.

which has solutions that can be expressed using solutions to the (usual) time independent Schrödinger's equation:

.

Now we can produce time dependent solutions using (time varying) linear combinations of time independent solutions like:

.

See that each term in the above sum satisfies the time dependent Schrödinger's equation. Since the time dependent Schrödinger's equation is a linear equation, the sum of solutions is a solution (i.e., superposition).

Note that the above Schrödinger's equations have been particularized
to 1d: *x*. I assume it is clear how to generalize these results
to more dimensions. For example it should be clear that in our particular 2d-fall
problem the time dependent solutions are linear combinations of solutions like:

or in dimensionless form:

where the time unit for dimensionless time *t'* is /*e*,
so *E t*/ = *E' t'*.

OK, our *y'* time dependent Schrödinger's equation has produced
unnormalizable solutions like:

= exp(*i k**y'*)
exp(-*i k*^{2} *t'*)

We seek a linear combination of these solutions to make
a normalizable lump of probability. Classical physics suggests
that this lump of probability
should move at constant speed. Basically the idea is to have
all the simple waves constructively interfere to produce the lump,
and destructively interfere everywhere else. Finding this desired
linear combination at *t'*=0 is easy once you remember
Fourier transforms:

Thus if at *t'*=0 is to be a Gaussian
lump of probability, say:

exp(-*y'*^{2})

we can use Fourier analysis to express such a function as a superposition
(integral)
of simple waves like exp(*i k*_{y}*y'*). At future times
each simple wave has time dependence
exp(-*i k*_{y}^{2} *t'*))
given by the time dependent Schrödinger's equation. Thus:

Three details: (1) According to Heisenberg's Uncertainty, the size
of the wavefunction in space (*y*) affects
the range of momentums (*p*_{y})
that make up the wavefunction. If the particle has a large range of speeds, the wavefunction
must separate out, i.e., expand. Thus to get relatively constant-sized wavefunctions
we will want to enlarge the initial spread of the wavefunction. (2) Our wavefunction
should be normalized so the total probability of it being someplace on the
*y'* axis is one -- indeed producing such normalized solutions was
the point in starting all this Fourier analysis in the first place. Thus we
will want to consider *t'*=0 wavefunctions of the form:

= *N* exp(-*y'*^{2}/4^{2})

where *N* is to be determined by the normalization condition:

and turns out to be the standard deviation
of the *y'* positions (i.e., *y*).
(3) The above wavefunction is just a lump: it lacks waves. Of course
we want a lump of *probability*, but we still want oscillation
in the wavefunction. (Remember de Broglie's relationship between the
wavelength of the wavefunction
and the particle's momentum.) Something like:

= *N* exp(-*y'*^{2}/4^{2})
exp(*i k*_{0}*y'*)

for constant wavenumber *k*_{0}, has the desired oscillation in
wavefunction with no oscillation (just a lump) in
probability density.
*g(k)* can now be determined:

The amplitude of the simple wave in our superposition (i.e., *g(k)*)
is large only when *k* is near *k*_{0}. If
is large, *k* must be very near *k*_{0}
or *g(k)* will be small. This is Heisenberg Uncertainty:
*y'* is proportional to ;
*k* is inversely proportional to ;
the product of the two is constant (in fact, ½).
Note that
we have employed the useful gaussian integral:

We can now calculate the fourier transform of *g(k)*
producing a formula for (*y',t'*).

where we have calculated the integral using the change of variables:

*q* = *k* - *k*_{0}

and the above gaussian integral. I claim that this kind of messy solves our time-dependent Schrödinger's equation. This task is left for a problem.

To better understand the nature of this solution, it is helpful
to calculate the probability density ||^{2}:

Note that the result is a lump of probability located
at the moving position: 2*k*_{0}*t'*
with a time-dependent width :

I'm getting tired of carefully distinguishing between the
normal and dimensionless versions of variables (e.g.,
*y* vs *y'*). On future pages I'll drop the
primes, but everything is to be considered the
dimensionless quantity.