A.27 Details of the animations

This note explains how the wave packet animations of chapter 7.11 and 7.12 were obtained. If you want a better understanding of unsteady solutions of the Schrödinger equation and their boundary conditions, this is a good place to start. In fact, deriving such solutions is a popular item in quantum mechanics books for physicists.

First consider the wave packet of the particle in free space, as shown
in chapter 7.11.1. An energy eigenfunction with energy

where

To study a single wave packet coming in from the far left, the
coefficient

With only the coefficient

A typical example is shown in figure A.9. Plus and minus the magnitude of the eigenfunction are shown in black, and the real part is shown in red. This wave function is an eigenfunction of linear momentum, with

To produce a coherent wave packet, eigenfunctions with somewhat
different energies

Next consider the animation of chapter 7.11.2, where
the particle accelerates along a downward potential energy ramp
starting from point A. A typical energy eigenfunction is shown in
figure A.10. Since to the left of point A, the potential
energy is still zero, in that region the energy eigenfunction is still
of the form

where

In this case, it can no longer be argued that the coefficient

In fact, it is known that the solution of the Hamiltonian eigenvalue
problem in a region with a linearly varying potential is a combination
of two weird functions Ai and Bi that are called the
Airy

functions. The bad news is that if you are
interested in learning more about their properties, you will need an
advanced mathematical handbook like [1] or at
least look at addendum {A.29}. The good news is that
free software to evaluate these functions and their first derivatives
is readily available on the web. The general solution for a linearly
varying potential is of the form

Note that

It may be deduced from the approximate analysis of addendum
{A.28} that to prevent a second wave packet coming in from
the far right, Ai and Bi must appear together in the combination

To complete the determination of the eigenfunction for a given value
of

That is equivalent to two equations for the two constants

As the final step, it is desirable to normalize the eigenfunction

For larger times, the
reflected

wave packet that returns toward
the far left. Note that

Next consider the animation of chapter 7.11.3, where the
particle is turned back by an upward potential energy ramp. A typical
energy eigenfunction for this case is shown in figure
A.11. Unlike in the previous example, where the
argument

The further determination of the energy eigenfunctions proceeds along
the same lines as in the previous example: give

For the harmonic oscillator of chapter 7.11.4, the
analysis is somewhat different. In particular, chapter
4.1.2 showed that the energy levels of the one-dimensional
harmonic oscillator are discrete,

so that unlike the motions just discussed, the solution of the Schrödinger equation is a sum, rather than the integral (A.206),

However, for large

Also, while the energy eigenfunctions

For the particle of chapter 7.12.1 that encounters a brief
accelerating force, an example eigenfunction looks like figure
A.13. In this case, the solution in the far right region
is similar to the one in the far left region. However, there cannot
be a term of the form

For the tunneling particle of chapter 7.12.2, an example
eigenfunction is as shown in figure A.14. In this case,
the solution in the middle part is not a combination of Airy
functions, but of real exponentials. It is essentially the same
solution as in the left and right parts, but in the middle region the
potential energy is greater than the total energy, making

For the particle tunneling through the delta function potential in
chapter 7.12.2, an example energy eigenfunction is shown
in figure A.15. The potential energy in this case is

For a delta function potential, a modification must be made in the
analysis as used so far. As figure A.15 illustrates, there
are kinks in the energy eigenfunction at the location A of the delta
function. The left and right expressions for the eigenfunction
*do not* predict the same value for its derivative

The integral in the right hand side is zero because of the vanishingly small interval of integration. But the delta function spike in the left hand side integrates to one regardless of the small integration range, so

For vanishingly small

So the correct equations for the provisional constants are in this case

Compared to the analysis as used previously, the difference is the final term in the second equation that is added by the delta function.

The remainder of this note gives some technical details for if you are
actually planning to do your own animations. It is a good idea to
assume that the units of mass, length, and time are chosen such that

where the values of

It should be noted that to select a good function

where

And that is essentially the function

The actual difference in numerical values is small, but it does make

In doing the numerical integrations to find

The animations in this book used the numerical implementations ` daie.f, dbie.f, daide.f,` and `dbide.f` from netlib.org for the
Airy functions and their first derivatives. These offer some basic
protection against underflow and overflow by splitting off an
exponential for positive

For the harmonic oscillator, the larger the nominal energy is compared to the ground state energy, the more the wave packet can resemble a single point compared to the limits of motion. However, the computer program used to create the animation computed the eigenfunctions by evaluating the analytical expression given in derivation {D.12}, and explicitly evaluating the Hermite polynomials is very round-off sensitive. That limited it to a maximum of about hundred times the ground state energy when allowing for enough uncertainty to localize the wave packet. Round-off is a general problem for power series, not just for the Hermite polynomials. If you want to go to higher energies to get a smaller wave packet, you will want to use a finite difference or finite element method to find the eigenfunctions.

The plotting software used to produce the animations was a mixture of
different programs. There are no doubt much simpler and better ways
of doing it. In the animations presented here, first plots were
created of