Quantum Mechanics question, technical.

[daggaz]

Ok I am dying from a nasty cold right now and my brain is too melted to make any use of my google-foo, or my own math books for that matter.

How do I find the hermitian conjugate of the derivative operator, d/dx?

[unstattedCommoner]

Integrate by parts (bearing in mind that the complex conjugate of df/dx is df/dx)

[daggaz]

Yeah, I figured it had something to do with integration by parts.. but please, more specific please? My brain is simply on shut down today and I need to have this answer for monday.

Im really having a hard time seeing how to prove that the integral of (operator on g(x)*f(x)) = integral of (g(x)*df(x)/dx))...

In other words, with any two functions g(x) and f(x), how can you possibly define an operator that works on g which is the same as simply taking the derivative of f??? :smalleek:

[unstattedCommoner]

In other words, with any two functions g(x) and f(x), how can you possibly define an operator that works on g which is the same as simply taking the derivative of f??? :smalleek:

Two operators L, M are hermitian conjugate if

http://i705.photobucket.com/albums/ww51/unstattedCommoner/def.png

where

http://i705.photobucket.com/albums/ww51/unstattedCommoner/def2.png

If you integrate df/dx * g by parts, you get (with D = d/dx):

http://i705.photobucket.com/albums/ww51/unstattedCommoner/parts.png

[fg] vanishes because both f and g must tend to zero as |x| -> infinity sufficiently fast for <f|f> and <g|g> to be finite.

Therefore, the hermitian conjugate operator of d/dx is -d/dx.

[Mr. Mud]

What commoner said, but louder. :smalltongue:.

Also, just wondering, is this a High School or College class you're taking?

[daggaz]

Its a second year university course in the nanotech program. I know its a pretty basic question (considering..) but I am absolutely fried from this cold virus and I cant read a page in a book before my eyes start burning and I need to lay down.

Thanks a lot for the explanation, very clear and concise. I understand very well now. :smallsmile: (and god.. it was so easy too :smallfrown:)