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Cambridge University Press eBooks

Bound States in a Central Potential: Applications

Rocco Schiavilla

Published Dec 5, 2024• 0 citations
Publisher

Abstract

Explicit solution of the hydrogen-like atom and isotropic harmonic oscillator radial equations by the technique of power series expansion; WKB derivation of the hydrogen-like spectrum; virial theorem; the two-dimensional isotropic harmonic oscillator in plane polar coordinates; the two-body problem and the center-of-mass and relative position and momentum operators.

DOI
10.1017/9781009473637.011

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Cambridge University Press eBooks•May 11, 2023•0 citations
Explicit solution of the hydrogen-like atom and isotropic harmonic oscillator radial equations by the technique of power series expansion; WKB derivation of the hydrogen-like spectrum; virial theorem; the two-dimensional isotropic harmonic oscillator in plane polar coordinates; the two-body problem and the center-of-mass and relative position and momentum operators.
1. Wave-Particle Duality and Schroedinger Equation. 2. Introduction to Bound States. 3. Rotationally Invariant Potentials: Hydrogen Atom and Beyond. 4. Wave Packets and Uncertainty Relations. 5. Scattering by Simple Barriers. 6. WKB Approximations. 7. Expectation Values and Operators. 8. Electrons in a Magnetic Field. 9. Beyond Hermitian Operators. 10. Harmonic Oscillator: Full Operator Treatment. 11. Composite Systems. 12. Variational Principle. 13. Expansion Principle and Matrix Formulation. 14. Perturbation Theory, I: Degenerate Perturbation Theory. 15. Perturbation Theory, II: Non-Degenerate Perturbation Theory. 16. Symmetry. 17. Electrons in Periodic Crystal Potentials. 18. Rotational Invariance and Angular Momentum. 19. Time-Dependent Perturbation Theory. 20. Elements of Field Quantization. 21. Electron Spin. 22. Indistinguishable Particles: Fermions and Bosons. Appendices: Dirac ...d-Function. Poisson-Distributed Events. Spherical Harmonics. Hydrogen Radial Eigenfunctions. Fourier Integral. Construction of Two Group Character Tables. Selected General References. Fundamental Constants. Index.
The solution of non-central PT -symmetric potentials is discussed by the separation of the variables in polar and angular coordinates. Conditions are formulated to guarantee the separation of the variables and the PT symmetry of the potential. The original eigenvalue equation is separated into one-dimensional Schrödinger-type differential equations. The importance of the boundary conditions, especially that of the periodic boundary condition of the azimuthal equation is pointed out. Further conditions leading to exact solutions of the whole problem are also formulated. An example combining the harmonic oscillator and the Scarf I potential in the radial and polar equation is discussed in detail, and the bound-state wave functions and the energy eigenvalues are derived. The spectrum exhibits partial degeneracies similar to those observed in the spectrum of the isotropic harmonic oscillator.
The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.