dorsal/arxiv
View SchemaMetric Operator in Pseudo-Hermitian Quantum Mechanics and the Imaginary Cubic Potential
| Authors | Ali Mostafazadeh |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0508195 |
| URL | https://arxiv.org/abs/quant-ph/0508195 |
| DOI | 10.1088/0305-4470/39/32/S18 |
| Journal | J.Phys. A39 (2006) 10171-10188 |
Abstract
We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, H= p^2/2 + v(x), in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1+1)-dimensions. For the case that v(x) is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v(x)=i \epsilon x^3. This reveals an infinite class of previously unknown CPT- as well as non-CPT-inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in real line that is consistent with the usual choice for the quantum Hamiltonian. Finally we discuss subtleties involved with higher dimensions and systems having a fixed length scale.
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"abstract": "We present a systematic perturbative construction of the most general metric\noperator (and positive-definite inner product) for quasi-Hermitian Hamiltonians\nof the standard form, H= p^2/2 + v(x), in one dimension. We show that this\nproblem is equivalent to solving an infinite system of iteratively decoupled\nhyperbolic partial differential equations in (1+1)-dimensions. For the case\nthat v(x) is purely imaginary, the latter have the form of a nonhomogeneous\nwave equation which admits an exact solution. We apply our general method to\nobtain the most general metric operator for the imaginary cubic potential,\nv(x)=i \\epsilon x^3. This reveals an infinite class of previously unknown CPT-\nas well as non-CPT-inner products. We compute the physical observables of the\ncorresponding unitary quantum system and determine the underlying classical\nsystem. Our results for the imaginary cubic potential show that, unlike the\nquantum system, the corresponding classical system is not sensitive to the\nchoice of the metric operator. As another application of our method we give a\ncomplete characterization of the pseudo-Hermitian canonical quantization of a\nfree particle moving in real line that is consistent with the usual choice for\nthe quantum Hamiltonian. Finally we discuss subtleties involved with higher\ndimensions and systems having a fixed length scale.",
"arxiv_id": "quant-ph/0508195",
"authors": [
"Ali Mostafazadeh"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
"math.MP"
],
"doi": "10.1088/0305-4470/39/32/S18",
"journal_ref": "J.Phys. A39 (2006) 10171-10188",
"title": "Metric Operator in Pseudo-Hermitian Quantum Mechanics and the Imaginary Cubic Potential",
"url": "https://arxiv.org/abs/quant-ph/0508195"
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