The intriguing thing about the solutions 
of the
previous section is that they
form a complete set, which means that
all valid solutions to (1) maybe made out of sums of
the particular solutions 
we have identified. This is a general
feature of the solutions generated by the procedure of the separation
of variables.
The proof of this as a mathematical statement
is beyond
the scope of this course. We can see, however, on physical grounds
that the solutions we have identified must be complete provided that we have
the correct mathematical representation of our pure physical
quantum states.
The physical reason for our confidence in the completeness of our solutions comes from the very special form of equation (9). It is just the eigenvalue equation for the energy operator!
The 
are eigenstates of the Hamiltonian operator 
.
Each solution 
to this equation is a pure state of the
energy observable with energy 
. The only values which may be
observed when the measure the energy of the system are the 
where
(10) has a valid solution. This is the reason for our choice of
the name E for the constant which appeared in our separation of
variables analysis.
The completeness of our solutions may be understood through the physical principle of superposition, which states that we may write all quantum states at a given time t=0 as an amplitude-weighted sum of the pure states of the physical observable of the energy,
In the position representation, this means that
we may expand any
wavefunction
at time t=0 in terms of our solutions to the TISE
,
From our separation of variables, we do know that 
is a solution to the Time Dependent
Schrödinger Equation (1). Because the TDSE is
linear, 
is also a solution. Because, as we saw
above, the 
may be chosen so that 
matches an arbitrary 
at t=0, the
general solution to the TDSE may be written
The interested student should note (and try as an exercise) that one may go (in three or four lines of algebra!) from the more abstract Schrödinger equation
and the principle of superposition (11) to derive almost immediately the central result of this note (13).
