Harmonic Oscillators Revisited: Relativistic Formulations, Confined Particles, and Several Other Aspects

Karwowski, Jacek

doi:10.1142/9789812704474_0030

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The Coulomb Potential1

Relativistic Models1

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2004

2007

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“…Consequently, if p =0then ζ r =0 , ω r (0,l)=ω is arbitrary and As one can show, condition (30) determines values of ωζ 2 , i.e. it is independent of the sign of ζ.…”

Section: The Coulomb Potentialmentioning

confidence: 99%

“…A similar analysis may be performed in the case of the Dirac equation and its non-relativistic counterpart, the Lévy-Leblond equation. The only significant difference is that in this case spin-dependent terms, corresponding to spin-orbit interaction, would appear [30]. The corresponding eigenvalue equation (valid in both Lévy-Leblond and Dirac cases) reads:…”

Section: Relativistic Modelsmentioning

confidence: 99%

See 1 more Smart Citation

Harmonium

Karwowski¹,

Cyrnek²

2004

Annalen der Physik

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The eigenvalue problem of a Hamiltonian describing two interacting particles confined by an external harmonic‐oscillator potential is analyzed. If the particles are identical, the system is referred to as harmonium. Harmonium belongs to very few non‐trivial two‐particle systems for which the Schrödinger equation is not only separable but also, for several interaction potentials and for a set of the coupling constants, quasi‐exactly solvable. In particular, the complete spectrum of harmonium has been obtained and analyzed. Also some aspects or relativistic generalizations of the model are briefly discussed.

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“…Consequently, if p =0then ζ r =0 , ω r (0,l)=ω is arbitrary and As one can show, condition (30) determines values of ωζ 2 , i.e. it is independent of the sign of ζ.…”

Section: The Coulomb Potentialmentioning

confidence: 99%

Section: Relativistic Modelsmentioning

confidence: 99%