## Linear Combinations of Eigenfunctions

An eigenfunction which represents a possible state of a particle is any solution of the Schrodinger equation

which may be written in the formwhere

A linear combination of any number of eigenfunctions is also a possible wavefunction.

Proof:

Hence the general state of a particle may be represented as a linear combination of eigenfunctions.

The inner productwhereifand 1 if

We can use this to prove the following result: Ifthen

Proof

The significance of have a wavefunction as a linear combination of eigenfunctions is this: If a measurement is made of a quantity then the probability of reading the eigenvalueof the quantity Q associated with the eigenfunctioniswhereis the quantum amplitude of the eigenfunction in the overall wavefunction. The only possible readings of the quantity Q are the eigenvalues and any value of Q may be read if the eigenfunction associated with that valueof Q is present in the wavefunction. Before the reading is taken, the particle in general exists in a superposition of states eigenfunctions, but when the reading is taken the wavefunction 'collapses' to occupy the eigenfunction corresponding to the eigenvalueof the quantity Q that has been read.