Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics

Christiane Quesne ; Volodymyr M. Tkachuk

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

Two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered. The first one is the most general quadratic commutation relation. The corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectrum and eigenfunctions of the harmonic oscillator in an electric field is studied. The second extension is a function-dependent generalization of the simplest quadratic commutation relation with only a nonzero minimal uncertainty in position. Such an uncertainty now becomes dependent on the average position. With each function-dependent commutation relation we associate a family of potentials whose spectrum can be exactly determined through supersymmetric quantum mechanical and shape invariance techniques. Some representations of the generalized Heisenberg algebras are proposed in terms of conventional position and momentum operators $x$, $p$. The resulting Hamiltonians contain a contribution proportional to $p^4$ and their $p$-dependent terms may also be functions of $x$. The theory is illustrated by considering Pöschl–Teller and Morse potentials.

Détail

Keywords: deformed algebras; uncertainty relations; supersymmetric quantum mechanics; shape invariance.

Bibliographie

[1] Gross D. J., Mende P. F., “String theory beyond the Planck scale”, Nuclear Phys. B, 303 (1988), 407–454 | DOI | MR

[2] Amati D., Ciafaloni M., Veneziano G., “Can spacetime be probed below the string size?”, Phys. Lett. B, 216 (1989), 41–47 | DOI | MR

[3] Maggiore M., “The algebraic structure of the generalized uncertainty principle”, Phys. Lett. B, 319 (1993), 83–86 ; hep-th/9309034 | DOI | MR

[4] Connes A., “Gravity coupled with matter and the foundation of non-commutative geometry”, Comm. Math. Phys., 182 (1996), 155–176 ; hep-th/9603053 | DOI | MR | Zbl

[5] Amelino-Camelia G., Mavromatos N. E., Ellis J., Nanopoulos D. V., “On the space-time uncertainty relations of Liouville strings and D-branes”, Modern Phys. Lett. A, 12 (1997), 2029–2036 ; hep-th/9701144 | DOI | MR

[6] Seiberg N., Witten E., “String theory and noncommutative geometry”, JHEP, 09 (1999), 032, 93 pp., ages ; hep-th/9908142 | DOI | MR | Zbl

[7] Kempf A., “Uncertainty relation in quantum mechanics with quantum group symmetry”, J. Math. Phys., 35 (1994), 4483–4496 ; hep-th/9311147 | DOI | MR | Zbl

[8] Hinrichsen H., Kempf A., “Maximal localization in the presence of minimal uncertainties in positions and in momenta”, J. Math. Phys., 37 (1996), 2121–2137 ; hep-th/9510144 | DOI | MR | Zbl

[9] Kempf A., Mangano G., Mann R. B., “Hilbert space representation of the minimal length uncertainty relation”, Phys. Rev. D, 52 (1995), 1108–1118 ; hep-th/9412167 | DOI | MR

[10] Kempf A., “Non-pointlike particles in harmonic oscillators”, J. Phys. A: Math. Gen., 30 (1997), 2093–2102 ; hep-th/9604045 | DOI | MR

[11] Kempf A., Quantum field theory with nonzero minimal uncertainties in position and momentum, hep-th/9405067 | MR

[12] Brau F., “Minimal length uncertainty relation and the hydrogen atom”, J. Phys. A: Math. Gen., 32 (1999), 7691–7696 ; quant-ph/9905033 | DOI | MR | Zbl

[13] Benczik S., Chang L. N., Minic D., Takeuchi T., “Hydrogen-atom spectrum under a minimal-length hypothesis”, Phys. Rev. A, 72 (2005), 012104, 4 pp., ages ; hep-th/0502222 | DOI

[14] Stetsko M. M., Tkachuk V. M., “Perturbation hydrogen-atom spectrum in deformed space with minimal length”, Phys. Rev. A, 74 (2006), 012101, 5 pp., ages | DOI

[15] Brau F., Buisseret F., “Minimal length uncertainty relation and gravitational quantum well”, Phys. Rev. D, 74 (2006), 036002, 5 pp., ages ; hep-th/0605183 | DOI

[16] Nouicer K., “An exact solution of the one-dimensional Dirac oscillator in the presence of minimal lengths”, J. Phys. A: Math. Gen., 39 (2006), 5125–5134 | DOI | MR | Zbl

[17] Chang L. N., Minic D., Okamura N., Takeuchi T., “Effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem”, Phys. Rev. D, 65 (2002), 125028, 7 pp., ages ; hep-th/0201017 | DOI | MR

[18] Benczik S., Chang L. N., Minic D., Okamura N., Rayyan S., Takeuchi T., “Short distance versus long distance physics: The classical limit of the minimal length uncertainty relation”, Phys. Rev. D, 66 (2002), 026003, 11 pp., ages ; hep-th/0204049 | DOI

[19] Chang L. N., Minic D., Okamura N., Takeuchi T., “Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations”, Phys. Rev. D, 65 (2002), 125027, 8 pp., ages ; hep-th/0111181 | DOI | MR

[20] Fityo T. V., Vakarchuk I. O., Tkachuk V. M., “One-dimensional Coulomb-like problem in deformed space with minimal length”, J. Phys. A: Math. Gen., 39 (2006), 2143–2150 ; quant-ph/0507117 | DOI | MR

[21] Fityo T. V., Vakarchuk I. O., Tkachuk V. M., “WKB approximation in deformed space with minimal length”, J. Phys. A: Math. Gen., 39 (2006), 379–388 ; quant-ph/0510018 | DOI | MR

[22] Kempf A., “Quantum group symmetric Bargmann–Fock space: integral kernels, Green functions, driving forces”, J. Math. Phys., 34 (1993), 969–987 | DOI | MR | Zbl

[23] Quesne C., Tkachuk V. M., “Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework”, J. Phys. A: Math. Gen., 36 (2003), 10373–10390 ; math-ph/0306047 | DOI | MR

[24] Cooper F., Khare A., Sukhatme U., “Supersymmetry and quantum mechanics”, Phys. Rep., 251 (1995), 267–385 ; hep-th/9405029 | DOI | MR

[25] Gendenshtein L. E., “Derivation of exact spectra of the Schrödinger equation by means of supersymmetry”, JETP Lett., 38 (1983), 356–359

[26] Dabrowska J., Khare A., Sukhatme U., “Explicit wavefunctions for shape-invariant potentials by operator techniques”, J. Phys. A: Math. Gen., 21 (1988), L195–L200 | DOI | MR

[27] Spiridonov V., “Exactly solvable potentials and quantum algebras”, Phys. Rev. Lett., 69 (1992), 398–401 ; hep-th/9112075 | DOI | MR | Zbl

[28] Spiridonov V., “Deformed conformal and supersymmetric quantum mechanics”, Modern Phys. Lett. A, 7 (1992), 1241–1252 ; hep-th/9202013 | DOI | MR

[29] Khare A., Sukhatme U. P., “New shape-invariant potentials in supersymmetric quantum mechanics”, J. Phys. A: Math. Gen., 26 (1993), L901–L904 ; hep-th/9212147 | DOI | MR | Zbl

[30] Barclay D. T., Dutt R., Gangopadhyaya A., Khare A., Pagnamenta A., Sukhatme U., “New exactly solvable Hamiltonians: shape invariance and self-similarity”, Phys. Rev. A, 48 (1993), 2786–2797 ; hep-ph/9304313 | DOI | MR

[31] Lutzenko I., Spiridonov V., Zhedanov A., “On the spectrum of a $q$-oscillator with a linear interaction”, Phys. Lett. A, 204 (1995), 236–242 | DOI | MR | Zbl

[32] Loutsenko I., Spiridonov V., Vinet L., Zhedanov A., “Spectral analysis of $q$-oscillator with general bilinear interaction”, J. Phys. A: Math. Gen., 31 (1998), 9081–9094 | DOI | MR | Zbl

[33] Quesne C., Tkachuk V. M., “More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum”, J. Phys. A: Math. Gen., 37 (2004), 10095–10114 ; math-ph/0312029 | DOI | MR

[34] Quesne C., Tkachuk V. M., “Dirac oscillator with nonzero minimal uncertainty in position”, J. Phys. A: Math. Gen., 38 (2005), 1747–1766 ; math-ph/0412052 | DOI | MR

[35] Pillin M., “On the deformability of Heisenberg algebras”, Comm. Math. Phys., 180 (1996), 23–38 ; q-alg/9508014 | DOI | MR | Zbl

[36] Bagchi B., Banerjee A., Quesne C., Tkachuk V. M., “Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass”, J. Phys. A: Math. Gen., 38 (2005), 2929–2946 ; quant-ph/0412016 | DOI | MR

[37] Abramowitz M., Stegun I. A., Handbook of mathematical functions, Dover, New York, 1965

[38] Quesne C., Tkachuk V. M., “Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem”, J. Phys. A: Math. Gen., 37 (2004), 4267–4282 ; math-ph/0403047 | DOI | MR

[39] Geller M. R., Kohn W., “Quantum mechanics of electrons in crystals with graded composition”, Phys. Rev. Lett., 70 (1993), 3103–3106 | DOI

[40] Serra Ll., Lipparini E., “Spin response of unpolarized quantum dots”, Europhys. Lett., 40 (1997), 667–672 | DOI

[41] Barranco M., Pi M., Gatica S. M., Hernández E. S., Navarro J., “Structure and energetics of mixed ${}^4$He-$^3$He drops”, Phys. Rev. B, 56 (1997), 8997–9003 | DOI

[42] Ekenberg U., “Nonparabolicity effects in a quantum well: sublevel shift, parallel mass, and Landau levels”, Phys. Rev. B, 40 (1989), 7714–7726 | DOI

[43] Ring P., Schuck P., The nuclear many body problem, Springer, New York, 1980 | MR

[44] Arias de Saavedra F., Boronat J., Polls A., Fabrocini A., “Effective mass of one ${}^4$He atom in liquid ${}^3$He”, Phys. Rev. B, 50 (1994), 4248–4251 | DOI

[45] Puente A., Serra Ll., Casas M., “Dipole excitation of Na clusters with a non-local energy density functional”, Z. Phys. D, 31 (1994), 283–286 | DOI

[46] Jones H. F., “On pseudo-Hermitian Hamiltonians and their Hermitian counterparts”, J. Phys. A: Math. Gen., 38 (2005), 1741–1746 | DOI | MR | Zbl

[47] Mostafazadeh A., “$\mathcal PT$-symmetric cubic anharmonic oscillator as a physical model Corrigendum”, J. Phys. A: Math. Gen., 38 (2005), 8185–8185 ; J. Phys. A: Math. Gen., 38 (2005), 6557–6570 ; quant-ph/0411137 | DOI | MR | DOI | MR

[48] Bagchi B., Quesne C., Roychoudhury R., “Pseudo-Hermitian versus Hermitian position-dependent-mass Hamiltonians in a perturbative framework”, J. Phys. A: Math. Gen., 39 (2006), L127–L134 ; quant-ph/0511182 | DOI | MR | Zbl