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The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach. II
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10.1063/1.4757604
/content/aip/journal/jmp/53/10/10.1063/1.4757604
http://aip.metastore.ingenta.com/content/aip/journal/jmp/53/10/10.1063/1.4757604
View: Figures

## Figures

FIG. 1.

Several polynomials with n = 4 and values of L = 0, 1, 2, 3, 4, 5, 6, 8. The polynomial with L = 0 is in dark grey, thick, and increasing L corresponds to lighter grey. The variable r in abscissas is the geodesic length on a sphere with curvature κ = 1, so r = 0 corresponds to the North Pole and r = π corresponds to the South pole. Notice all these polynomials have precisely 4 zeros and all are even around the equator, r = π/2.

FIG. 2.

Several polynomials with n = 5 and values of L = 0, 1, 2, 3, 4, 5, 6, 8. The polynomial with L = 0 is in dark grey, thick. The variable r in abscissas is the geodesic length on a sphere with curvature κ = 1, so r = 0 corresponds to the North Pole and r = π corresponds to the South pole. Notice all these polynomials have precisely 5 zeros and all are odd around the equator, r = π/2.

FIG. 3.

Several polynomials with n = 12 and values of L = 0, 1, 2, 3, 4, 5, 6, 8. Same conventions as in preceding figures. Notice all these polynomials have precisely 12 zeros and all are even around the equator, r = π/2; the amplitude of oscillations at the equatorial band decreases when L grows, and to allow this to be seen clearly the range displayed has been reduced.

FIG. 4.

Several polynomials with n = 13 and values of L = 0, 1, 2, 3, 4, 5, 6, 8. Notice all these polynomials have precisely 13 zeros and all are odd around the equator, r = π/2.

FIG. 5.

Approaching the Euclidean radial function for L = 0 (the spherical Bessel function j 0(kr) for k = 10 with a sequence of the solutions for the sphere, corresponding to the values n = 20, 24, 32, 40 and L = 0. In the limit κ → 0, the upper hemisphere goes to the full Euclidean plane and the sphere equator goes to Euclidean infinity, so here only the left-hand side of the graphics in Figures 1–4 (which corresponds to the upper hemisphere) is pertinent.

FIG. 6.

Same as Figure 5 but for L = 3, approaching the spherical Bessel function j 3(kr).

/content/aip/journal/jmp/53/10/10.1063/1.4757604
2012-10-17
2014-04-18

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