How do I determine if it's a Hermitian Operator or not

[MixilPlixit]

First post so please go easy on me, here goes:

I have looked over the basic definition of what is a Hermitian operator such as: <f|Qf> = <Qf|f> but I still am unclear what to do with this definition if I am asked prove whether or not i(d/dx) or (d^2)/(dx^2) for example are Hermitian operators. I am using Griffiths' "Intro to Quantum Mechanics" and it really seems like he skips a lot of steps. Steps I need to make sure I understand. Can someone help enlighten me.

 

[Meir Achuz]

You have to put the operator in the integral \int\psi*[id/dx]\psi dx.
Then integrate by parts, using the BC at the endpoints. If you get the same result after integratilng by part, the operator is hermitian.

 

[inha]

You have to put the operator in the integral \int\psi*[id/dx]\psi dx.
Then integrate by parts, using the BC at the endpoints. If you get the same result after integratilng by part, the operator is hermitian.

This is exactly how it's done in Griffiths too. Maybe Mixil had some specific part of this in mind?

 

[MixilPlixit]

I think I am a little hazy on integration by parts, so that apeears to be where my problem lies. Can someone recommend a good refresher on the matter?

 

[ghosts_cloak]

http://archives.math.utk.edu/visual.calculus/4/int_by_parts.3/
I just found this on google, and it appears to be a nice introduction, and takes you through each step. Perhaps a little basic for what you want though?
Hope it helps,
~Gareth

 

[MixilPlixit]

OK, that somewhat helps but if I have $$\int \frac{d^2}{d \phi^2}d \phi$$ how do I evaluate that? It's been a while since calc obviously. It's the 2nd derivative thing that's throwing me. Hopefully I am not tiring you guys, cause I am certainly getting bleary. Thanks for your help this far. I am thankful for your continued assistance.

 

[George Jones]

OK, that somewhat helps but if I have $$\int \frac{d^2}{d \phi^2}d \phi$$ how do I evaluate that?

You can't evaluate that. On what is $\frac{d^2}{d \phi^2}$ operating?

Regards,
George

 

[MixilPlixit]

I am given $$Q = \frac{d^2}{d \phi^2}$$ And am asked to prove if eigenvalues are real. In order to do that I need to determine if Q is in fact a Hermitian operator. So I am at: $$\int f*(\frac{d^2}{d \phi^2})g d\phi$$ This is integrated over 0 to 2 pi.

 

[George Jones]

You need to use integration by parts twice. For the first integration by parts, it may be helpful to think of

$$\left( \frac{d^2}{d \phi^2} \right)g$$

as

$$\frac{dg'}{d \phi}$$

where

$$g' = \frac{dg}{d \phi}.$$

Regards,
George

 

[MixilPlixit]

For the first integration then would $$v = \frac{dg}{d \phi}$$ and $$du = (\frac{df}{d\phi})^*$$?

[edit] Well I think I solved it, but it feels wrong. I got $$-\int f^*(\frac{d^2}{d \phi^2}) g d\phi$$ It the initial integral but minus. Can that be right?

 

[dextercioby]

Assuming the operator is densly defined, can u at least identify the operator's adjoint...? Next thing to worry you is to find whether your operator is bounded or not...

Daniel.