Position in Models of Quantum Mechanics with a Minimal Length

Abstract

Candidate theories of quantum gravity predict the presence of a minimal measurable length at high energies. Such feature is in contrast with the Heisenberg Uncertainty Principle. Therefore, phenomenological approaches to quantum gravity introduced models spelled as modifications of quantum mechanics including a minimal length. The effects of such modification are expected to be relevant at large energies/small lengths. One first consequence is that position eigenstates are not included in such models due to the presence of a minimal uncertainty in position. Furthermore, depending on the particular modification of the position-momentum commutator, when such models are considered from momentum space, the position operator is changed and a measure factor appears to let the position operator be self-adjoint. As a consequence, the (quasi-)position representation acquires numerous issues. For example, the position operator is no longer a multiplicative operator and the momentum of a free particle does not correspond directly to its wavenumber. Here, based on the work [1], we will review such issues, clarifying aspects of minimal length models, with particular reference to the representation of the position operator. Furthermore, we will show how such a (quasi-)position description of quantum mechanical models with a minimal length affects results concerning simple systems.

GUP modified Wigner function using classical-quantum unified framework

“…In this paper, we have studied the Wigner functions for the GUP given by the equation (23). The GUP-modified wave function for the harmonic oscillator in configuration space is (34) and, with the help of the Bopp transformation [37] and the use of the equation (37), it allows us to study the corresponding Wigner function W (q, Π). The plots presented in Fig.…”

“…The proposal that a theory of quantum gravity can indeed deform the usual Heisenberg uncertainty relation does exist in the literature for some time [14,[29][30][31]. The existence of GUP implies a modification of quantum mechanics [32,34], and the same goes for classical mechanics too although there are some mathematical subtleties associated with it [35]. A modification of quantum mechanics due to GUP in phase space implies the corresponding modification of the Wigner function [36].…”

“…Using the Eq (33), ( 35) and (36) in the Eq (34), and by doing some manipulations, one can easily find the final expression of the wave-function Ψ(q 1 , q 2 , t) to the first order in β. In Appendix VII A, we have shown the wavefunctions for the ground state, first excited state and second excited state.…”

GUP modified Wigner function using classical-quantum unified framework

“…The proposal that a theory of quantum gravity can indeed deform the usual Heisenberg uncertainty relation does exist in the literature for some time [14,[27][28][29]. The existence of GUP implies a modification of quantum mechanics [30,32], and the same goes for classical mechanics too although there are some mathematical subtleties associated with it [33]. A modification of quantum mechanics due to GUP in phase space implies the corresponding modification of the Wigner function [34].…”

“…Note that ρ m (u, v, t) = ρ m (q 1 , q 2 , t) = Ψ(q 1 , q 2 , t) = Ψ * (q 2 , t)Ψ(q 1 , t). The expression for Ψ(q i , t) can be found in Eq (32). Now that we have the modified density matrix ρ m (u, v, t) for a quantum system with Hamiltonian (25), by implementing the transformations of the form u = q + Q/2, v = q − Q/2, the ρ m (u, v, t) transforms to ρ m (q, Q, t) which is nothing but ψ(q, Q, t).…”