dorsal/arxiv
View SchemaSum rules and the domain after the last node of an eigenstate
| Authors | C. V. Sukumar |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0610207 |
| URL | https://arxiv.org/abs/quant-ph/0610207 |
| DOI | 10.1088/0305-4470/39/45/023 |
| Journal | J.Phys. A: Math. Gen. 39 (2006) 14153-14163 |
Abstract
It is shown that it is possible to establish sum rules that must be satisfied at the nodes and extrema of the eigenstates of confining potentials which are functions of a single variable. At any boundstate energy the Schroedinger equation has two linearly independent solutions one of which is normalisable while the other is not. In the domain after the last node of a boundstate eigenfunction the unnormalisable linearly independent solution has a simple form which enables the construction of functions analogous to Green's functions that lead to certain sum rules. One set of sum rules give conditions that must be satisfied at the nodes and extrema of the boundstate eigenfunctions of confining potentials. Another sum rule establishes a relation between an integral involving an eigenfunction in the domain after the last node and a sum involving all the eigenvalues and eigenstates. Such sum rules may be useful in the study of properties of confining potentials. The exactly solvable cases of the particle in a box and the simple harmonic oscillator are used to illustrate the procedure. The relations between one of the sum rules and two-particle densities and a construction based on Supersymmetric Quantum Mechanics are discussed.
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"abstract": "It is shown that it is possible to establish sum rules that must be satisfied\nat the nodes and extrema of the eigenstates of confining potentials which are\nfunctions of a single variable. At any boundstate energy the Schroedinger\nequation has two linearly independent solutions one of which is normalisable\nwhile the other is not. In the domain after the last node of a boundstate\neigenfunction the unnormalisable linearly independent solution has a simple\nform which enables the construction of functions analogous to Green\u0027s functions\nthat lead to certain sum rules. One set of sum rules give conditions that must\nbe satisfied at the nodes and extrema of the boundstate eigenfunctions of\nconfining potentials. Another sum rule establishes a relation between an\nintegral involving an eigenfunction in the domain after the last node and a sum\ninvolving all the eigenvalues and eigenstates. Such sum rules may be useful in\nthe study of properties of confining potentials. The exactly solvable cases of\nthe particle in a box and the simple harmonic oscillator are used to illustrate\nthe procedure. The relations between one of the sum rules and two-particle\ndensities and a construction based on Supersymmetric Quantum Mechanics are\ndiscussed.",
"arxiv_id": "quant-ph/0610207",
"authors": [
"C. V. Sukumar"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/39/45/023",
"journal_ref": "J.Phys. A: Math. Gen. 39 (2006) 14153-14163",
"title": "Sum rules and the domain after the last node of an eigenstate",
"url": "https://arxiv.org/abs/quant-ph/0610207"
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