All Hermitian Hamiltonians Have Parity

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.

References in corpus (7)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• PT-Symmetric Quantum Mechanics
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Supersymmetry and the spontaneous breakdown of PT symmetry
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• On the eigenproblems of PT-symmetric oscillators

Cited by in corpus (23)

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• Must a Hamiltonian be Hermitian?
• Synthetic parity-time symmetry breaking in a single microcavity
• PT symmetry and necessary and sufficient conditions for the reality of energy eigenvalues
• The C Operator in PT-Symmetric Quantum Theories
• Biorthogonal Quantum Systems
• Finite-Dimensional PT-Symmetric Hamiltonians
• PT/Non-PT Symmetric and Non-Hermitian Poschl-Teller-Like Solvable Potentials via Nikiforov-Uvarov Method
• $\mathcal{P}\mathcal{T}$-symmetric, non-Hermitian quantum many-body physics -- a methodological perspective
• PT Symmetry as a Generalization of Hermiticity
• Non-Hermitian quantum systems and their geometric phases
• P,T,PT, and CPT invariance of Hermitian Hamiltonians
• A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models
• C, PT, and CPT invariance of pseudo-Hermitian Hamiltonians
• Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries
• Pseudo-reality and pseudo-adjointness of Hamiltonians
• Exact solution of the Schrodinger equation with the spin-boson Hamiltonian
• PT-Symmetric Quantum Electrodynamics and Unitarity
• Counterdiabatic driving for pseudo- and antipseudo- Hermitian systems
• Stationary states of two-level open quantum systems
• Two dimensional non-Hermitian harmonic oscillator: coherent states
• Monopoles in non-Hermitian systems
• Notes on the Riccati operator equation in open quantum systems