Orthogonality of sines

[/0]

In (physics) lectures we discovered that

    (with

And from a physics lecturer, the conclusion came quickly...



But isn't it true that while the limit exists, the function at is discontinuous? Why can we say it is 'equal' in this case?


(Also, is orthogonal with anything?)

[Mao]

In derivation of the first one, you would have done something that have forbidden m=n, so we can't really say n,m element of Z. However, for the sake of simplicity, it is written as such.

[/0]

But, say you have the wavefunction in an infinite square well,



In this case, must be an integer since it is a quantum number.

If you have a superposition of eigenstates:



You must use Fourier's trick to find the coefficients :



But this requires

I still don't understand why this works without a limit

[humph]

I'm guessing it's as Mao says, when you prove this it's not true when . You have to split it up into different cases (i.e. when and when ) and show that each is true. The fact that you get the same result as when you take limits isn't too surprising, but it doesn't follow directly.

[Mao]

One word: laziness.

To get things done, many forget the absolute rigor of mathematics, and simply 'apply' it.


Now, as for the actual mathematical side of things, the derivation would have been done in two parts, one where it exists, one where it does not [as humph has said]. When both are done, we join them back together to give the piecewise form of the answer. The workings given by your lecturers would have omitted a lot of things, simply because it is 'trivial' and not worth the effort [again, laziness].

[/0]

Sorry, I might be a bit slow here... but when trying for a piecewise solution, gives "undefined"

[humph]

Sorry, I might be a bit slow here... but when trying for a piecewise solution, gives "undefined"

Well yes, it's mathematically incorrect to say that

In (physics) lectures we discovered that

    (with

You would not see this in a maths lecture (I hope). It's just sloppy notation, because it's not actually valid for all .

[/0]

Oh right... thanks

*facepalms* self