Why are even and odd functions orthogonal in quantum mechanics?

[hasanhabibul]

shai n and shai m are mutually orthogonal ...where n and m can any numerical value...but i can't imagine it how they can be perpendicular to one another ... (to me the worst thing is to think shai4 and shai100 are perpendicular) and what is the advantage or reason of it...can anyone help me to make me imagine orthogonality of wave function??

 

[f95toli]

Do you understand what it means when two mathematical (forget physics for the moment) functions are orthogonal?

 

[hasanhabibul]

I know about orthogonality of two vectors... Does it contain more?

 

[hasanhabibul]

I can easily picture shai4 is perpendicular to shai5 ... But is it also Perpendicular to shai100? Then it makes me difficult to picture up.

 

[f95toli]

I know about orthogonality of two vectors... Does it contain more?

Yes, functions can also be orthogona, it is a generalization of orthogonality of vectorsl. A good example would be Hermite polynomials; but there are many, many others. This is something you would study in a course in linear algebra.

Also, I don't think you can -in general- "visualize" orthogonality.
My suggestion would be that you start by taking a look at the wiki for "orthogonal functions".

 

[Matterwave]

Since wave-functions exist in the infinite-dimensional Hilbert space, I don't think you can actually visualize them being orthogonal...

 

[sol66]

Well, in simple terms you can't imagine a wave function. Usually if your dealing with a vector space, the concept of orthogonal is merely the fact that the dot products between two such vectors equal 0. Heres the killer, in quantum mechanics you would be wrong to think that you can even imagine a physical interpretation of the abstract vectors that represent the state of a wave function. Much of the mathematical abstraction that you get with quantum mechanics were made up by rules that have no physical sense except in very specific cases.

In your case you are wondering why an even and odd function are considered orthogonal or even what it means. When two functions are orthogonal it means that the integral of the two functions product over a period equals 0. It has nothing to do with how you'd imagine something to be perpendicular in the traditional sense. That is a mathematical definition.

The reason why all odd functions are orthogonal to all even functions that are periodic can be explained by simple Fourier analysis. I'm sure wiki would have an ok explanation.