## Properties of the Eigenfunctions

The eigenfunctions are the solutions of the eigenfunction equation the solutions for the one dimensional simple harmonic oscillator case, are polynomials in multiplied by a gaussian If the are normalized to unity they have the following properties:

1. The are orthonormal This means that (1) where is the Kronecker delta, defined by 1. The form an abstract vector space. Any arbitrary wavefunction can be expressed in terms of the eigenfunctions We can write (2) or using the abstract vector space properties of the we can write as a column vector: with the property that In this expression the are the coefficients of the eqienfunctions in the expression (2), and the position in the ith row signifies the ith eigenvector. It must be understood here that the vector space usually has infinite dimension. Because we can write the wavefunction as a column vector, we can operate on it with a matrix like any other vector.

1. For the one dimensional case, as This is necessary so that the eigenfunction can be normalized.

2. is real for the one dimensional harmonic oscillator, is real for the two dimensional harmonic oscillator and is real for the three dimnsional harmonic oscillator. is real for the hydrogen atom, and in fact for all stationary state eigenfunctions. Each of these may be multiplied by a time term and in the case of the hydrogen atom, an angular term which is complex in general, to obtain the state function which is a function of t and other space variables Each of these factors is normalised to 1.

3. Given a wavefunction we can find the coefficients using that all the are orthogonal. Take the dot product with  since being normalized, hence The dot product here is the generalized dot product – it could be an integral for example as in (1). 