, how do we find the time dependence of
? Since 
is the coefficient vector for
the eigenfunctions of 
, we know the time dependence is
accounted for by multiplying each coefficient by 
. So, we now create a
vector E, indexed by q.
Now that we have , how do we find the time dependence of
? Since is the coefficient vector for
the eigenfunctions of , we know the time dependence is
accounted for by multiplying each coefficient by . So, we now create a
vector E, indexed by q.
Now, if we take the exponential of 
times each element, where t
is the time we wish to evaluate 
at, we get
Now, if we perform element by element multiplication ( Matlab command is `` 
'') on
(q) and 
we get:
Now, we have taken account for the time dependence by modifying our
amplitude function (note the subscript t to denote that this is
at a particular time t). The last chore is to now compute the
wave function in x-space from the modified amplitude function.