Within the framework of 𝒫𝒯-symmetric quantum mechanics of bound states (which works with parity-pseudo-Hermitian Hamiltonians H = 𝒫 H †𝒫 and real spectra) we mimic some effects of the double-well structure of potentials by a pair of δ functions with mutually complex conjugate strengths. The model is solvable by the standard matching technique and exhibits several interesting features. We observe an amazingly robust absence of any 𝒫𝒯-symmetry breaking. A quasi-degeneracy which occurs in the high-energy domain is interpreted as a manifestation of certain ‘quantum beats’.
Within the framework of 𝒫𝒯-symmetric quantum mechanics of bound states (which works with parity-pseudo-Hermitian Hamiltonians H = 𝒫 H †𝒫 and real spectra) we mimic some effects of the double-well structure of potentials by a pair of δ functions with mutually complex conjugate strengths. The model is solvable by the standard matching technique and exhibits several interesting features. We observe an amazingly robust absence of any 𝒫𝒯-symmetry breaking. A quasi-degeneracy which occurs in the high-energy domain is interpreted as a manifestation of certain ‘quantum beats’.
We consider the interaction between the Hermitian world, represented by a real delta-function potential $\ensuremath{-}\ensuremath{\alpha}\ensuremath{\delta}(x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $i\ensuremath{\beta}(\ensuremath{\delta}(x\ensuremath{-}L)\ensuremath{-}\ensuremath{\delta}(x+L))$. In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave function is small for L large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be studied instead in the context of quasi-Hermiticity, whereby quantum mechanics is endowed with a new metric $\ensuremath{\eta}$, and consequently a new wave function $\ensuremath{\Psi}(x)$, defined in terms of the original wave function $\ensuremath{\psi}(x)$ by means of $\ensuremath{\eta}$. In this picture, working perturbatively in $\ensuremath{\beta}$, the bound-state wave function is actually unchanged from its unperturbed form for $|x|\ensuremath{\ll}L$. However, the scattering wave function, for $|x|\ensuremath{\gg}L$, is changed in a significant manner. In particular, there are incoming and outgoing waves on both sides of the potential. One can then no longer talk in terms of reflection and transmission coefficients, but the total right-moving flux is now conserved.
The concept of point interactions offers one of the most suitable guides towards a quantitative analysis of properties of certain specific non-Hermitian (so-called -symmetric) quantum-mechanical systems. This is illustrated with a double-well model, the easy solvability of which is shown to lead to a clear picture of the mechanisms of the unavoided level crossing and of the spontaneous -symmetry breaking at a certain strong-non-Hermiticity boundary. Below this limit the model is shown to be suitable for an explicit illustration of technicalities related to the standard probabilistic physical interpretation of bound states in -symmetric quantum mechanics.
Changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions. Here we relax the standard Hermiticity requirement and find an innovative construction which leads to unusual, complex potentials. Their energy spectrum is shown to stay real after a weakening of the Hermiticity of the Schrödinger equation to its mere invariance under the combined (parity) and (time-reversal) symmetry. This ultimately results in richer bound-state spectra. Some of our new exactly solvable potentials generalize the current textbook models. Details are given for constructions based on the hypergeometric and confluent hypergeometric special functions.
This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H‡=H, where ‡ 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=p2+x2(ix)ε of the harmonic oscillator Hamiltonian, where ε is a real parameter. The system exhibits two phases: When ε⩾0, the energy spectrum of H is real and positive as a consequence of 𝒫𝒯 symmetry. However, when −1<ε<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 ε=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=p2+x2N(ix)ε with N integer and ε>−N; each of these complex Hamiltonians exhibits a phase transition at ε=0. These 𝒫𝒯-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
PT Symmetric Hamiltonian Model and Exactly Solvable Potentials
Searching for non-Hermitian (parity-time)PT-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a PT symmetric non-Hermitian Hamiltonian model which is given as H^=ω(b^†b^+21)+α(b^2−(b^†)2) where ω and α are real constants, b^ and b†^ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of PT symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian H is pseudo-Hermitian, we have obtained the Hermitian equivalent of H which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. H is called pseudo-Hermitian, if there exists a Hermitian and invertible operator η satisfying H†=ηHη−1. For the Hermitian Hamiltonian h, one can write h=ρHρ−1 where ρ=η is unitary. Using this ρ we have obtained a physical Hamiltonian h for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function ρ is obtained for each potential case.