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.
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 Principle:
to the Generalized Uncertainty Principle (GUP) [3, 4]:
where , is a dimensionless constant usually assumed to be of order unity, is the Planck length , and may depend on but not on . The second term on the RHS above is important at very high energies/ small length scales (i.e. ).
In this article, we study two forms of the GUP. The first (GUP1) [5, 6] is:
which follows from the modified commutation relation :
The second (GUP2) [7, 8] is:
which follows from the proposed modified commutation relation :
where , is a constant usually assumed to be of order unity. In addition to a minimum measurable length, GUP2 implies a maximum measurable momentum.
The commutation relation (4) admits the following representation in position space [9, 10]:
where satisfy the canonical commutation relation This definition modifies any Hamiltonian near the Planck scale to [9, 10]:
Similarly, (6) admits the definition [7, 8]:
leading to the perturbed Hamiltonian:
The aim of this article is to study the impact of GUP1 and GUP2 on the energy of the harmonic oscillator and 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.
2. Harmonic Oscillator
The harmonic oscillator is a good model for many systems, so it is important to calculate its energy accurately to compare it with future experiments. Recently a quantum optics experiment was proposed  to probe the commutation relation of a mechanical oscillator with mass close to the Planck mass.
The effect of GUP1 on the eigenvalues of the harmonic oscillator was calculated exactly in . The effect of GUP2 was considered in  to first and second order for the ground energy only. In this section, we consider first and second order corrections to all energy levels for both GUPs to compare them, and we use the ladder operator method, which is simpler than the other methods.
Figure 1. The relative change in energy due to GUP1 and GUP2 as a function of , assuming .
Figure 1 is a plot of (25) and (26), as a function of . It is clear that the difference between the corrections of GUP1 and GUP2...