## 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 form where 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 product where if and 1 if We can use this to prove the following result: If then 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 eigenvalue of the quantity Q associated with the eigenfunction is where is 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 value of 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 eigenvalue of the quantity Q that has been read. 