University Physics Notes: Quantum Mechanics – 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 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
eigenvalue
of
the quantity Q that has been read.