Verify Orthonormality of exp(2πimx/L)/sqrt(L) for Integer m and 0 ≤ x ≤ L

[Delta what]

Homework Statement

The question given to me was:
Verify the orthonormality of exp(2πimx/L)/sqrt(L) for m an integer and 0<or= to x <or= L.

Homework Equations

I really have no idea where to start on this one. I am taking a math refresher course and we are learning about linear vector spaces to include Fourier transforms and such. Can anyone give me a clue as to how to initially attack this problem?

 

[S.G. Janssens]

I assume that you work in the space $H := L^2(0,L)$. This is a vector space (recall the definition of a vector space from your notes or book) with an inner product (recall the definition of "inner product") given by
$$
\langle f, g\rangle := \int_0^L{f(x)\overline{g(x)}\,dx}
$$
where $:=$ means "is defined to equate", $f, g \in H$ are arbitrary and the bar over $g(x)$ denotes complex conjugation. Thus $H$ becomes an inner product space (= a vector space with an inner product). You can also show that $H$ enjoys a property called "completeness", which implies that $H$ is a Hilbert space (= an inner product space that is complete).

Now, look up the definitions of orthogonality and orthonormality in an inner product space (or a Hilbert space). How is this related to the inner product? Once you have this clarified, you should be able to proceed.

 

[Delta what]

I assume that you work in the space $H := L^2(0,L)$. This is a vector space (recall the definition of a vector space from your notes or book) with an inner product (recall the definition of "inner product") given by
$$
\langle f, g\rangle := \int_0^L{f(x)\overline{g(x)}\,dx}
$$
where $:=$ means "is defined to equate", $f, g \in H$ are arbitrary and the bar over $g(x)$ denotes complex conjugation. Thus $H$ becomes an inner product space (= a vector space with an inner product). You can also show that $H$ enjoys a property called "completeness", which implies that $H$ is a Hilbert space (= an inner product space that is complete).

Now, look up the definitions of orthogonality and orthonormality in an inner product space (or a Hilbert space). How is this related to the inner product? Once you have this clarified, you should be able to proceed.

I think I have a lot of work to do then hahaha

 

[S.G. Janssens]

I think I have a lot of work to do then hahaha

Forget about the completeness for the moment. It is important from a mathematical viewpoint, but it is not important to solve your problem. Then there is not so much work left: Recall what an inner product space is, recall what the definitions of orthogonality and orthonormality are and you are good to go.

If you call your function $f_m$ then ultimately you will need to show that $\langle f_m, f_n \rangle = 1$ when $m = n$ and zero otherwise. However, it is good to know the context of the problem, so it becomes more than a mechanical exercise and you will be able to solve similar problems in the future.

 

[Delta what]

Forget about the completeness for the moment. It is important from a mathematical viewpoint, but it is not important to solve your problem. Then there is not so much work left: Recall what an inner product space is, recall what the definitions of orthogonality and orthonormality are and you are good to go.

If you call your function $f_m$ then ultimately you will need to show that $\langle f_m, f_n \rangle = 1$ when $m = n$ and zero otherwise. However, it is good to know the context of the problem, so it becomes more than a mechanical exercise and you will be able to solve similar problems in the future.

Please bear with me hahah, I would like to put this all into words to ensure I am understanding the process. So I am finding that the integral of the products of bra(f) and ket(g) (which is the inner product) which is the equation your wrote in the first post correct? Also, due to the complex number my ket (g) needs to conjugate bra(f). When the integral is evaluated it must to equal one to show orthonormality. Alright so I did all of that and got 1 for our answer so have the process down for this specific problem. However, I am still having trouble connecting the dots and how this is showing orthonormality. I will read some more and try to figure it out. Thanks for all of your help!

 

[S.G. Janssens]

You are welcome. A few remarks, also in relation to your comments:

1. Mathematicians usually define
$$
\langle f, g\rangle := \int_0^L{f(x)\overline{g(x)}\,dx}
$$
(as I did), while physicists and chemists instead define
$$
\langle f, g\rangle := \int_0^L{\overline{f(x)}g(x)\,dx}
$$
conjugating the first function, not the second. This is purely a matter of convention and it does not make any difference beyond that. It is best to see what your book goes with and then stick with that consistently.

2. Just to be sure, for orthonormality it is not sufficient to show that $\langle f_m, f_m\rangle = 1$, you also need to show that $\langle f_m, f_n\rangle = 0$ when $m \neq n$. (The first shows that $f_m$ has length one, since the length is the square root of the inner product of a vector with itself. The second shows that different $f_m$ are orthogonal to each other.)

3. In a sense, there is nothing to understand about why this shows orthonormality, it is purely a matter of definition. Of course you can ask why the inner product that appears here is a reasonable choice. The intuitive answer would be that this inner product most naturally generalizes the familiar inner products on $\mathbb{R}^n$ and $\mathbb{C}^n$. (This intuition is made mathematically precise in a statement called "Parseval's theorem".)