Dimensional Hydrogenlike Orbitals

John Avery H.C. 0rsted Institute University of Copenhagen DK-2100 Copenhagen, Denmark 

The Schrodinger equation for the D-dimensional analogue of hydrogen (equation (88)) can be solved exactly, both in direct space and in reciprocal space and in both cases the solutions involve hyperspheri-cal harmonics. In this section we shall discuss the close relationship between hyperspherical harmonics, harmonic polynomials, and exact D-dimensional hydrogenlike wave functions. We shall also discuss the importance of these functions in dimensional scaling and in the hyperspherical method. 

The familiar 3-dimensional hydrogenhke wave functions in direct space can be expressed as confluent hypergeometric functions multiplied by spherical harmonics. We shall see that the jD-dimensional hydrogenlike wave functions can also be expressed as confluent hypergeometric functions, but in this case they axe multipUed by hyper-spherical harmonics. 

In 1935, V.A. Fock [27,28] solved the Schrodinger equation for hydrogen in momentum space by a remarkable and beautiful method He was able to show that when momentmn space is mapped onto the surface of a 4-dimensional hypersphere by a suitable transformation, the hydrogen wave functions are proportional to 4-dimensional 

Alliluev [29,30] was able to obtain exact D-dimensional hydrogenlike wave functions in momentum space by a generalization of Fock s method. In Alliluev s treatment, Fock s transformation was generalized in such a way as to project D-dimensional momentum space onto the surface of a (DH-l)-dimensional hypersphere [24]. The momentum-space hydrogenlike wave functions could then be shown to be proportional to (D-f-l)-dimensional hyperspherical harmonics. 

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The Fourier transforms of the hydrogenlike orbitals were shown by Fock [18] to be expressible in terms of 4-dimensional hyperspherical harmonics when momentum space is mapped onto the surface of a 4-dimensional unit hypersphere by the transformation ... 

The orbital has two nodal cones. The orbital has two nodal planes. Note that the view shown is not the same for the various orbitals. The relative signs of the wave functions are indicated. The other three real M orbitals in Table 6.2 have the same shape as the 3d -y orbital but have different orientations. The 3djcy orbital has its lobes lying between the x and y axes and is obtained by rotating the 3d -f orbital by 45° about the z axis. The 3dy and 3dx orbitals have their lobes between the y and z axes and between the X and z axes, respectively. (Online three-dimensional views of the real hydrogenlike orbitals are at www.falstad.com/qmatom these can be rotated nsing a mouse.)... 

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