Inner product Integral

[Ewie]

Yoo, I'm having trouble finding the complex conjugate for the following question : "Find the inner product of the functions sin(x) and cos(x) on the interval 0=<x=<1".
This question was retired from the free pdf "Quantum mechanics for engineers" in section 1.3.
How do I even approach the sin*(x) thing? Is the relation sin*(z)=sin(z*) valid there?
Since you need to integrate it alongside with cos(x) I posted it here, sorry if it was in the wrong section.
Any help would be appreciated.
Thanks for reading and have a nice day.

 

[Steven G]

Can you define inner product? Can you find the inner product of sin(x) and cos(x)?

 

[Ewie]

Can you define inner product? Can you find the inner product of sin(x) and cos(x)?

I know that these functions acts like orthogonal vectors, so their inner product is supposed to be zero (at interval 0rad, and 2πrad).
The hard part is using the conjugated sine in the formula for functions.
Even if we assume that the conjugated of x (x*) is simply x the integral is still is going to be different than zero.
I'm pretty sure that you need to solve it using the definition of the inner product for functions : <sin(x)|cos(x)>= ∫ sin*(x)cos(x)dx
The main problem is that x is supposed to be a variable, rather than a complex function of it's own and the book don't have any answer at all.

 

[topsquark]

I know that these functions acts like orthogonal vectors, so their inner product is supposed to be zero (at interval 0rad, and 2πrad).
The hard part is using the conjugated sine in the formula for functions.
Even if we assume that the conjugated of x (x*) is simply x the integral is still is going to be different than zero.
I'm pretty sure that you need to solve it using the definition of the inner product for functions : <sin(x)|cos(x)>= ∫ sin*(x)cos(x)dx
The main problem is that x is supposed to be a variable, rather than a complex function of it's own and the book don't have any answer at all.

You have the correct inner product for QM. (Please be aware, though, that there are many different kinds of inner product definitions.) The integral measures the "overlap" of the sine and cosine functions, so we would expect a number and not a function of x. I suspect the main problem you are having, and it's a common one, is that sin(x) and cos(x) are vectors in a Hilbert space. (Let's call it an infinite dimensional vector space and leave the other details of it alone for right now.) If you were dealing with "Euclidean vectors" such as \(\displaystyle 3 \hat{i} -2 \hat{j}\) then I suspect you would understand it much more easily.

-Dan

 

[Ewie]

You have the correct inner product for QM. (Please be aware, though, that there are many different kinds of inner product definitions.) The integral measures the "overlap" of the sine and cosine functions, so we would expect a number and not a function of x. I suspect the main problem you are having, and it's a common one, is that sin(x) and cos(x) are vectors in a Hilbert space. (Let's call it an infinite dimensional vector space and leave the other details of it alone for right now.) If you were dealing with "Euclidean vectors" such as \(\displaystyle 3 \hat{i} -2 \hat{j}\) then I suspect you would understand it much more easily.

-Dan

Thanks for the response, I was considering using the Hilbert space for this, even so, I still can't figure ot the answer, because of the conjugate of sine thing, do you mind solving the problem (integral)? I would really appreciate it.
Have a nice day.

 

[Ewie]

Nevermind, I just (probably) figured it out.
The conjugate of a real variable it's the variable itself,by solving the integral you get to evaluate sin^2(x)/2 at the lower limit 0 and upper limit 1, this turns out to be sin^2(1)/2 wich is approx 0,354 radians.
Not that sure on this one , please correct me if I'm wrong, again ,thanks everyone for the help.