We formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P , Q satisfy the R -deformed Heisenberg relations, which depend on an operator ν. We find representations of P , Q in which the dimension d and angular momentum ℓ appear as parameters related to the eigenvalues of ν. We analyse the domain of P and find conditions which ensure that P is Hermitian. We investigate plane wave solutions and also free particle wavefunctions in fractional dimensions, and show that as a consequence of wavefunction continuity ℓ is quantized. The representations of P , Q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d , both algebraically and analytically. We demonstrate that the formalism extends also to time-dependent Hamiltonians by solving the time-dependent harmonic oscillator in any dimension d > 0 using the method of Lewis and Riesenfeld.We formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P , Q satisfy the R -deformed Heisenberg relations, which depend on an operator ν. We find representations of P , Q in which the dimension d and angular momentum ℓ appear as parameters related to the eigenvalues of ν. We analyse the domain of P and find conditions which ensure that P is Hermitian. We investigate plane wave solutions and also free particle wavefunctions in fractional dimensions, and show that as a consequence of wavefunction continuity ℓ is quantized. The representations of P , Q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d , both algebraically and analytically. We demonstrate that the formalism extends also to time-dependent Hamiltonians by solving the time-dependent harmonic oscillator in any dimension d > 0 using the method of Lewis and Riesenfeld.

Journal of Physics A Mathematical and GeneralMay 25, 200438 citations

Time Evolution of the Wave Function

Rhett J. Allain

Synthesis lectures on wave phenomena in the physical sciencesNov 26, 20240 citations

The Wavefunction of Continuous-Time Recurrent Neural Networks

Ikjyot Singh Kohli, Michael C. Haslam

In this paper, we explore the possibility of deriving a quantum wavefunction for continuous-time recurrent neural network (CTRNN). We did this by first starting with a two-dimensional dynamical system that describes the classical dynamics of a continuous-time recurrent neural network, and then deriving a Hamiltonian. After this, we quantized this Hamiltonian on a Hilbert space $\mathbb{H} = L^2(\mathbb{R})$ using Weyl quantization. We then solved the Schrodinger equation which gave us the wavefunction in terms of Kummer's confluent hypergeometric function corresponding to the neural network structure. Upon applying spatial boundary conditions at infinity, we were able to derive conditions/restrictions on the weights and hyperparameters of the neural network, which could potentially give insights on the the nature of finding optimal weights of said neural networks.In this paper, we explore the possibility of deriving a quantum wavefunction for continuous-time recurrent neural network (CTRNN). We did this by first starting with a two-dimensional dynamical system that describes the classical dynamics of a continuous-time recurrent neural network, and then deriving a Hamiltonian. After this, we quantized this Hamiltonian on a Hilbert space $\mathbb{H} = L^2(\mathbb{R})$ using Weyl quantization. We then solved the Schrodinger equation which gave us the wavefunction in terms of Kummer's confluent hypergeometric function corresponding to the neural network structure. Upon applying spatial boundary conditions at infinity, we were able to derive conditions/restrictions on the weights and hyperparameters of the neural network, which could potentially give insights on the the nature of finding optimal weights of said neural networks.

arXiv:2102.09399Feb 13, 20210 citations

TIME DEPENDENCE OF THE WAVE FUNCTION AND MULTIPLE EVENTS

WORLD SCIENTIFIC eBooksMay 1, 19920 citations

Sources of quantum waves

Andrés D. Baute, I. L. Egusquiza, Juan Gonzalo Muga

Due to the space and time dependence of the wavefunction in the time-dependent Schrödinger equation, different boundary conditions are possible. The equation is usually solved as an `initial-value problem', by fixing the value of the wavefunction in all space at a given instant. We compare this standard approach to `source boundary conditions' that fix the wave at all times in a given region, in particular at a point in one dimension. In contrast to the well known physical interpretation of the initial-value-problem approach, the interpretation of the source approach has remained unclear, since it introduces negative energy components, even for `free motion', and a time-dependent norm. This work provides physical meaning to the source method by finding the link with equivalent initial-value problems.Due to the space and time dependence of the wavefunction in the time-dependent Schrödinger equation, different boundary conditions are possible. The equation is usually solved as an `initial-value problem', by fixing the value of the wavefunction in all space at a given instant. We compare this standard approach to `source boundary conditions' that fix the wave at all times in a given region, in particular at a point in one dimension. In contrast to the well known physical interpretation of the initial-value-problem approach, the interpretation of the source approach has remained unclear, since it introduces negative energy components, even for `free motion', and a time-dependent norm. This work provides physical meaning to the source method by finding the link with equivalent initial-value problems.

arXiv:quant-ph/0007066Journal of Physics A Mathematical and GeneralMay 9, 200129 citations

Time-Dependent Schrödinger Equation

Jędrzej Śniatycki

Applied mathematical sciencesJan 1, 19803 citations

\mathbb{H} = L^2(\mathbb{R})

using Weyl quantization. We then solved the Schrodinger equation which gave us the wavefunction in terms of Kummer's confluent hypergeometric function corresponding to the neural network structure. Upon applying spatial boundary conditions at infinity, we were able to derive conditions/restrictions on the weights and hyperparameters of the neural network, which could potentially give insights on the the nature of finding optimal weights of said neural networks.