Orthonormal Basis: Showing Wave Functions are Orthonormal

[asi123]

Homework Statement

Hey guys.

http://img39.imageshack.us/img39/2345/27760913.jpg

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.

Homework Equations

The Attempt at a Solution

 

[phsopher]

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.

 

[asi123]

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]

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 $2$.

 

[asi123]

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]

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

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

 

[asi123]

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.

 

[asi123]

Well, here is the second part of the question

http://img207.imageshack.us/img207/879/95899388.jpg

I also posted there answer.
I think they have a mistake, I marked it in the red box.
Shouldn't it be A^2=1/2 ?
Am I missing something?

Thanks a lot.