Orthonormal basis

**asi123** · May 24th 2009, 02:08 PM

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.

**the_doc** · May 25th 2009, 04:12 AM

The i's and j's in the integrand are just the indices of the wave functions.

This appears to be a quantum mechanics based problem so you should already know that the schrodinger equation has an infinite set of eigenfunctions corresponding to the eigenvalues (or energy levels) of the Schrodinger equation. The subscript i then just denotes which eigenfunction out of the set you are referring to. In the integrand the j is simply a different letter to i to signify the integral over any two eigenfunctions from the set.

Note that the i in the exponent of each wave function is the imaginary number i not the subscript.

Firstly no matter what you do you cannot just ignore the exponential term - you must include it.

Now, what you need to do depends on what exactly you have been asked to find.

If you just need to show that these two functions $\psi_1$ and $\psi_2$ are orthonormal then you need to carry out three integrals. Carry out the integral in your jpeg for i=j=1 and i=j=2 and show that both are equal to 1. This shows the functions are normalised. Finally carry out the same with i=1 and j=2 and show the result to be zero. This shows they are orthogonal and hence they are orthonormal.

If you actually have to show that all the wave functions are orthonormal to one another then you need to carry out the general integral where the wavefunctions are functions of j and then show the integral is equal to the Kronekker delta (as you said).

Hope that helped!

**asi123** · May 25th 2009, 04:40 AM

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.

**the_doc** · May 25th 2009, 07:47 AM

Yes, you are correct it should be 1/2 since, at the very least, the magnitude of a complex number must be $\ge 0$ !

For whatever reason the person who set the problem seems to want A to be an imaginary number. Hence there is the bizarre inequality at the start that can only be meaningful if A is an imaginary number. Anyway, for that reason they seem to have then squared their imaginary result to get their answer when in fact the magnitude of their A is 1/2.

So to sum up their value of $\color[rgb]{0,0,1}A$ is correct (given the constraint of the inequality at the start) but their value for $\color[rgb]{0,0,1}|A|^2$ is not!

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