Question

what is orthonormal eigenfunctions and eigenvalues?? define and explain

Answer 1

If you have an operator A, eigenfunctions are those functions f such that
\[\hat{A}f = \lambda f \]
where lambda is a constant. If two functions f and g are orthogonal (over an interval), then their inner product is zero, usually defined as
\[\int f\cdot g\space dx = 0\]
with the integral being taken over the interval in question. If the functions are also normalized, i.e.
\[ \int f^2 \space dx = \int g^2 \space dx = 1\]
then they are called orthonormal.

The importance of orthonormal eigenfunctions is that they can be used as a basis in which one might expand other functions, in a way similar to a power series. In quantum mechanics, for instance, one might expand the wavefunction of a particle in terms of the orthonormal eigenfunctions of the Hamiltonian operator.

Answer 2

thank u

Answer 3

orthonormal can be visualised by the value of the kronecker delta value(\[\delta i,j\]). If the value of the kronecker delta becomes zero for the indices, then the functions represented by the indices are orthogonal. If the two functions are normalised then the functions are orthonormal.

These functions are eigen functions if they obey the relation
Aˆf=λf

These functions can be used as the basis functions to expand any function in terms of these orthonormal functions. For instant, in vector algebra, we express the vectors in terms of the vectors i,j,k. Likewise it happens here also.