Periodic effective potentials for spin systems and new exact solutions of the one-dimensional Schroedinger equation for the energy bands

Journal Article · Sun Nov 01 00:00:00 EST 1987 · Theor. Math. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01028652· OSTI ID:5234221

Zaslavskii, O B; Ul'yanov, V V

The technique developed in an earlier paper by the authors is used in conjunction with a representation of generalized coherent states to find new effective periodic potential fields that rigorously describe stationary states of (pseudo) spin systems of the type of a two-axis paramagnet in a magnetic field. The potentials depend strongly on several parameters, their profiles are rich in distinctive features of the type of double wells, two-hump barriers, fourfold minima and maxima, and in the bands interesting structural transformation take place (finite-gap property, band pairing, etc.). It is shown that the spin system corresponds to periodic and antiperiodic solutions with extremal energy levels in the 2S + 1 lowest bands (S is the spin). On the basis of the established spin-coordinate correspondence, new classes of exact solutions of the Schroedinger equation are found for energy bands with simple explicit expressions for the energy levels and wave functions for S = 0, 1/2, 1, 5/2, 3, 7/2, 4, 9/2, 5. The potentials are expressed in terms of elliptic functions and contain as special cases the finite-gap Lame-Ince potential and the Eckart and Morse potentials. Effective potentials are also constructed for Hamiltonians of the group SU(1,1).

• Research Organization: Khar'kov State Univ. (USSR)
• OSTI ID: 5234221
• Journal Information: Theor. Math. Phys.; (United States), Vol. 71:2; Other Information: Translated from Teor. Mat. Fiz.; 71: No. 2, 260-271(May 1987)
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CRYSTAL MODELS
PARAMAGNETISM
POTENTIALS
SCHROEDINGER EQUATION
SPIN
BAND THEORY
ENERGY GAP
ENERGY LEVELS
HAMILTONIANS
MAGNETIC FIELDS
ONE-DIMENSIONAL CALCULATIONS
QUANTUM MECHANICS
RENORMALIZATION
SU GROUPS
WAVE FUNCTIONS
ANGULAR MOMENTUM
DIFFERENTIAL EQUATIONS
EQUATIONS
FUNCTIONS
LIE GROUPS
MAGNETISM
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
PARTICLE PROPERTIES
QUANTUM OPERATORS
SYMMETRY GROUPS
WAVE EQUATIONS
657002* - Theoretical & Mathematical Physics- Classical & Quantum Mechanics