arXiv subjects

Manjit Bhatia

Publications and source records attributed to Manjit Bhatia.

2 recordsLinked to original sources

arXiv2010

Elementary Quantum Mechanics in a Space-time Lattice

Studies of quantum fields and gravity suggest the existence of a minimal length, such as Planck length \cite{Floratos,Kempf}. It is natural to ask how the existence of a minimal length may modify the results in elementary quantum mechanics (QM) problems familiar to us \cite{Gasiorowicz}. In this paper we address a simple problem from elementary non-relativistic quantum mechanics, called "particle in a box", where the usual continuum (1+1)-space-time is supplanted by a space-time lattice. Our lattice consists of a grid of $λ_0 \times τ_0 $ rectangles, where $λ_0$, the lattice parameter, is a fundamental length (say Planck length) and, we take $τ_0$ to be equal to $λ_0/c$. The corresponding Schrodinger equation becomes a difference equation, the solution of which yields the $q$-eigenfunctions and $q$-eigenvalues of the energy operator as a function of $λ_0 $. The $q$-eigenfunctions form an orthonormal set and both $q$-eigenfunctions and $q$-eigenvalues reduce to continuum solutions as $ λ_0 \rightarrow 0 .$ The corrections to eigenvalues because of the assumed lattice is shown to be $O(λ_0^2).$ We then compute the uncertainties in position and momentum, $Δx, Δp$ for the box problem and study the consequent modification of Heisenberg uncertainty relation due to the assumption of space-time lattice, in contrast to modifications suggested by other investigations such as \cite{Floratos}.

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arXiv2009

Free Particle Eigenfunctions of Schrodinger Equation with Quantized Space-time

It is well-known that the coordinate as a continuous variable, consisting of a set of all points between 0 and $L$ contradicts the observability of measurement. In other words there might exist a fundamental length in nature, such as the Planck length $λ_P$, so that it is not possible to measure a position coordinate with accuracy smaller than this fundamental length. It is therefore necessary to investigate the formulation of quantum mechanics using only discrete variables as coordinates. To investigate all of quantum mechanics or any branch of physics from this approach is of course a daunting task and thus it is worthwhile to consider a specific simple problem in order to formulate the basic ideas. In this note we compare the solutions of Schrodinger equation for one-dimensional free particle under the usual space-time continuum with those that are obtained when space-time is assumed to be quantized using a simple model. For this purpose, we replace the derivatives occurring in Schrodinger equation with the corresponding discrete derivatives. We compute the probability density (and the probability current) under the two scenarios; they turn out to be quite different in the two cases. We also obtain the operator identity for the commutator $[p,x]^q$ under the assumption of quantized space-time and contrast it with the usual commutator $[p,x]$.

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