To separate the spatial from time dependencies in the TDSE, we must
assume that the potential 
is constant in time. This
will cover all of the cases of interest to us. It is valid for any
isolated system. If some external influence acts on the system then
V will have a time dependence, and more sophisticated procedures must
then be employed to solve the TDSE.
We begin the separation procedure by
noting that (for those V which are constant in time)
all of the
operators on the left hand side of (1) involve
only the time variable t and none of
the spatial variables 
, whereas all of the operators on the right
hand side involve the spatial variables 
only. The only
appearance of spatial variables on the left hand side is as
arguments to the wave function and the only appearance of the time
variable on right hand side is also as an argument to the wave
function. We could separate the dependencies in the equation entirely
if we could only break up the variable dependencies in 
. This
observation leads us to seek solutions to
(1) of the separated form
Substituting this ansatz (guess) into (1) gives
Finally, dividing both sides by 
,
In (4), the variables are
now completely separated. Nothing on the left hand side depends on
the spatial variables 
and nothing on the right hand side depends
on the time variable t. Therefore, if the left hand side is to
equal the right hand side
for all 
and t, the two sides must both equal something which
depends on neither 
nor t. If we call this constant,
which
is independent of 
and t, E,
we now have two separate equations,
one for 
and one for 
,
(We chose the letter E for this constant because it will later turn
out to be the energy of the quantum state 
.)
(5) is a simple differential equation
easily solved for the time dependent part 
,
(6) is the Time Independent Scrödinger Equation for
. At this point we have identified a whole series of
solutions because generally there will be multiple values of the
constant E for which (6) has solutions for
. To distinguish the different solutions, we shall refer
to them as 
and call the corresponding values of E in
(6) 
. It will be true for most of the systems we
shall study that the allowed values of E form a discrete set, as the
notation 
suggests. There are exceptions where there is a
continuous range of allowed values of E. All of the developments
below will still hold so long as summations over the discrete index
n are replaced by appropriate integrations.
By the method of separation of variables, we have now identified a whole series of solutions to the TDSE of the form
(We have absorbed the integration constant C from the solution to
into the function 
, a valid procedure because
the TISE which 
satisfies is linear and so multiplying
by a constant factor C has no effect on its validity as
a solution.)
