In this report we examine the concept of PT -symmetric quantum mechanics and analyze two such systems. PT -symmetric quantum mechanics is a new branch of quantum mechanics, first proposed by Bender and Boettcher in 1998 and has been the subject of much research since then. Various important aspects of PT -symmetric quantum mechanics is explained, such as asymptotic behaviour of solutions of the Schrödinger equation, as well as the concept of the C operator that is unique to PT -symmetric quantum mechanics. We briefly discuss both the concept of broken and unbroken PT -symmetry, and the relation between PT -symmetric and Hermitian quantum mechanics through similarity transforms. We specifically direct our attention to two theoretical quantum mechanical systems described by Hamiltonians with the property of being PT -symmetric. For these systems we determine the energy levels, one by analytical means and the other by use of numerical methods. To determine the energy levels numerically we have used the builtin ODE-solver ode45 in MATLAB in conjunction with the equation solver fzero. Our results are compared to those of previous studies and our methods and results are discussed.
In this report we examine the concept of PT -symmetric quantum mechanics and analyze two such systems. PT -symmetric quantum mechanics is a new branch of quantum mechanics, first proposed by Bender and Boettcher in 1998 and has been the subject of much research since then. Various important aspects of PT -symmetric quantum mechanics is explained, such as asymptotic behaviour of solutions of the Schrödinger equation, as well as the concept of the C operator that is unique to PT -symmetric quantum mechanics. We briefly discuss both the concept of broken and unbroken PT -symmetry, and the relation between PT -symmetric and Hermitian quantum mechanics through similarity transforms. We specifically direct our attention to two theoretical quantum mechanical systems described by Hamiltonians with the property of being PT -symmetric. For these systems we determine the energy levels, one by analytical means and the other by use of numerical methods. To determine the energy levels numerically we have used the builtin ODE-solver ode45 in MATLAB in conjunction with the equation solver fzero. Our results are compared to those of previous studies and our methods and results are discussed.
PT-symmetric quantum mechanics is an alternative to the usual hermitian quantum mechanics. We will start this thesis by taking an overview of the subject, seeing some of the elementary consequences of this different approach. The main part of the work will be an depth study of a specific Hamiltonian(diagram) This is a generalisation of the well understood harmonic oscillator with angular momentum. By making this generalisation we break the hermiticity of the problem. This leads to some intriguing results. We will be particularly interested in the merging of eigenvalues for M <1.We study the problem using a number of techniques. First the Hamiltonian is studied at the classical level and the behaviour of a particle moving in the corresponding potential is studied. Having seen the consequences at the classical level we return to the quantum case. The Hamiltonian is first solved perturbativly. This method is shown to be valid for PT-symmetric quantum mechanics. It is shown that asymptotic limits of the matrix do not capture the full behaviour of the energy levels. We then move on to study the problem considering techniques arising from the ODE/IM correspondence. Using this approach we are able to give an analytic description of the phenomena and explain the merging of eigenvalues.
It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.
In the seminal paper (Bender & Boettcher, 1998) a new view of quantum mechanics was proposed. This new view differs from the old one in that the restriction on the Hamiltonian to be Hermitian is relaxed: now the Hamiltonian is PT -symmetric. Here P is parity and T is time reversal. Since 1998, PT -symmetric Hamiltonians have been analyzed intensively by many authors. In Mostafazadeh (2002) PT -symmetry was embedded into the more general mathematical framework of pseudo-Hermiticity or, what is the same, self-adjoint operators in Kre˘ın spaces, see (Langer & Tretter, 2004; Azizov & Trunk, 2012; Hassi & Kuzhel, 2013; Leben & Trunk, 2019). For a general introduction to PT -symmetric quantum mechanics we refer to the overview paper of Mostafazadeh (2010) and to the books of Moiseyev (2011) and Bender (2019).
Space-time reflection symmetry, or PT symmetry, first proposed in quantum mechanics by Bender and Boettcher in 1998 [2], has become an active research area in fundamental physics. This dissertation contains several research problems which are more or less related to this field of study. After an introduction on complementary topics for the main projects in Chap.1, we discuss about an idea which is originated from the remarkable paper by Chandrasekar et al in Chap.2. They showed that the (second-order constant-coefficient) classical equation of motion for a damped harmonic oscillator can be derived from a Hamiltonian having one degree of freedom. We gives a simple derivation of their result and generalizes it to the case of an nth-order constant-coefficient differential equation. In Chap.3 we studied the analytical continuation of the coupling constant g of a coupled quantum theory. We get to this conclusion that one can, at least in principle, arrive at a state whose energy is lower than the ground state of the theory. The idea is to begin with the uncoupled g = 0 theory in its ground state, to analytically continue around an exceptional point (square-root singularity) in the complex-coupling-constant plane, and finally to return to the point g = 0. In the course of this analytic continuation, the uncoupled theory ends up in an unconventional state whose energy is lower than the original ground-state energy. However, it is unclear whether one can use this analytic continuation to extract energy from the conventional vacuum state; this process appears to be exothermic but one must do work to vary the coupling constant g. PT-symmetric quantum mechanics began with a study of the Hamiltonian H=p^2+x^2(ix)^\epsilon. When \epsilon\geq 0, this portion of parameter space is known as the region of unbroken PT symmetry. The region of unbroken PT symmetry has been studied but the region of broken PT symmetry which is related to the negative epsilon has thus far been unexplored. In Chap.4 we present a detailed numerical and analytical examination of the behavior of the eigenvalues for _4<\epsilon<0. In particular, it reports the discovery of an infinite-order exceptional point at \epsilon =_1, a transition from a discrete spectrum to a partially continuous spectrum at \epsilon=_2, a transition at the Coulomb value \epsilon =_3, and the behavior of the eigenvalues as \epsilon approaches the conformal limit \epsilon =_4. Finally in Chap.5 we devised a simple and accurate numerical technique for finding eigenvalues, node structure, and expectation values of PT-symmetric potentials. The approach involves expanding the solution to the Schrdinger equation in series involving powers of both the coordinate and the energy. The technique is designed to allow one to impose boundary conditions in PT-symmetric pairs of Stokes sectors. The method is illustrated by using many examples of PT-symmetric potentials in both the unbroken- and broken-PT-symmetric regions.
Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C d , it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.
Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.