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.
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
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.