How to interpret the orthonormality of the wave functions, physically?

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How to interpret the orthonormality of the wave functions, physically?

Post by Confusions allowed on Sat Sep 17, 2016 3:31 pm

We all know that the wave functions are orthonormal, i.e.,
(Integral,-inf,inf)ψm*ψndτ = 1 if m=n, otherwise, it's 0.
Also, some texts suggest that the integral is nothing but the inner (Dot) product of the wave functions. Considering this, and also considering the wave functions as vectors, one may say that the wave functions are all normal to each other, which doesn't seem to be making sense, as we're in a 3D, and, if not, 4D, space. Can anyone, therefore, give a physical sense of this? Also, where did the fact, that the wave functions are orthonormal, arise from? (Please don't say that it's a postulate, rather give some logical steps from which one may arrive at this fact.)
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Re: How to interpret the orthonormality of the wave functions, physically?

Post by physicistsid on Mon Sep 19, 2016 7:05 pm

First thing that it is not as if all wave functions are orthogonal.
Schrodinger Equation says
HΨ = EΨ
Different solutions of this equation with different eigenvalues are guaranteed to be orthogonal since H is Hermitian.

Check my derivation.If you don't get something ask me.

https://i.servimg.com/u/f97/19/53/89/85/20160910.jpg

Vectors you generally think about are called Euclidean vectors which have length(magnitude) and directions.
But mathematics of vectors can be generalized where vectors are objects which obey certain properties and form a vector space together with scalars(complex nos. in general).

The Properties  for a real vector space are:-

A real vector space is a set X with a special element 0, and three operations:

Addition:  Given two elements x, y in X, one can form the sum x+y, which is also an element of X.

Inverse: Given an element x in X, one can form the inverse -x, which is also an element of X
.
Scalar multiplication:  Given an element x in X and a real number c, one can form the product cx, which is also an element of X.
These operations must satisfy the following axioms:

Additive axioms.  For every x,y,z in X, we have
x+y = y+x.
(x+y)+z = x+(y+z).
0+x = x+0 = x.
(-x) + x = x + (-x) = 0.

Multiplicative axioms.  For every x in X and real numbers c,d, we have
0x = 0
1x = x
(cd)x = c(dx)

Distributive axioms.  For every x,y in X and real numbers c,d, we have
c(x+y) = cx + cy.
(c+d)x = cx +dx.

Vectors can be generalised to any no. of dimensions(even infinite).
Consider a set A of all polynomials with degree less than n(don't worry about degree of 0 polynomial.Leave it undefined and include 0 in A.)
It satisfies all the above properties. So A is a vector space where polynomials are n dimensional vectors.
To gain more insight read Griffiths's QM Ch.3.

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