Abstract
In the present chapter, we broaden the quantum mechanical treatment of the problem of a single particle moving in three-dimensional space to incorporate the Hamiltonian. We only treat central potentials \(V (\vec{r}) = V (r)\). In particular, we study the Coulomb and the three-dimensional harmonic oscillator potentials, including Rydberg atoms. We also present the spin–orbit interaction and elements of scattering theory.
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- 1.
This is another application of the separation of variables method for solving partial differential equations.
- 2.
Solutions for the Coulomb potential applying matrix algebra can be found in [39].
- 3.
Criteria which are frequently used to construct interactions involving pure quantum variables are (1) simplicity and (2) invariance under transformations, such as rotations, parity and time-reversal operations. The interaction (6.12) satisfies all these criteria. Moreover, it may also be obtained in the non-relativistic limit of the Dirac equation.
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Bes, D.R. (2012). Three-Dimensional Hamiltonian Problems. In: Quantum Mechanics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20556-9_6
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