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
in free space takes the general form

where is the momentum of the particle and and are constants.

To study a single wave packet coming in from the far left, the coefficient has to be set to zero. The reason was worked out in chapter 7.10: combinations of exponentials of the form produce wave packets that propagate backwards in , from right to left. Therefore, a nonzero value for would add an unwanted second wave packet coming in from the far right.

With only the coefficient of the forward moving part
left, you may as well scale the eigenfunction so that
1, simplifying it to

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 the linear momentum.

To produce a coherent wave packet, eigenfunctions with somewhat
different energies have to be combined together. Since the
momentum is given by , different energy
means different momentum ; therefore the wave packet can be
written as

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 is the momentum that a classical particle of energy would have in the left region. (Quantum mechanics looks at the complete wave function, not just a single point of it, and would say that the momentum is uncertain.)

In this case, it can no longer be argued that the coefficient must be zero to avoid a packet entering from the far right. After all, the term does not extend to the far right anymore. To the right of point , the potential changes linearly with position, and the exponentials are no longer valid.

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 is the -position measured from the point where . Also note that the cube root is negative, so that is.

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 as shown in figure A.10. The fact that no second packet comes in from the far right in the animation can be taken as an experimental confirmation of that result, so there seems little justification to go over the messy argument.

To complete the determination of the eigenfunction for a given value of , the constants , and must still be determined. That goes as follows. For now, assume that has the provisional value 1. Then provisional values and for the other two constants may be found from the requirements that the left and right regions give the same values for and at the point A in figure A.10 where they meet:

That is equivalent to two equations for the two constants and , since everything else can be evaluated, using the mentioned software. So and can be found from solving these two equations.

As the final step, it is desirable to normalize the eigenfunction so that 1. To do so, the entire provisional eigenfunction can be divided by , giving and . The energy eigenfunction has now been found. And since 1, the term is exactly the same as the free space energy eigenfunction of the first example. That means that if the eigenfunctions are combined into a wave packet in the same way as in the free space case, (A.206) with replaced by , the terms produce the exact same wave packet coming in from the far left as in the free space case.

For larger times, the
terms
produce a reflected

wave packet that returns toward
the far left. Note that is the
complex conjugate of , and
it can be seen from the unsteady Schrödinger equation that if the complex
conjugate of a wave function is taken, it produces a reversal of time.
Wave packets coming in from the far left at large negative times
become wave packets leaving toward the far left at large positive
times. However, the constant turns out to be
very small in this case, so there is little reflection.

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 of the Airy functions was negative at the far right, here it is positive. Table books that cover the Airy functions will tell you that the Airy function Bi blows up very strongly with increasing positive argument . Therefore, if the solution in the right hand region would involve any amount of Bi, it would locate the particle at infinite for all times. For a particle not at infinity, the solution in the right hand region can only involve the Airy function Ai. That function decays rapidly with positive argument , as seen in figure A.11.

The further determination of the energy eigenfunctions proceeds along the same lines as in the previous example: give a provisional value 1, then compute and from the requirements that the left and right regions produce the same values for and at the point A where they meet. Finally divide the eigenfunction by . The big difference is that now is no longer small; turns out to be of unit magnitude just like . It means that the incoming wave packet is reflected back completely.

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 the difference between summation and integration is small.

Also, while the energy eigenfunctions are not exponentials as for the free particle, for large they can be pairwise combined to approximate such exponentials. For example, eigenfunction , shown in figure A.12, behaves near the center point much like a cosine if you scale it properly. Similarly, behaves much like a sine. A cosine plus times a sine gives an exponential, according to the Euler formula (2.5). Create similar exponential combinations of eigenfunctions with even and odd values of for a range of values, and there are the approximate exponentials that allow you to create a wave packet that is at the center point at time 0. In the animation, the range of values was centered around 50, making the nominal energy hundred times the ground state energy. The exponentials degenerate over time, since their component eigenfunctions have slightly different energy, hence time evolution. That explains why after some time, the wave packet can return to the center point going the other way.

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 in the far right region, because when the eigenfunctions are combined, it would produce an unwanted wave packet coming in from the far right. In the middle region of linearly varying potential, the wave function is again a combination of the two Airy functions. The way to find the constants now has an additional step. First give the constant of the far right exponential the provisional value 1 and from that, compute provisional values and by demanding that the Airy functions give the same values for and as the far-right exponential at point B, where they meet. Next compute provisional values and by demanding that the far-left exponentials give the same values for and as the Airy functions at point A, where they meet. Finally, divide all the constants by to make 1.

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 an imaginary number. Therefore the arguments of the exponentials become real when written in terms of the absolute value of the momentum . The rest of the analysis is similar to that of the previous example.

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 , where is a spike at point A that integrates to one and the strength is a chosen constant. In the example, was chosen to be with the nominal energy. For that strength, half the wave packet will pass through.

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
at point A. To find the difference, integrate
the Hamiltonian eigenvalue problem from a point a very short distance
before point A to a point the same very short distance
behind it:

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 , at becomes what the right hand part of the eigenfunction gives for at , while at becomes what the left hand part gives for it. As seen from the above equation, the difference is not zero, but .

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
and the nominal energy are one, while the mass of the particle
is one-half. That avoids having to guesstimate suitable values for
all sorts of very small numbers. The Hamiltonian eigenvalue problem
then simplifies to

where the values of of interest cluster around 1. The nominal momentum will be one too. In those units, the length of the plotted range was one hundred in all but the harmonic oscillator case.

It should be noted that to select a good function in
(A.206) is somewhat of an art. The simplest idea would be to
choose equal to one in some limited range around the nominal
momentum, and zero elsewhere, as in

where is the relative deviation from the nominal momentum below which is nonzero. However, it is know from Fourier analysis that the locations where jumps from one to zero lead to lengthy wave packets when viewed in physical space. {D.44}. Functions that do lead to nice compact wave packets are known to be of the form

And that is essentially the function used in this study. The typical width of the momentum range was chosen to be 0.15, or 15%, by trial and error. However, it is nice if becomes not just very small, but exactly zero beyond some point, for one because it cuts down on the number of energy eigenfunctions that have to be evaluated numerically. Also, it is nice not to have to worry about the possibility of being negative in writing energy eigenfunctions. Therefore, the final function used was

The actual difference in numerical values is small, but it does make exactly zero for negative momenta and those greater than twice the nominal value. Strictly speaking, should still be multiplied by a constant to make the total probability of finding the particle equal to one. But if you do not tell people what numbers for are on the vertical axes, you do not need to bother.

In doing the numerical integrations to find , note that the mid point and trapezium rules of numerical integration are exponentially accurate under the given conditions, so there is probably not much motivation to try more advanced methods. The mid point rule was used.

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 . It may be a good idea
to check for underflow and overflow in general and use 64 bit
precision. The examples here did.

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 versus for a large number of closely spaced times covering the duration of the animation. These plots were converted to gifs using a mixture of personal software, netpbm, and ghostview. The gifs were then combined into a single movie using gifsicle.