dorsal/arxiv
View SchemaA Convergent Iterative Solution of the Quantum Double-well Potential
| Authors | R. Friedberg, T. D. Lee, W. Q. Zhao, A. Cimenser |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0105142 |
| URL | https://arxiv.org/abs/quant-ph/0105142 |
| DOI | 10.1006/aphy.2001.6187 |
Abstract
We present a new convergent iterative solution for the two lowest quantum wave functions $\psi_{ev}$ and $\psi_{od}$ of the Hamiltonian with a quartic double well potential $V$ in one dimension. By starting from a trial function, which is by itself the exact lowest even or odd eigenstate of a different Hamiltonian with a modified potential $V+\delta V$, we construct the Green's function for the modified potential. The true wave functions, $\psi_{ev}$ or $\psi_{od}$, then satisfies a linear inhomogeneous integral equation, in which the inhomogeneous term is the trial function, and the kernel is the product of the Green's function times the sum of $\delta V$, the potential difference, and the corresponding energy shift. By iterating this equation we obtain successive approximations to the true wave function; furthermore, the approximate energy shift is also adjusted at each iteration so that the approximate wave function is well behaved everywhere. We are able to prove that this iterative procedure converges for both the energy and the wave function at all $x$.
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"abstract": "We present a new convergent iterative solution for the two lowest quantum\nwave functions $\\psi_{ev}$ and $\\psi_{od}$ of the Hamiltonian with a quartic\ndouble well potential $V$ in one dimension. By starting from a trial function,\nwhich is by itself the exact lowest even or odd eigenstate of a different\nHamiltonian with a modified potential $V+\\delta V$, we construct the Green\u0027s\nfunction for the modified potential. The true wave functions, $\\psi_{ev}$ or\n$\\psi_{od}$, then satisfies a linear inhomogeneous integral equation, in which\nthe inhomogeneous term is the trial function, and the kernel is the product of\nthe Green\u0027s function times the sum of $\\delta V$, the potential difference, and\nthe corresponding energy shift. By iterating this equation we obtain successive\napproximations to the true wave function; furthermore, the approximate energy\nshift is also adjusted at each iteration so that the approximate wave function\nis well behaved everywhere. We are able to prove that this iterative procedure\nconverges for both the energy and the wave function at all $x$.",
"arxiv_id": "quant-ph/0105142",
"authors": [
"R. Friedberg",
"T. D. Lee",
"W. Q. Zhao",
"A. Cimenser"
],
"categories": [
"quant-ph"
],
"doi": "10.1006/aphy.2001.6187",
"title": "A Convergent Iterative Solution of the Quantum Double-well Potential",
"url": "https://arxiv.org/abs/quant-ph/0105142"
},
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