Abstract
Going from one to two or three dimensions significantly increases the difficulty of solving Schrödinger’s equation. In general the problem must be solved numerically. Moreover, the numerical solutions may be difficult to obtain. However there are classes of separable potentials for which the solution can be obtained easily.
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This problem in number theory is related to the so-called Ramanujan or “taxi cab” numbers. The famous number theorist Srinivasa Ramanujan is said to have commented on a taxi-cab number, 1729, as a very interesting number, since it is the smallest number expressible as the sum of two cubes in two different ways, 13 + 123 or 93 + 103. Generalized Ramanujan numbers are different integral solutions \(\left \{ n_{1},n_{2}\right \} \) to the equation \( n_{1}^{m}+n_{2}^{m}=n\), for integer n and m.
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Berman, P.R. (2018). Problems in Two and Three-Dimensions: General Considerations. In: Introductory Quantum Mechanics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68598-4_8
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DOI: https://doi.org/10.1007/978-3-319-68598-4_8
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