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Electronic Conference on Universe (ECU)

Position in Models of Quantum Mechanics with a Minimal Length

Pasquale Bosso

Published Feb 22, 2021• 3 citations
Publisher

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 wave number. Here, 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.

DOI
10.3390/ecu2021-09275

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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 wave number. Here, 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.
Several approaches to quantum gravity imply the presence of a minimal measurable length at high energies. This is in tension with the Heisenberg Uncertainty Principle. Such a contrast is then considered in phenomenological approaches to quantum gravity by introducing a minimal length in quantum mechanics via the Generalized Uncertainty Principle. Several features of the standard theory are affected by such a modification. For example, position eigenstates are no longer included in models of quantum mechanics with a minimal length. Furthermore, while the momentum-space description can still be realized in a relatively straightforward way, the (quasi-)position representation acquires numerous issues. Here, we will review such issues, clarifying aspects regarding models with a minimal length. Finally, we will consider the effects of such models on simple quantum mechanical systems.
Abstract Quantum mechanical models with a minimal length are often described by modifying the commutation relation between position and momentum. Although this represents a small complication when described in momentum space, at least formally, the (quasi-)position representation acquires numerous issues, source of misunderstandings. In this work, we review these issues, clarifying some of the aspects of minimal length models, with particular reference to the representation of the position operator.
The different proposals for quantum gravity, almost in their entirety, lead to the existence of a minimum length. The introduction of a minimum length causes radical changes in the mathematical description, as well as the physical concepts involved in a theory. In a quantum theory the implementation of a minimum length scenario can be achieved by imposing a minimum uncertainty on the position, obtained through a generalization of the Heisenberg uncertainty principle (GUP). There are different proposals for GUPs. As a result of the minimal uncertainty in position, the operator’s eigenstates are not physical states, which makes the use of position space representation inappropriate. The most natural alternative is to use the moment-space representation. However, to retrieve the information about the position, the quasi-position representation obtained by projecting the state vector onto the states of maximum location is used. The objective of this work is, then, to make a functional analysis of these representation spaces for the main proposals of GUPs, in particular to determine the representations of the position and moment operators in the quasi-position space.
Although gravity could not be quantized yet, there are good theoretical evidences that the unification of General Relativity and Quantum Mechanics should lead to the existence of a minimal observable length, a length scale below which the very notion of length looses meaning. Such effect proves to be worth some further analysis, for it acts as a natural regulator parameter, thus avoiding the divergences that plague Quantum Field Theories. In order to investigate some of the physical consequences of the existence of this peculiar effect, we propose to include it in the framework of Quantum Mechanics by modifying the Heisenberg algebra (i.e., the commutation relation of position and momentum operators, Xˆ and Pˆ), so that a minimum non-zero value for the uncertainty ?x emerges, which, being a limitation to the localizability of particles, acts as a minimal length. The Hilbert space of the theory must be modified accordingly. As we will see, the changes are not merely quantitative. On the contrary, our main result is that the familiar concept of “position measurement” must be reformulated, as well as other concepts related to it. We present a proposal for such reformulation.
in ordinary quantum mechanics for the position uncertainty since the minimum value for the uncertainty can be tend to zero. Different theories of quantum gravity, such as string theory, loop quantum gravity, and black hole physics, confirm a minimal observable length. Kempf et al have formulated The minimal length is a fundamental length for position measurement, although there is no lower bound a generalized uncertainty principle (GUP) by modifying the Heisenberg uncertainty principle to include an appearance of the length in quantum mechanics. In this study, we have found out the influence of the GUP on the particle for its confinement in a 1D potential box and calculated energy eigenvalues. The modified Schrödinger equation for the particle is factorized to be of second order to obtain the eigenvalues in a handy way. The obtained energy levels are modified from the usual result with GUP parameter dependency. The modification in the energy spectrum due to the GUP characterized by the presence of the minimal length has been also explored graphically.