PT-symmetric quantum mechanics
- Department of Physics, Washington University, St. Louis, Missouri 63130 (United States)
- Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Maxico 87545 (United States)
This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H{sup {dagger}}=H on the Hamiltonian, where {dagger} represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian {ital H} has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H{sup {double_dagger}}=H, where {double_dagger} represents combined parity reflection and time reversal PT, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H=p{sup 2}+x{sup 2}(ix){sup {epsilon}} of the harmonic oscillator Hamiltonian, where {epsilon} is a real parameter. The system exhibits two phases: When {epsilon}{ge}0, the energy spectrum of {ital H} is real and positive as a consequence of PT symmetry. However, when {minus}1{lt}{epsilon}{lt}0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken. The phase transition that occurs at {epsilon}=0 manifests itself in both the quantum-mechanical system and the underlying classical system. Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H=p{sup 2}+x{sup 2N}(ix){sup {epsilon}} with {ital N} integer and {epsilon}{gt}{minus}N; each of these complex Hamiltonians exhibits a phase transition at {epsilon}=0. These PT-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. {copyright} {ital 1999 American Institute of Physics.}
- OSTI ID:
- 337511
- Journal Information:
- Journal of Mathematical Physics, Vol. 40, Issue 5; Other Information: PBD: May 1999
- Country of Publication:
- United States
- Language:
- English
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