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Title: Modified commutation relations
Keywords: Modified Commutation Relations, Generalized Uncertainty Principles, Quantum Gravity Phenomenology, Modified Dispersion Relations, Maximum Momentum
Issue Date: 29-May-2014
Citation: CHING CHEE LEONG (2014-05-29). Modified commutation relations. ScholarBank@NUS Repository.
Abstract: We examine deformed quantum mechanics in which the commutator between coordinates and momenta is a function of momenta. We obtained a two-parameter class of such MCR's which encode an intrinsic maximum momentum favored by deformed special relativity; a sub-class of which also imply a minimum position uncertainty/minimal length. Maximum momentum causes the bound state spectrum of the one-dimensional harmonic oscillator (and other potentials) to terminate at finite energy, whereby classical characteristics are observed. The leading order energy shifts of bound states are of opposite sign compared to those obtained using string-theory motivated MCR's. The formation of bound states in a finite potential well is delayed. We construct generalized coherent states for deformed harmonic oscillator as the quantum simulator and study their probability distribution, entropy of states exactly. Entropy and GUP of these states turn out to increase generally. However, for certain families of the GUP is possible to vanish and hence exhibits the classical characteristic. Mandel Q-number is calculated and shown that the statistics can be Poissonian, super/sub-Poissonian. Equation of motion is studied and both Ehrenfest's theorem, Correspondence Principle are recovered. Fractional revival times are obtained through the auto-correlation and they indicate that the superposition of classical-like sub-wave packet is natural. Non-perturbative effect of maximum momentum on the relativistic wave equations was examined. Beside than modified dispersion relations (MDR), we obtain the exact eigen-energies and wavefunctions of one-dimensional Klein-Gordon and Dirac equation with linear confining potentials, and the Dirac oscillator. Bound state solutions are only possible when the strength of scalar potential are stronger than vector potential. The energy spectrum of the systems studied are bounded from above and there is a truncation in the maximum number of bound states that is allowed. With the advances of quantum optics, ultra-cold atomic and ion traps technology, some of these distinguishing quantum-gravitational features might be possible to be realized within the domain of future experiments.
Appears in Collections:Ph.D Theses (Open)

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