Radial coherent states for central potentials: The isotropic harmonic oscillator

Gerry, Christopher C.; Kiefer, J.

doi:10.1103/physreva.38.191

Cited by 15 publications

(13 citation statements)

References 20 publications

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“…On the other hand, for the SU (1, 1) coherent states of the radial oscillator, some results were already obtained in e.g. [9] and [10]. These are recovered as particular cases in our model at the time that some of the aspects that were unclosed in [9,10] are now clarified in terms of the factorization method.…”

Section: Introductionsupporting

confidence: 54%

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SU(1,1) and SU(2) app

2016

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It is shown that each one of the Lie algebras su(1, 1) and su(2) determine the spectrum of the radial oscillator. States that share the same orbital angular momentum are used to construct the representation spaces of the non-compact Lie group SU (1, 1). In addition, three different forms of obtaining the representation spaces of the compact Lie group SU (2) are introduced, they are based on the accidental degeneracies associated with the spherical symmetry of the system as well as on the selection rules that govern the transitions between different energy levels. In all cases the corresponding generalized coherent states are constructed and the conditions to squeeze the involved quadratures are analyzed.

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Section: Introductionsupporting

confidence: 54%

“…In general, according to (31), the variance of one quadrature is reduced at the expense of the other. In Figure 10 we can appreciate that the phase φ alternates such squeezing between the quadratures (compare with [9]). In particular, L 1 is squeezed for φ = π 2 , 3π 2 , .…”

Section: Su (1 1) Perelomov Coherent Statesmentioning

confidence: 91%

SU(1,1) and SU(2) app

2016

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“…The uncertainties of the SU(1, 1) variables for these modes are given by with ⟨K 0 ⟩ 0 = K ⟨k, k|K 0 |k, k⟩ K = k [32,50]. Thus…”

Section: Variances and Squeezingmentioning

confidence: 99%

Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

Cruz

Gress

2017

Annals of Physics

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h i g h l i g h t s• Ladder and shift operators for the Hermite-Gaussian modes are constructed.• These operators are used to obtain the generators of the dynamical algebras.• The Barut-Girardello and Perelomov coherent states are determined.• The uncertainty relations are considered in connection to the beam quality factor. a r t i c l e i n f o A group-theoretical approach to the paraxial propagation of Hermite-Gaussian modes based on the factorization method is presented. It is shown that the su(1, 1) and the su(2) algebras generate the spectrum of propagation constants at any fixed transversal plane. The complete set of HG modes is decomposed into hierarchies that are used to establish the representation spaces of SU(1, 1) and SU(2). The corresponding families of generalized coherent states are constructed and the variances of the quadratures and canonical variables are determined.

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“…However a subgroup of this is Sp(2) (g) 0(3) which corresponds to a separation in spherical polar coordinates: 0(3) being the usual rotational symmetry group and Sp(2) being the radial spectrum generating group. Since Sp(2) ~ 51/(1,1) we may use the previously developed formalism to write radial coherent states for the isotropic oscil lator including the centrifugal barrier terms 50 .…”

Section: Radial Isotropic Oscillatormentioning

confidence: 99%

PART III SU(1, 1) COHERENT STATES AND PATH INTEGRALS FOR SU(1, 1)

Gerry

1992

Path Integrals and Coherent States of SU(2) and SU(1,1)

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Path Integrals and Coherent States of SU(2) and SU(1,1) Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/20/15. For personal use only. This page is intentionally left blank Path Integrals and Coherent States of SU(2) and SU(1,1) Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/20/15. For personal use only. Path Integrals and Coherent States of SU(2) and SU(1,1) Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/20/15. For personal use only. 222are used to construct a generalized uncertainty relation; the generalized coherent states (CS) being those states for which the uncertainty product is a minimum (MUCS). The states developed in this fashion when sub jected to time evolution are not completely dispersionless as they move in the classical potential but do appear to be capable of a least partly reforming after a few oscillations although, eventually, an initial wave packet will disperse throughout the classically allowed region 10 .An alternate method of creating generalized coherent states for sys tems other than the oscillator is associated with the unitary irreducible representations (UIRs) of Lie groups and Lie algebras 11 . This approach may be considered as complementary to the MUCS discussed above. The most familiar Lie group, SU(2), of course describes a number of systems such as two level atoms, spin systems etc. Coherent states of SU(2) are the subject of Professor Kuratsuji's lectures. On the other hand the noncompact group SU(1,1), which has the following local isomorphisms: SU(1,1) ~ SO(2,l) ~~ Sp(2R) ~ S1(2R), plays the role of dynamical group for a number of systems of general interest. The term dynamical group as used here should be taken as synonymous with the terms noninvariance group and spectrum generating group. Actually it is the spectrum generating algebra (SGA) of the associated dynamical group that is mainly of interest. With the Hamiltonian composed of the generators of the group, the UIRs yield the spectrum from the energy eigenvalue prob lem. Among the systems for which SU(1,1), or one of its isomorphic forms, provides an SGA, are, the harmonic oscillator 12 , the Coulomb problem 13 , Hartmann's 14 ring potential 15 , the s-states of the Morse oscilla tor 16 , the s-states of the Hulthen potential 17 , the Dirac-Coulomb and Klein-Gordon-Coulomb problems 18 . Some many-particle problems are also included. Examples are a model of a superfluid bose system 19 and coupled harmonic oscillators 20 . There recently has been a great deal of interest in the algebraic approach to scattering problems 21 and in the cal culation of resonance widths and positions 22 . Now in constructing coherent states for SU(1,1) we recall that ordi nary oscillator coherent states, {\z>} where z is complex, may be con structed in three ways: i) as states which minimize the uncertainty rela tion AxAp >7T/2, ii) as eigenstates of the annihilation operator a, i.e. a \z> «= z \z> and III) as states displaced from the vacuum |0> by the operation of the displacement o...

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Radial coherent states for central potentials: The isotropic harmonic oscillator

Cited by 15 publications

References 20 publications

SU(1,1) and SU(2) app

SU(1,1) and SU(2) app

Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

PART III SU(1, 1) COHERENT STATES AND PATH INTEGRALS FOR SU(1, 1)

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