Coherent states and scattering for exactly solvable quantum systems
Scott E. Hoffmann
This thesis concerns quantum states that are superpositions of discrete or of continuum energy levels. It develops mathematical methods to characterize the physical properties of these states and finds important consequences. Regarding rational extensions of the harmonic oscillator, no other authors had constructed the coherent states associated with the ladder operators of Marquette and Quesne. It was our aim to proceed with this construction, then to calculate physical properties of these coherent states, looking for classical or quantum behaviour. The method of partial wave analysis was known to be not applicable to the Coulomb potential for the scattering of plane waves. It was my aim to repeat the calculation for wavepacket scattering, to investigate whether the method became applicable. Other authors have constructed relativistic probability amplitudes and compared them to the scalar Klein-Gordon amplitude. In this context, it was my aim to question the conclusion of a minimum width for a massive particle. In the same context, I investigated the possibility of a four-vector probability current with the Newton-Wigner position probability density as the zero component. In Chapter 12, my aim was to construct a measure of localization for the photon, knowing from the work of other authors that a point-localized photon state is not possible.In Chapters 3, 4 and 5, our measures of classicality for the Barut-Girardello coherent states included plotting energy expectations as functions of the magnitude of the coherent state para- meter, z, position probability densities for the coherent states and the even and odd cat states, the Wigner functions for particular choices of z, the two-photon number probability distribu- tion and the linear entropy for a coherent state on a beamsplitter, position spreads to look for squeezing and the Mandel Q parameter. In Chapter 6, it was necessary to approximate an integral containing a particular Wigner rotation matrix. An approximation of general Wigner rotation matrices, at low angle, was derived and its precision evaluated numerically in Chapter 7. The exact values for the Coulomb phase shifts were used to construct the incoming wavefunction. Time reversal and a rotation were used to construct the outgoing wavefunction. The sums over l were evaluated numerically. In Chapter 10, four arguments were used against the minimum width. Comparison of the scalar amplitude and the relativistic probability amplitude for a Newton-Wigner localized state showed the root of the problem. A Lorentz transformation was applied to the scalar function to see if the “minimum” width became even smaller. A simple construction produced scalar functions with arbitrarily small widths. A relevant argument from Almeida and Jabs was included here. In Chapter 11, the algebra involving the four components of a four-vector current (at the origin) and the boost generators was derived. All of these commutators must be satisfied by a four-vector, providing a test for the claimed four-vector probability currents of other authors. In Chapter 12, field expectations were taken in a coherent state with a mean photon number of unity, built from identical single-photon states.In Chapters 3, 4 and 5, we found a mix of classical and quantum behaviour for the coherent states. In none of the systems studied did we see the high degree of classicality of the unmodified harmonic oscillator. The investigation of cat states was relevant to the fundamental problem of measurement in quantum mechanics. In Chapter 6, the use of wavepackets introduced a convergence factor into the sum over l, which diverges in the plane wave treatment. The elimination of a divergence is the most significant result of this thesis. Scattering probabilities remained finite at all angles, in disagreement with the diverging Rutherford formula, but only on small angular regions around forward scattering. Future work will investigate the measurability of the phenomena predicted here. In Chapter 10, all four arguments pointed to no minimum width for a massive particle. In Chapter 11, it was found that there can be no four-vector current density with the Newton-Wigner position probability density as the zero component, for any spin. Other authors were found to be in error when they claimed to have produced such a four-vector current density. This result strengthens the position of the Dirac current as the appropriate current for an electron. In Chapter 12, the coherent state field expectations were found to provide a meaningful measure of localization. In summary, wavepackets give a realism to the description of a scattering experiment that is absent with the use of plane waves. Phenomena not captured by the plane wave scattering amplitude, such as time delays, are calculable in the wavepacket context. Relativistic probability amplitudes provide a complete description of free particles, consistent with quantum mechanics and with special relativity. They predict agreement with the Heisenberg uncertainty principle and relativistic effects such as Lorentz contraction. The absence of relativistic probability amplitudes from standard textbooks on relativistic quantum mechanics is a serious omission.
•Jun 8, 2020•0 citations