# Orthonormal basis

## Homework Statement

Hey guys.

http://img39.imageshack.us/img39/2345/27760913.jpg [Broken]

I need to show that these wave functions are orthonormal.
I'm a bit confuse, what's i and what's j?
I mean, do I need to take both of the functions, put them in the integral and to show that the result is the Kronecker delta?
Can I neglect the exponent for this?

Thanks a lot.

## The Attempt at a Solution

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i and j are just arbitrary indices. Yes you have to calculate the integrals for different cases. When i=j the exponential cancels (when you take the complex conjugate it changes sign). When i and j aren't equal you'll get some exponential dependence as well but in that case you should get zero anyway.

i and j are just arbitrary indices. Yes you have to calculate the integrals for different cases. When i=j the exponential cancels (when you take the complex conjugate it changes sign). When i and j aren't equal you'll get some exponential dependence as well but in that case you should get zero anyway.
Well, where are i and j in my problem?
I mean, this is not a series, it's a function.

gabbagabbahey
Homework Helper
Gold Member
Well, where are i and j in my problem?
I mean, this is not a series, it's a function.
You have two wave functions, $\psi_1$ and $\psi_2$, so the indices $i$ and $j$ can each take on the values $1$ and $2$.

You have two wave functions, $\psi_1$ and $\psi_2$, so the indices $i$ and $j$ can each take on the values $1$ and $2$.
Yeah, but I don't have i and j inside the functions so how can I come up with the kronecker delta?

How can I show that if i=j then it's 1 and if i does not equal to j, it's 0 if I don't have i and j?

Thanks.

gabbagabbahey
Homework Helper
Gold Member
Yeah, but I don't have i and j inside the functions so how can I come up with the kronecker delta?

How can I show that if i=j then it's 1 and if i does not equal to j, it's 0 if I don't have i and j?

Thanks.
Showing that

$$\int \psi_i \psi_j dx =\delta_{ij}$$

just means that you need to show:

$$\int \psi_1 \psi_1 dx =\int \psi_2 \psi_2 dx =1$$

and

$$\int \psi_1 \psi_2 dx=\int \psi_2 \psi_1 dx =0$$

Showing that

$$\int \psi_i \psi_j dx =\delta_{ij}$$

just means that you need to show:

$$\int \psi_1 \psi_1 dx =\int \psi_2 \psi_2 dx =1$$

and

$$\int \psi_1 \psi_2 dx=\int \psi_2 \psi_1 dx =0$$
Oh, now I get it.

Thanks a lot.

Well, here is the second part of the question

http://img207.imageshack.us/img207/879/95899388.jpg [Broken]