Two Forms of the Generalized Uncertainty Principle

1323 WordsJun 16, 20186 Pages
Various theories of quantum gravity predict the existence of a minimum length scale, which leads to the modification of the standard uncertainty principle to the Generalized Uncertainty Principle (GUP). In this paper, we study two forms of the GUP and calculate their implications on the energy of the harmonic oscillator and the Hydrogen atom more accurately than previous studies. In addition, we show how the GUP modifies the Lorentz force law and the time-energy uncertainty principle. 1. Introduction Developing a theory of quantum gravity is currently one of the main challenges in theoretical physics. Various approaches predict the existence of a minimum length scale [1, 2] that leads to the modification of the Heisenberg Uncertainty…show more content…
4. Modified Lorentz force law Because the GUP modifies the Hamiltonian, one expects that any system with a well-defined Hamiltonian is perturbed [9], perhaps even classical Hamiltonians. The impact of the GUP2-corrected classical Hamiltonian on Newton's gravitational force law was examined in [17]; here, we derive a modified Lorentz force law. For a particle in an electromagnetic field, the GUP1-modified Hamiltonian is [5]: (41) differentiating w.r.t : (42) Using inversion of series: (43) Substitution in leads to: (44) Simplifying: . (45) Applying the Euler-Lagrange equation we obtain: . (46) The RHS is , which means that the Lorentz force law becomes: (47) which is approximately: . (48) Using the same method as above, the GUP2-corrected Hamiltonian takes the form [8]: (49) differentiating w.r.t and using inversion of series: (50) leading to the Lagrangian: . (51) from which we obtain: (52) which is approximately: . (53) The new term in (48) and (53) depends on , which means that its effect in high energy physics will be too small even at relativistic speeds. For example, in a proton-proton scattering experiment: . (54) Experimental tests of Coulomb’s law use large, but usually static, masses [18]. For example, coulomb’s torsion balance experiment measures the torsion force needed to balance the
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