and
.
Solutions constructed
are single-valued, finite, and continuous functions in spherical space and correspond to discrete energy levels.
Key words: plane wave, non-Euclidean geometry, complex coordinates, Schrödinger equation
2009, Vol.12, No.1, pp.1-15
The goal of the paper is to clarify the status of Shapiro plane
wave solutions of the Schrödinger's equation in the frames of
the well-known general method of separation of variables. To
solve this task, we use the well-known cylindrical coordinates in
Riemann and Lobachevsky spaces, naturally related with Euler
angle-parameters. Conclusion may be drawn: the general method
of separation of variables embraces the all plane wave
solutions; the plane waves in Lobachevsky and Riemann space
consist of a small part of the whole set of basis wave functions
of the Schrödinger equation.
In space of constant positive curvature S3, a complex analog
of horispherical coordinates of Lobachevsky space H3 is
introduced. To parameterize real space S3, two complex
coordinates (r,z) must obey additional restriction in the form
of the equation r2 = ez-z* - e2z. The metrical
tensor of the space S3 is expressed in terms of (r,z) with
additional constraint, or through pairs of conjugate variables
(r,r*) or (z,z*); correspondingly there exist three different
representations for the Schrödinger Hamiltonian.
Shapiro plane waves are determined and explored as solutions of the Schrödinger
equation in complex horisperical coordinates of S3.
In particular, two oppositely directed plane waves may be presented
as exponentials in conjugated coordinates.
and
.
Solutions constructed
are single-valued, finite, and continuous functions in spherical space and correspond to discrete energy levels.
Key words:
plane wave, non-Euclidean geometry, complex
coordinates, Schrödinger equation
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