Overcritical PT-symmetric square well potential in the Dirac equation (original) (raw)
Scattering off PT -symmetric upside-down potentials
The upside-down −x 4 , −x 6 , and −x 8 potentials with appropriate PT-symmetric boundary conditions have real, positive, and discrete quantum-mechanical spectra. This paper proposes a straightforward macroscopic quantum-mechanical scattering experiment in which one can observe and measure these bound-state energies directly. In 1998 the class of PT-symmetric Hamiltonians H = p 2 + x 2 (ix) ε (1) was introduced and it was shown numerically and per-turbatively that even though these Hamiltonians are not Hermitian, their spectra are real, positive, and discrete if ε > 0 [1, 2]. A rigorous proof was given in Ref. [3]. This spectral positivity is particularly surprising for the case ε = 2 because on the real axis the −x 4 potential is upside-down and therefore appears to be unstable. Nevertheless , for this special value of ε there is an elementary transformation which establishes that the Hamiltonians H = p 2 − x 4 and H = 2p 2 + 4x 4 − 2x are isospectral; that is, the eigenvalues of these two Hamiltonians are identical [4–6]. Since the potential of the latter Hamiltonian is conventionally right-side-up, its bound-state eigenvalues are indeed real, positive, and discrete. The numerical values of the first three bound-state energies are E 0 = 1.477150, E 1 = 6.003386, E 2 = 11.802434. (2)
Various scattering properties of a new PT-symmetric non-Hermitian potential
Annals of Physics, 2013
We complexify a 1-d potential V (x) = V 0 cosh 2 µ{tanh[(x − µd)/d] + tanh(µ)} 2 which exhibits bound, reflecting and free states to study various properties of a non-Hermitian system. This potential turns out a PT-symmetric non-Hermitian potential when one of the parameters (µ, d) becomes imaginary. For the case of µ → iµ, we have entire real bound state spectrum. Explicit scattering states are constructed to show reciprocity at certain discrete values of energy even though the potential is not parity symmetric. Coexistence of deep energy minima of transmissivity with the multiple spectral singularities (MSS) is observed. We further show that this potential becomes invisible from left (or right) at certain discrete energies. The penetrating states in the other case (d → id) are always reciprocal even though it is PT-invariant and no spectral singularity (SS) is present in this case. Presence of MSS and reflectionlessness are also discussed for the free states in the later case.
Quantum Particle in a PT-symmetric Well
2014
In this thesis, we study the role of boundary conditions via-symmetric quantum mechanics. Where denotes parity operator and denotes time reversal operator. We present the boundary conditions so that the-symmetry remains unbroken. We give exact solvable solutions for a free particle in a box. In the first approach, we consider one dimensional Schrödinger Hamiltonian for a free particle in an infinite well. The energy equation is obtained and the results for the Eigenfunctions of the-symmetry are observed completely different form the usual textbooks ones. The second approach is the solution of the Klein Gordon equation in 11 dimensions for the free particle in an infinite well. For both cases, the-symmetric eigenfunctions are normalized and plotted. The asymptotic behavior of the eigenfunction is provided. We consider a variational principle for-symmetric quantum system and examine an invertible linear operator ̂ for a weak-pseudo-hermicity generators for non-Hermitian Hamiltonian.
The Woods–Saxon potential in the Dirac equation
Journal of Physics A: Mathematical and General, 2002
The two-component approach to the one-dimensional Dirac equation is applied to the Woods-Saxon potential. The scattering and bound state solutions are derived and the conditions for a transmission resonance (when the transmission coefficient is unity) and supercriticality (when the particle bound state is at E = −m) are then derived. The square potential limit is discussed. The recent result that a finite-range symmetric potential barrier will have a transmission resonance of zero-momentum when the corresponding well supports a half-bound state at E = −m is demonstrated.
Low momentum scattering in the Dirac equation
Journal of Physics a Mathematical and General, 2002
It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is −1 and hence the transmission coefficient T = 0 in general. If however the potential supports a half-bound state at momentum k = 0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be non-zero whilst for a symmetric potential T = 1. Therefore in some circumstances a Dirac particle of arbitrarily small momentum can tunnel without reflection through a potential barrier.
Dirac particle in a spherical scalar potential well
Physical Review D, 2011
In this paper we investigate a solution of the Dirac equation for a spin-1 2 particle in a scalar potential well with full spherical symmetry. The energy eigenvalues for the quark particle in s 1 2 states (with κ = −1) and p 1 2 states (with κ = 1) are calculated. We also study the continuous Dirac wave function for a quark in such a potential, which is not necessarily infinite. Our results, at infinite limit, are in good agreement with the MIT bag model. We make some remarks about the sharpness value of the wave function on the wall. This model, for finite values of potential, also could serve as an effective model for the nucleus where U (r) is the effective single particle potential.
Chinese Physics Letters, 2009
By using two-component approach to the one-dimensional effective mass Dirac equation bound states are investigated under the effect of two new non-P T-symmetric, and non-Hermitian, exponential type potentials. It is observed that the Dirac equation can be mapped into a Schrödinger-like equation by rescaling one of the two Dirac wave functions in the case of the position dependent mass. The energy levels, and the corresponding Dirac eigenfunctions are found analytically.
Journal of Physics A: Mathematical and Theoretical, 2007
Jia and Dutra (J. Phys. A: Math. Gen. 39 (2006) 11877) have considered the one-dimensional non-Hermitian complexified potentials with real spectra in the context of position-dependent mass in Dirac equation. In their second example, a smooth step shape mass distribution is considered and a non-Hermitian non-P T-symmetric Lorentz vector potential is obtained. They have mapped this problem into an exactly solvable Rosen-Morse Schrödinger model and claimed that the energy spectrum is real. The energy spectrum they have reported is pure imaginary or at best forms an empty set. Their claim on the reality of the energy spectrum is fragile, therefore.
Bound states, scattering states, and resonant states in PT -symmetric open quantum systems
Physical Review A, 2015
We study a simple open quantum system with a PT -symmetric defect potential as a prototype in order to illustrate a number of general features of PT -symmetric open quantum systems; however, the potential itself could be mimicked by a number of PT systems that have been experimentally studied quite recently. One key feature is the resonance in continuum (RIC), which appears in both the discrete spectrum and the scattering spectrum of such systems. The RIC wave function forms a standing wave extending throughout the spatial extent of the system, and in this sense represents a resonance between the open environment associated with the leads of our model and the central PT -symmetric potential. We also illustrate that as one deforms the system parameters, the RIC may exit the continuum by splitting into a bound state and a virtual bound state at the band edge, a process which should be experimentally observable. We also study the exceptional points appearing in the discrete spectrum at which two eigenvalues coalesce; we categorize these as either EP2As, at which two real-valued solutions coalesce before becoming complex-valued, and EP2Bs, for which the two solutions are complex on either side of the exceptional point. The EP2As are associated with PT -symmetry breaking; we argue that these are more stable against parameter perturbation than the EP2Bs. We also study complex-valued solutions of the discrete spectrum for which the wave function is nevertheless spatially localized, something that is not allowed in traditional open quantum systems; we illustrate that these may form quasi-bound states in continuum (QBICs) under some circumstances. We also study the scattering properties of the system, including states that support invisible propagation and some general features of perfect transmission states. We finally use our model as a prototype for the construction of scattering states that satisfy PT -symmetric boundary conditions; while these states do not conserve the traditional probability current, we introduce the PT -current which is preserved. The perfect transmission states appear as a special case of the PT -symmetric scattering states. arXiv:1505.04267v1 [quant-ph] 16 May 2015
States of the Dirac Equation in Confining Potentials
Physical Review Letters, 2008
We study the Dirac equation in confining potentials with pure vector coupling, proving the existence of metastable states with longer and longer lifetimes as the non-relativistic limit is approached and eventually merging with continuity into the Schrödinger bound states. We believe that the existence of these states could be relevant in high energy model construction and in understanding possible resonant scattering effects in systems like Graphene. We present numerical results for the linear and the harmonic cases and we show that the the density of the states of the continuous spectrum is well described by a sum of Breit-Wigner lines. The width of the line with lowest positive energy, as expected, reproduces very well the Schwinger pair production rate for a linear potential: we thus suggest a different way of obtaining informations on the pair production in unbounded, non uniform electric fields, where very little is known.