We determine families of spherically symmetrical $D$ -dimensional quantum potential functions $V(r)$ having ground state wavefunctions that exhibit, either in configuration or in momentum space, the form of an isotropic $q$ -Gaussian. These wavefunctions admit a maximum entropy description in terms of $S_q$ power-law entropies. We show that the potentials with a ground state of the $q$ -Gaussian form in momentum space admit the Coulomb potential $-1/r$ as a particular instance. Furthermore, all these potentials behave asymptotically as the Coulomb potential for large $r$ for all values of the parameter $q$ such that $0We determine families of spherically symmetrical $D$-dimensional quantum potential functions $V(r)$ having ground state wavefunctions that exhibit, either in configuration or in momentum space, the form of an isotropic $q$-Gaussian. These wavefunctions admit a maximum entropy description in terms of $S_q$ power-law entropies. We show that the potentials with a ground state of the $q$-Gaussian form in momentum space admit the Coulomb potential $-1/r$ as a particular instance. Furthermore, all these potentials behave asymptotically as the Coulomb potential for large $r$for all values of the parameter $q$ such that $0

arXiv:1011.3459Nov 15, 20102 citations

Self-similar potentials and q-coherent states

Takahiro Fukui

arXiv:hep-th/9309121Physics Letters AJun 1, 19947 citations

Hermite–Gaussian model for quantum states

Marcelo Losada, Ignacio S. Gomez, Federico Holik

arXiv:1811.01207Physica A Statistical Mechanics and its ApplicationsJun 19, 20191 citation

Effects of finite-range Gaussian repulsion on trapped many-boson system

Mohd. Imran, M. A. H. Ahsan

We investigate a rotating system of $N$ spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive finite-range Gaussian interaction potential of large $s$ -wave scattering length. Exact diagonalization of the Hamiltonian matrix in subspaces of quantized total angular momenta $L_{z}$ is carried out to obtain the $N$ -body ground states in slow rotating regime of $0 \le L_{z} \le 2N$ . We study the finite-range effects of the Gaussian repulsion potential on the many-body ground state properties in different subspaces of quantized $L_{z}$ . The study ranges from the few-body ( $N=2$ ) system to the many-body ( $N=16$ ), system where quantum (Bose) statistics becomes perceptible. In particular, we analyze the effect of interaction range of the Gaussian potential on the ground state energy of different $L_{z}$ states as well on the critical angular velocity $\Omega_{\bf c1}$ of first vortex ( $L_{z}=N$ ) state. For an optimal value of the range of the Gaussian interaction potential, the nucleation of the first centered vortex may begin at a lower value of the rotational angular velocity as compared to the zero-range interaction potential. Moreover, we explore the role of interaction potential range on the quantum correlation (measured in terms of the degree of condensation and the von Neumann entropy), for small systems ( $N=2$ to $8$ ) where an understanding of few-body effects provide valuable insight into large systems ( $N=8$ to $16$ ) which admit the possibility of many-body correlation. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential.We investigate a rotating system of $N$ spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive finite-range Gaussian interaction potential of large $s$-wave scattering length. Exact diagonalization of the Hamiltonian matrix in subspaces of quantized total angular momenta $L_{z}$ is carried out to obtain the $N$-body ground states in slow rotating regime of $0 \le L_{z} \le 2N$. We study the finite-range effects of the Gaussian repulsion potential on the many-body ground state properties in different subspaces of quantized $L_{z}$. The study ranges from the few-body ($N=2$) system to the many-body ($N=16$), system where quantum (Bose) statistics becomes perceptible. In particular, we analyze the effect of interaction range of the Gaussian potential on the ground state energy of different $L_{z}$ states as well on the critical angular velocity $\Omega_{\bf c1}$ of first vortex ($L_{z}=N$) state. For an optimal value of the range of the Gaussian interaction potential, the nucleation of the first centered vortex may begin at a lower value of the rotational angular velocity as compared to the zero-range interaction potential. Moreover, we explore the role of interaction potential range on the quantum correlation (measured in terms of the degree of condensation and the von Neumann entropy), for small systems ($N=2$ to $8$) where an understanding of few-body effects provide valuable insight into large systems ($N=8$ to $16$) which admit the possibility of many-body correlation. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential.

Feb 11, 20160 citations

Ground state properties of trapped boson system with finite-range Gaussian repulsion: exact diagonalization study

Mohd. Imran, M. A. H. Ahsan

Abstract We use exact diagonalization to study an interacting system of N spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum L z is carried out to obtain the N -body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body ( N = 2) to many-body ( N = 16) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter σ ) of Gaussian potential, the ground state energy increases for few-boson (2 ⩽ N ⩽ 8) systems but decreases for many-boson ( N > 8) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons N . For a given N , there is found an optimal value of the range of Gaussian potential for which the first vortex (with L z = N ) nucleates at a lower value of the rotational angular velocity Ω c1 compared to the zero-range ( δ -function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former. With increase in the range of Gaussian potential, the value of von Neumann entropy decreases and the degree of condensation increases implying an enhanced quantum correlation and phase rigidity.Abstract We use exact diagonalization to study an interacting system of N spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum L z is carried out to obtain the N -body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body ( N = 2) to many-body ( N = 16) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter σ ) of Gaussian potential, the ground state energy increases for few-boson (2 ⩽ N ⩽ 8) systems but decreases for many-boson ( N > 8) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons N . For a given N , there is found an optimal value of the range of Gaussian potential for which the first vortex (with L z = N ) nucleates at a lower value of the rotational angular velocity Ω c1 compared to the zero-range ( δ -function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former. With increase in the range of Gaussian potential, the value of von Neumann entropy decreases and the degree of condensation increases implying an enhanced quantum correlation and phase rigidity.

arXiv:1602.03811Journal of Physics B Atomic Molecular and Optical PhysicsMar 13, 20203 citations

Quantum Systems With Constant Potential

Kenneth S. Schmitz

Elsevier eBooksNov 19, 20160 citations

We determine families of spherically symmetrical

D

-dimensional quantum potential functions

V(r)

having ground state wavefunctions that exhibit, either in configuration or in momentum space, the form of an isotropic

q

-Gaussian. These wavefunctions admit a maximum entropy description in terms of

S_q

power-law entropies. We show that the potentials with a ground state of the

q

-Gaussian form in momentum space admit the Coulomb potential

-1/r

as a particular instance. Furthermore, all these potentials behave asymptotically as the Coulomb potential for large

r

for all values of the parameter

q

such that $0

We investigate a rotating system of

N

spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive finite-range Gaussian interaction potential of large

s

-wave scattering length. Exact diagonalization of the Hamiltonian matrix in subspaces of quantized total angular momenta

L_{z}

is carried out to obtain the

N

-body ground states in slow rotating regime of

0 \le L_{z} \le 2N

. We study the finite-range effects of the Gaussian repulsion potential on the many-body ground state properties in different subspaces of quantized

L_{z}

. The study ranges from the few-body (

N=2

) system to the many-body (

N=16

), system where quantum (Bose) statistics becomes perceptible. In particular, we analyze the effect of interaction range of the Gaussian potential on the ground state energy of different

L_{z}

states as well on the critical angular velocity

\Omega_{\bf c1}

of first vortex (

L_{z}=N

) state. For an optimal value of the range of the Gaussian interaction potential, the nucleation of the first centered vortex may begin at a lower value of the rotational angular velocity as compared to the zero-range interaction potential. Moreover, we explore the role of interaction potential range on the quantum correlation (measured in terms of the degree of condensation and the von Neumann entropy), for small systems (

N=2

8

) where an understanding of few-body effects provide valuable insight into large systems (

N=8

16

) which admit the possibility of many-body correlation. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential.