2013
DOI: 10.1080/10236198.2012.658384
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On a hidden symmetry of quantum harmonic oscillators

Abstract: Abstract. We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the He… Show more

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Cited by 26 publications
(52 citation statements)
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References 60 publications
(113 reference statements)
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“…These explicit solutions that are omitted in all textbooks on quantum mechanics (see [13,15]) provide a new multiparameter family of oscillating Gaussian-Hermitian beams in parabolic (self-focusing fiber) waveguides, which deserve an experimental observation; special cases were theoretically studied earlier in [8,16] …”
Section: −1∕2mentioning
confidence: 99%
See 1 more Smart Citation
“…These explicit solutions that are omitted in all textbooks on quantum mechanics (see [13,15]) provide a new multiparameter family of oscillating Gaussian-Hermitian beams in parabolic (self-focusing fiber) waveguides, which deserve an experimental observation; special cases were theoretically studied earlier in [8,16] …”
Section: −1∕2mentioning
confidence: 99%
“…The real-or complex-valued parameters α 0 , β 0 ≠ 0, γ 0 0, δ 0 , ε 0 , and κ 0 0 are initial data of the corresponding Ermakov-type system [2,12,13]. A direct Mathematica verification can be found in Media 1.…”
Section: −1∕2mentioning
confidence: 99%
“…18) provides a measure of the difference between the above states and the corresponding stationary state with the same N.…”
mentioning
confidence: 99%
“…We have done coordinate transformations and reparametrization from the original expressions in Ref [18]…”
mentioning
confidence: 99%
“…Extensions of this approach have been presented in [26] and [28]. Applications include nonlinear optics, Bose-Einstein condensates, integrability of NLS and quantum mechanics, see for example [3], [31], [32] and [33], and references therein. E. Marhic in 1978 introduced (probably for the first time) a one-parameter {α(0)} family of solutions for the linear Schrödinger equation of the one-dimensional harmonic oscillator, where the use of an explicit formulation (classical Melher's formula [34]) for the propagator was fundamental.…”
Section: Introductionmentioning
confidence: 99%