The articles on the uncertainty principle and on the ladder operators need to be read carefully, but there isn't much in them which can't be verified by following the algebra and reading their conclusions. However, in presenting the uncertainty princple as an exercise in averages and variances of sums of operators relative to their commutation relations, a whole area of linear algebra is opened up. On the one hand it leads to crossing and non-crossing theoreme for eigenvalues of linear combinations of operators which had a role in quantum mechanical spectroscopy.

By scaling, the eigenvalue of a linear combination, or of a convex combination reduces to examining the eigenvalues and eigenvectors of a sum of matrices relative to the individual quantities, for which there have been two or three articles in recent years giving a very elaborate analysis of the possibilities. But all that is leading off in another direction, while all that is required for the course is to understand the basic idea, which is the relation between confined wave packets and plane wave solutions of a differential equation.

What doesn't seem to be so clear is what all that means with respect to the Dirac equation, which is an exercise in differential equation theory as well as in quantum mechanics. As was mentioned in class, no sooner than Dirac's paper on relativistic quantum mechanics was published than people began noticing anomalies. That stood in contrast to its outstanding success which was in getting a very accurate spectrum for the hydrogen atom, including the magnetic effects ascribed to spin.

One of the anomalies lay in interpreting the velocity of a free particle, which mostly arises from applying the operator algebra for the Schroedinger equation, including Dirac's interpretation on terms of Hamiltonian mechanics and the Poisson Bracket, directly to the Dirac equation, with its quaternion-like matrices and all. So much anticummuting guarantees negative pairs of eigenvalues, and so the notorious negative energy levels. From 1930 to 1950 the velocity problem remained as an unsettling curiosity.

So the paper of Newton and Wigner is significant because it attempted to assign a role to the negative energy states in producing the zitterbewegung and the apparent finite size of a particle obeying Dirac's equation. But it is hard to read because it relies on invariance principles and group representations, and is done directly in three dimensions. The nearly simultaneous paper of Foldy and Wouthuysen gives a much more explicit solution to the zitterbewegung and its removal, but again applies to three dimensions where spin complicates the negative energy problem.

Another of the anomalies was the Klein paradox, whereby a sufficiently large potential step didn't reflect parfticles arriving from the low potential side, but rather sent them on with an augmented reflection coefficient. In contrast to the three-dimensional treatment of the zitterbewegung, spin was hardly mentioned, although it would be necessary to look at the original article to see whether it wasn't just a separation of variables in three dimensions. In contrast to the central potential of the hydrogen atom favoring the use of polar coordinates and an inevitable discussion of angular momentum, cartesian coordinates suffice with a potential constant on planes and subject to a jump in just one direction.

In fact, it is interesting that in recent years there has been a discussion, in American Journal of Physics for example, of whether there can be s apin flip in one dimension for such a configuration, There can't, the concept isn't even relevant, but there an effect in three dimensions when the angular momentum of the spin of the oncoming particles isn't parallel to the direction of the potential jump.

Although the Klein paradox and the Zitterbewegung are two different things, they have a common origin in the behavior of the negative energy states. Although the two topics could be discussed separately, that article of Nitta, Kudo and Minowa invites some comparisons. Actually it shouldn't be too hard to reproduce their algebra, if not the full drawings (but I think that summing just a handful of waves, even two or three, should give a good idea of the packet).

To begin with, the matrix of coefficients in the Dirac equation reads

| 0 m - (E - V) |
| m + (E - V) 0 |,

whose square is a multiple of the identy matrix; negative for free waves and positive for exponential waves. Consequently the solution of the differential equation is just a matrix exponential with a matrix which leads to Euler's formula. If phi is sqrt(m^2 - (E-V)^2) and Z0 is something similar, the solution is

| cos(phi x) Z0 sin(phi x) |
| -1/Z0 sin (phi x) cos(phi x) |,

That means that we don't even need numerical integration to get a solution; it is already there in algebraic form.

In a reflection-transmission problem with a potential step, either a symmetric form could be used with a potential -V/2 to the left and V/2 to the right, and the appropriate matrix, x measured from the origin, used in the two regions. Alternatively, and less symmetrically, use potentials of 0 and V. Normally what is done is that the eigenvectors are extracted, because they correspond to waves travelling in opposite directions. A boundary condition could be "no particles incident from the left" so that the left travelling wave alone is used on the right (try initial condition (1, i) to get it). On the right a combination of the two waves with coefficients R and T can be used, and this leads to coefficients of reflection and transmission.

With this background, we can begin to analyze the NKM paper. One point which might not bother anybody is that they want a scalar potential, presumably referring to the fourth component of a Minkowski vector, which would be called a scalar in contrast to the rest, which is a vector. This is the "vector plus scalar" decomposition of quaternions, among other things. Byt scalar is used in another sense in recent papers on the Dirac Harmonic Oscillator, in which the Lorents group is the reference, and things like r^2-t^2 are scalars, meaning they don't change under a Lorentz transofmation. And not just the t, as the first usage would have it.

A minor point, but annoying when the article is hard to understand and one is looking for all the familiar concepts that can be found.

By constructing a "gaussian" wave packet whose energy lies near the center of the potential jump and whose variance lies within the jump,they can do the integration without worrying that it is wrong where the gaussian is already very small. No problem there.

However, this wave packet doesn't seem to exhibit zitterbewegung, as for example the one on Thaller's web page, which undergoes all manner of contortions. Is it a matter of time scale, or are the wave packets formed differently? Is zitterbewegung only noticeable in the vicinity of zero momentum, which is not the case for their waves?

Their paper gives three alternatives, which presumably amount to three different boundary conditions as they select the kind of solution in the right region. Here is another little difficulty, because either a momentum or an energy integral is ambiguous in the definition of the other quantity. That is because the dispersion relation is composed of square, in which either sign is possible. So a wave can travel in a given direction with positive energy, negative momentum; or negative energy, positive momentum.

I think the only way we are going to understand this is to repeat their calculations, even if it is only using a few waves. I've been looking at SERO to see how hard it is to do. Of course, it isn't hard, it's just laziness.

In this respect, recall that web page with the differences between phase velocities and group velocities. The dispersion relation gives phase velocity, because it tells where phases are equal in the space-time fiagram. But if you add two complex exponentials, you can take out a common factor which is the average frequency and be left with the sum of two reciprocals which define a cosing; that is the envelope, and it is the known formula for beats: the sum of two cosines in the product of the cosine of the sum by the cosine of the difference.

As more terms are added the average can shift, depending on what's added, and the envelope can become asymmetrical; these items combine to locate the maximum of the envelope, the point of zero derivative, which yields the group velocity. That is, it is wherte the sum has zero derivative, not the individual terms in the sum.

What seems to characterize the Dirac equation, or at least its relativistic part, is including the rest mass in the energy, so that in forming this sum to get the protowavepacket, the average even for zero momentum, will always include this high frequency term. The question seems to be, whether what is taken to define velocity? And then there's that energy gap, which was compared to a Gibb's phenomonon. At zero momentum, or at small momentum, there is either a large positive energy or a large negative energy to be chosen, either of which represent a small change and hence continuity in the momentum.

Another difficulty: In making up a Gaussian wave packet for the Schroedinger equation, the integral is extended over positive and negative momenta, without much regard to a boundary condition. On the other hand, the Schroedinger free wave packet spreads out in time (in contrast to Schroedinger's harmonic oscillator wave packet, which does not). Part of the spread is presumably due to those waves running in the wrong direction, as well as some on the oth3er side which may be running too fast.

For the Dirac equation, the situation is worse, because there are two distinct components, each with slightly different momentum depencences (the +,- a's in NKM's paper), so both components aren't going to show the same Gaussian. Then, it is the sum of the squares of the two components which gives the probability. The differences may be too small to dect in the NKM paper, although they may be more important if the discussion refers to pure zitterbewegung.

And is this isn't complicated enough, the Foldy-Wouthuysen transformation attempts to pick out just one of the components. The part which I am having trouble with in reading that paper is that they (as well as Newton-Wigner) talk about positive and negative energy components. But there aren't any such things; there are oscillatory solutions for positive E's and there are oscillatory solutions for negative E's, but in both cases the solutions have their own two components. It just happens that they are typically of different sizes, and that their locations in the spinor are reversed between the two signs of energy. So it is only a poetic identification to say that of the two components, one is positive energy and the other is negative energy.

What be done is to form linear combinations of solutions of the two momenta, or of the two components, or whatever, using a matrix coefficient rather than the scalar coefficients in the Schroedinger wave packets. In particular, the component exchange can be used to eliminate one component so there is just one square, not a sum of two, in making the probability. THAT seems to be what the Foldy-Wouthuysen transformation is all about. And it probably has nothing to do with the NKM paper.

Following up on that memorandum about citations, it seems that the University does have some kind of service, although it is not clear just what. It bears some investigation, because it may give access to some european and other journals, but somebody's going to have to work on it.

SERO doesn't do much with wells, barriers and steps, not even for the Schroedinger equation, but that's mainly because there hasn't been much demand for them. But now that the interest is more specific, it shouldn't be hard to fix it up.

In the meantime, the computation is so simple, given the explicit form of the solutions, that it ought to be possible to write it directly in Java or in C. Even in Mathematica, if anyone has any experience in using it!

- hvm

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