i'm supposed to know it. i just don't recognise the terms "dual correspondence" and "fundamental property of inner product", perhaps i do know them, just not by those names.

i tried googling them but to no avail...

malawi_glenn
Homework Helper
here is a good introduction:

http://www.physics.unlv.edu/~bernard/phy721_99/tex_notes/node6.html

Now using integrals and wave functions:

Operator A is hermitian if:
$$\int \psi _1 ^* (\hat{A} \psi _2 ) \, dx = \int (\hat{A}\psi _1 )^* \psi _2 \, dx$$

so we get
$$\int \psi _1 ^* (\hat{O}\hat{O}^{\dagger} \psi _2 ) \, dx = \int (\hat{O}^{\dagger}\psi _1 )^* \hat{O}^{\dagger} \psi _2 \, dx$$

verify that and continue

ok. im getting confused now.
my notes define hermitian conjugate as:

$(\int \psi^* \hat{O} \phi dx)^* = \int \phi^* \hat{O}^{\dagger} \psi dx$

the operator is hermitian if $\hat{O}=\hat{O}^{\dagger}$

how does that compare to what you have?

is your definition just saying $\hat{A}=\hat{A}^{\dagger}$

how would you write the first equation you wrote down in bra-ket notation?

malawi_glenn
Homework Helper
https://www.physicsforums.com/attachment.php?attachmentid=13008&d=1205092708 [Broken]

page 53.

eq. 4.58. since expectation value is real. (4.57) so the opreator O there is hermitian if O dagger = O.

That is to say:
$$\int \psi _1 ^* (\hat{O} \psi _2 ) \, dx = \int (\hat{O}\psi _1 )^* \psi _2 \, dx$$

If you want to do in bra-ket:

<b|O O^dagger |a> = <a|O O^dagger |b>* ??

consinder bra <b|O
DUAL is
O dagger |b>

Consider ket O^dagger |a>
DUAL is
<a| O

now use <r|t>* = <t|r>

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all we need to do is put together the two duals you derived into a bra-ket to give

<a| O O(dagger) |b>

so we can now say

<a| O O(dagger) |b>* = <b| O O(dagger) |a> which implies a hermitian operator

is that ok?

out of interest, for getting it to work in terms of integrals,

i took the conjugate of the integral you wrote at the bottom left of post 10 but then got lost?

malawi_glenn
Homework Helper
yes, that is correct done in bra-ket, as you saw, it was really easy.

$$\int \psi _1 ^* (\hat{O}\hat{O}^{\dagger} \psi _2 ) \, dx = \int (\hat{O}^{\dagger}\psi _1 )^* \hat{O}^{\dagger} \psi _2 \, dx$$

do this one more time.. recall that conjugating reverses the order (AB)* = B*A*

could we say

$(\int \psi^* \hat{O} \hat{O}^{\dagger} \phi dx)^*=(\int (\hat{O}^{\dagger} \psi)^* \hat{O}^{\dagger} \phi dx)^* = \int (\phi \hat{O}^{\dagger})^* \hat{O}^{\dagger} \psi dx = \int \phi^* \hat{O} \hat{O}^{\dagger} \psi dx$

so operator is hermitian?

when doing it in bra-ket notation, do you have to evaluate the bra and the ket dual seperately like you did or is it possible to do it all in one line?

malawi_glenn