dorsal/arxiv
View SchemaAll Hermitian Hamiltonians Have Parity
| Authors | Carl M. Bender, Peter N. Meisinger, Qinghai Wang |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0211123 |
| URL | https://arxiv.org/abs/quant-ph/0211123 |
| DOI | 10.1088/0305-4470/36/4/312 |
Abstract
It is shown that if a Hamiltonian $H$ is Hermitian, then there always exists an operator P having the following properties: (i) P is linear and Hermitian; (ii) P commutes with H; (iii) P^2=1; (iv) the nth eigenstate of H is also an eigenstate of P with eigenvalue (-1)^n. Given these properties, it is appropriate to refer to P as the parity operator and to say that H has parity symmetry, even though P may not refer to spatial reflection. Thus, if the Hamiltonian has the form H=p^2+V(x), where V(x) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has PT symmetry. This shows that PT symmetry is a generalization of Hermiticity: All Hermitian Hamiltonians of the form H=p^2+V(x) have PT symmetry, but not all PT-symmetric Hamiltonians of this form are Hermitian.
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"abstract": "It is shown that if a Hamiltonian $H$ is Hermitian, then there always exists\nan operator P having the following properties: (i) P is linear and Hermitian;\n(ii) P commutes with H; (iii) P^2=1; (iv) the nth eigenstate of H is also an\neigenstate of P with eigenvalue (-1)^n. Given these properties, it is\nappropriate to refer to P as the parity operator and to say that H has parity\nsymmetry, even though P may not refer to spatial reflection. Thus, if the\nHamiltonian has the form H=p^2+V(x), where V(x) is real (so that H possesses\ntime-reversal symmetry), then it immediately follows that H has PT symmetry.\nThis shows that PT symmetry is a generalization of Hermiticity: All Hermitian\nHamiltonians of the form H=p^2+V(x) have PT symmetry, but not all PT-symmetric\nHamiltonians of this form are Hermitian.",
"arxiv_id": "quant-ph/0211123",
"authors": [
"Carl M. Bender",
"Peter N. Meisinger",
"Qinghai Wang"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/36/4/312",
"title": "All Hermitian Hamiltonians Have Parity",
"url": "https://arxiv.org/abs/quant-ph/0211123"
},
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