## Falling p.13

### Falling in 2 Dimensions: *z* Solutions

Our *z* solutions are exactly as in the 1-d fall. Thus it may be useful
to recall what was said about the 1-d fall on page 7

Our normalized energy eigenfunctions are displaced Airy functions:

These eigenfunctions are orthogonal:

So if we try to express a given function *f(z')* in terms
of a sum of these functions:

orthogonality allows us to solve for the *b*_{n}:

To see how this probability lump moves
with time, we need only use the time dependence of the energy eigenfunctions:

The only change from the 1-d fall is the initial condition. Before we were
dealing with a particle dropped from rest. Here we seek a solution that
moves upwards, reaches a maximum height, and then falls; that is we seek
a solution with an upwards initial velocity. The previous pages have
given us a model for the initial wavefunction of a moving wave-packet:
= *N* exp(-*y'*^{2}/4^{2})
exp(*i k*_{0}*y'*)

We need to modify it only a bit to use it for the initial state of our
*z* motion. First, since *U*= for
*z*<0, =0 for
*z*<0. The gaussian wave=packet is never actually zero, just very
small several s away from the peak.
Thus if we continue to use a gaussian form, we must
raise its CM location by several . Thus we limit the discontinuity
at *z*=0 by making
small for positive *z* near 0 (whereas
=0 for *z*<0). Thus we are led to an
initial wavefunction:

*f(z)* = exp(-(*z*-*z*_{0})^{2}/4^{2})
exp(*i k*_{z0}*z*)

where *z*_{0}/ >> 1, so *f*(0^{+}) is
small. As before we must chose relatively large if we want
a narrow range of velocities in the wave-packet. We then proceed to calculate the
*b*_{n} exactly as above. I chose *k*_{z0}=12,
=5, *z*_{0}=20. I cannot actually find
and infinite number of *b*_{n}, so I restrict my sum about 150
*n* values whose eigenenergies are near the expected value of:

*k*_{z0}^{2} + *z*_{0} = 164

which occurs near *n*=450. Since I have not included
the complete set of orthonormal wavefunctions in the sum, my
solution to Schrödinger's equation will not exactly equal
the gaussian form. (Furthermore I did not normalize
my above *f(z)*.) Hence I will want to normalize
my finite-sum solution:

Thus we have found our solution to the time dependent Schrödinger's equation
for motion in the *z* direction
which is a particular a superposition of single-energy solutions
making a lump of probability that moves.

Here are some pictures of the probability density as it moves first
up and then down:

### t=0

### t=3

### t=6

### t=9

### t=12

### t=15

### t=18

### t=21

### t=24

All of these features can be seen in greater detail by
examining the following QuickTime movie
(0.2 Mb) on a frame-by-frame basis.

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