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 $\lambda_0 \times \tau_0 $ rectangles, where $\lambda_0$ , the lattice parameter, is a fundamental length (say Planck length) and, we take $\tau_0$ to be equal to $\lambda_0/c$ . 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 $\lambda_0 $. The $q$ -eigenfunctions form an orthonormal set and both $q$ -eigenfunctions and $q$ -eigenvalues reduce to continuum solutions as $ \lambda_0 \rightarrow 0 .$ The corrections to eigenvalues because of the assumed lattice is shown to be $O(\lambda_0^2).$ We then compute the uncertainties in position and momentum, $\Delta x, \Delta p$ 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}.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 $\lambda_0 \times \tau_0 $ rectangles, where $\lambda_0$, the lattice parameter, is a fundamental length (say Planck length) and, we take $\tau_0$ to be equal to $\lambda_0/c$. 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 $\lambda_0 $. The $q$-eigenfunctions form an orthonormal set and both $q$-eigenfunctions and $q$-eigenvalues reduce to continuum solutions as $ \lambda_0 \rightarrow 0 .$ The corrections to eigenvalues because of the assumed lattice is shown to be $O(\lambda_0^2).$ We then compute the uncertainties in position and momentum, $\Delta x, \Delta p$ 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}.

Aug 16, 20163 citations

Quantum lattice problems

Hans A. De Raedt, Wolfgang von der Linden

Topics in applied physicsJan 1, 19923 citations

Forced Vibrations of a Harmonic Lattice in Quantum Mechanics

William Band

Schrödinger's equation is solved in an elementary fashion for a harmonic lattice subject to external forces that are functions explicitly only of time, and to damping forces proportional to velocity expectation values. The result shows that the response is a purely classical motion of the quantum mechanical probability distribution associated with the free lattice. The energy of the lattice is Eq + Ec, where Eq is any energy eigenvalue of the free undisturbed lattice existing before the external forces were applied, and Ec is the energy that would be obtained by classical mechanics for the response of the lattice to the disturbance. The energy propagates through the lattice as a purely classical wave motion superposed on any quantum mechanical background that may happen to exist. Anharmonic coupling terms are needed to cause quantum excitations and produce thermal equilibrium within the lattice. The conventional treatment of energy transport as due to unidirectional normal modes is criticized.Schrödinger's equation is solved in an elementary fashion for a harmonic lattice subject to external forces that are functions explicitly only of time, and to damping forces proportional to velocity expectation values. The result shows that the response is a purely classical motion of the quantum mechanical probability distribution associated with the free lattice. The energy of the lattice is Eq + Ec, where Eq is any energy eigenvalue of the free undisturbed lattice existing before the external forces were applied, and Ec is the energy that would be obtained by classical mechanics for the response of the lattice to the disturbance. The energy propagates through the lattice as a purely classical wave motion superposed on any quantum mechanical background that may happen to exist. Anharmonic coupling terms are needed to cause quantum excitations and produce thermal equilibrium within the lattice. The conventional treatment of energy transport as due to unidirectional normal modes is criticized.

American Journal of PhysicsSep 1, 19620 citations

Quantum Lattice Problems

Hans A. De Raedt, Wolfgang von der Linden

Topics in applied physicsJan 1, 19926 citations

Title: Quantum-Enhanced Lattice Theory for Complex Systems

Jincheng Zhang

Quantum-enhanced lattice theory offers a novel approach to modeling complex systems, particularly those exhibiting strong correlations and dynamic behavior. This paper explores the potential of extending lattice theory to incorporate quantum mechanics principles, specifically focusing on the dynamics of highly correlated systems. We introduce a quantum algorithm designed to efficiently calculate entanglement and correlations within a lattice, aiming to significantly enhance simulation accuracy compared to classical methods. This research addresses the limitations of traditional lattice methods in capturing intricate patterns and predicting complex system evolution, providing a pathway towards more precise modeling and control. The core mechanism involves the integration of quantum mechanics into lattice dynamics, offering a promising shift towards a fundamentally different approach to complex system analysis.Quantum-enhanced lattice theory offers a novel approach to modeling complex systems, particularly those exhibiting strong correlations and dynamic behavior. This paper explores the potential of extending lattice theory to incorporate quantum mechanics principles, specifically focusing on the dynamics of highly correlated systems. We introduce a quantum algorithm designed to efficiently calculate entanglement and correlations within a lattice, aiming to significantly enhance simulation accuracy compared to classical methods. This research addresses the limitations of traditional lattice methods in capturing intricate patterns and predicting complex system evolution, providing a pathway towards more precise modeling and control. The core mechanism involves the integration of quantum mechanics into lattice dynamics, offering a promising shift towards a fundamentally different approach to complex system analysis.

Sep 3, 20260 citations

\lambda_0

, the lattice parameter, is a fundamental length (say Planck length) and, we take

\tau_0

to be equal to

\lambda_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 $\lambda_0 $. The

q

-eigenfunctions form an orthonormal set and both

q

-eigenfunctions and

q

-eigenvalues reduce to continuum solutions as $ \lambda_0 \rightarrow 0 .$ The corrections to eigenvalues because of the assumed lattice is shown to be

O(\lambda_0^2).

We then compute the uncertainties in position and momentum,

\Delta x, \Delta p

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}.