The Physical Interpretation of PT-invariant Potentials

A purely imaginary potential can provide a phenomenological description of creation and absorption of quantum mechanical particles. PT-invariance of such a potential ensures that the non-unitary phenomena occur in a balanced manner. In spite of wells and sinks which locally violate the conservation of quantum probability, there is no net get loss or gain of particles. This, in turn, is intuitively consistent with real energy eigenvalues.Comment: 4 page

Quantum parametric resonance

The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are shown to correspond to Floquet operators with qualitatively different properties. Their eigenfunctions, which are calculated exactly, exhibit a transition: for parameter values with classically stable solutions the eigenstates are normalizable while they cannot be normalized for parameter values with classically unstable solutions. Similarly, the spectrum of quasi energies undergoes a specific transition. These observations remain valid qualitatively for arbitrary linear systems exhibiting classically parametric resonance such as the paradigm example of a frequency modulated pendulum described by Mathieu's equation

Detecting broken PT-symmetry

A fundamental problem in the theory of PT-invariant quantum systems is to determine whether a given system 'respects' this symmetry or not. If not, the system usually develops non-real eigenvalues. It is shown in this contribution how to algorithmically detect the existence of complex eigenvalues for a given PT-symmetric matrix. The procedure uses classical results from stability theory which qualitatively locate the zeros of real polynomials in the complex plane. The interest and value of the present approach lies in the fact that it avoids diagonalization of the Hamiltonian at hand

Landscape of uncertainty in Hilbert space for one-particle states

The functional of uncertainty J[¿] assigns to each state ¿¿> the product of the variances of the momentum and position operators. Its first and second variations are investigated. Each stationary point is located on one of a countable set of three-dimensional manifolds in Hilbert space. For a harmonic oscillator with given mass and frequency the extremals are identified as displaced squeezed energy eigenstates. The neighborhood of the stationary states is found to have the structure of a saddle, thus completing the picture of the landscape of uncertainty in Hilbert space. This result follows from the diagonalization of the second variation of the uncertainty functional, which is not straightforward since J[¿] depends nonlinearly on the state ¿¿>

Quantum Particle on a Rotating Loop: Topological Quenching due to a Coriolis-Aharonov-Bohm Effect

A particle is assumed to move along a one-dimensional loop such as an ellipse that rotates in a plane. Because of the centrifugal force the particle is subjected to a symmetric double-well potential. Classically, the Coriolis force does not affect the motion of the particle, whereas the corresponding term in the Lagrangian influences the properties of the quantum system: its ground state turns out to be degenerate for a discrete set of angular velocities. The analogy between a constant magnetic field and a uniform rotation is used to propose, in addition, a variant of the Aharonov-Bohm experiment, which can be performed also with neutral particles

Completeness and orthonormality in PT-symmetric quantum systems

Some PT-symmetric non-Hermitian Hamiltonians have only real eigenvalues. There is numerical evidence that the associated PT-invariant energy eigenstates satisfy an unconventional completeness relation. An ad hoc scalar product among the states is positive definite only if a recently introduced "charge operator" is included in its definition. A simple derivation of the conjectured completeness and orthonormality relations is given. It exploits the fact that PT symmetry provides a link between the eigenstates of the Hamiltonian and those of its adjoint, forming a dual pair of bases. The charge operator emerges naturally upon expressing the properties of the dual bases in terms of one basis only, and it is shown to be a function of the Hamiltonian

Gauge transformations for a driven quantum particle in an infinite square well

Quantum mechanics of a particle in an infinite square well under the influence of a time-dependent electric field is reconsidered. In some gauge, the Hamiltonian depends linearly on the momentum operator which is symmetric but not self-adjoint when defined on a finite interval. In spite of this symmetric part, the Hamiltonian operator is shown to be self-adjoint. This follows from a theorem by Kato and Rellich which guarantees the stability of a self-adjoint operator under certain symmetric perturbations. The result, which has been assumed tacitly by other authors, is important in order to establish the equivalence of different Hamiltonian operators related to each other by quantum gauge transformations. Implications for the quantization procedure of a particle in a box are pointed out