The problem of d-dimensional Schrodinger equations with a position-dependent mass is analyzed in the framework of first-order intertwining operators. With the pair (H, H_1) of intertwined Hamiltonians one can associate another pair of second-order partial differential operators (R, R_1), related to the same intertwining operator and such that H (resp. H_1) commutes with R (resp. R_1). This property is interpreted in superalgebraic terms in the context of supersymmetric quantum mechanics (SUSYQM). In the two-dimensional case, a solution to the resulting system of partial differential equations is obtained and used to build a physically-relevant model depicting a particle moving in a semi-infinite layer. Such a model is solved by employing either the commutativity of H with some second-order partial differential operator L and the resulting separability of the Schrodinger equation or that of H and R together with SUSYQM and shape-invariance techniques. The relation between both approaches is also studied.
The Darboux transformation operator technique in differential and integral forms is applied to the generalized Schrodinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. Intertwining operators are obtained in an explicit form and used for constructing generalized Darboux transformations of an arbitrary order. A relation between supersymmetry and the generalized Darboux transformation is considered. The method is applied to generation of isospectral potentials with additional or removal bound states or construction of new partner potentials with…
General first- and higher-order intertwining relations between non-stationary one-dimensional Schrödinger operators are introduced. For the first-order case it is shown that the intertwining relations imply some hidden symmetry which in turn results in a $R$-separation of variables. The Fokker-Planck and diffusion equation are briefly considered. Second-order intertwining operators are also discussed within a general approach. However, due to its complicated structure only particular solutions are given in some detail.
The general solution of SUSY intertwining relations of first order for two-dimensional Schrödinger operators with position-dependent (effective) mass is built in terms of four arbitrary functions. The procedure of separation of variables for the constructed potentials is demonstrated in general form. The generalization for intertwining of second order is also considered. The general solution for a particular form of intertwining operator is found, its properties - symmetry, irreducibility, separation of variables - are investigated.
A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of $η$-weak-pseudo-Hermitian position-dependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target $η$-weak-pseudo-Hermitian PDM -- Hamiltonians' map is suggested. Some $η$-weak-pseudo-Hermitian PT -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-levels crossing and flown-away states phenomena are reported for the resulting Scarf II spectrum. Some of the corresponding $η$-weak-pseudo-Hermitian Scarf II- and periodic-type-isospectral models…
The behaviour of a Dirac electron in graphene, under magnetic fields which are orthogonal to the layer, is studied. The initial problem is reduced to an equivalent one, where two one-dimensional Schrödinger Hamiltonians $H^{\pm}$ are intertwined by a first order differential operator. Special magnetic field are initially chosen, in order that $V^{\pm}$ will be shape invariant exactly solvable potentials. When looking for more general first order operators, intertwining $H^-$ with a non-necessarily shape invariant Hamiltonian, new magnetic fields associated also to analytic solutions will be ge…
Here we have studied first and second-order intertwining approach to generate isospectral partner potentials of position-dependent (effective) mass Schroedinger equation. The second-order intertwiner is constructed directly by taking it as second order linear differential operator with position depndent coefficients and the system of equations arising from the intertwining relationship is solved for the coefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained which is valid for any arbitrary potential and mass function. The proposed technique allows us to …
The three-dimensional Schrödinger equation with a position-dependent (effective) mass is studied in the framework of Supersymmetrical (SUSY) Quantum Mechanics. The general solution of SUSY intertwining relations with first order supercharges is obtained without any preliminary constraints. Several forms of coefficient functions of the supercharges are investigated and analytical expressions for the mass function and partner potentials are found. As usual for SUSY Quantum Mechanics with nonsingular superpotentials, the spectra of intertwined Hamiltonians coincide up to zero modes of supercharge…
The intertwining technique has been widely used to study the Schrödinger equation and to generate new Hamiltonians with known spectra. This technique can be adapted to find the bound states of certain Dirac Hamiltonians. In this paper the system to be solved is a relativistic particle placed in a magnetic field with cylindrical symmetry whose intensity decreases as the distance to the symmetry axis grows and its field lines are parallel to the $x-y$ plane. It will be shown that the Hamiltonian under study turns out to be shape invariant.
The general solution of the intertwining relations between a pair of Schrödinger Hamiltonians by the supercharges of third order in derivatives is obtained. The solution is expressed in terms of one arbitrary function. Some properties of the spectrum of the Hamiltonian are derived, and wave functions for three energy levels are constructed. This construction can be interpreted as addition of three new levels to the spectrum of partner potential: a ground state and a pair of levels between successive excited states. Possible types of factorization of the third order supercharges are analysed, t…
The general solution of SUSY intertwining relations for three-dimensional Schrödinger operators is built using the class of second order supercharges with nondegenerate constant metric. This solution includes several models with arbitrary parameters. We are interested only in quantum systems which are not amenable to separation of variables, i.e. can not be reduced to lower dimensional problems. All constructed Hamiltonians are partially integrable - each of them commutes with a symmetry operator of fourth order in momenta. The same models can be considered also for complex values of parameter…
Using an isomorphism between Hilbert spaces $L^2$ and $\ell^{2}$ we consider Hamiltonians which have tridiagonal matrix representations (Jacobi matrices) in a discrete basis and an eigenvalue problem is reduced to solving a three term difference equation. Technique of intertwining operators is applied to creating new families of exactly solvable Jacobi matrices. It is shown that any thus obtained Jacobi matrix gives rise to a new exactly solvable non-local potential of the Schroedinger equation. We also show that the algebraic structure underlying our approach corresponds to supersymmetry. Sup…
Exact solutions of Schrodinger equation are obtained for the modified Kratzer and the corrected Morse potentials with the position-dependent effective mass. The bound state energy eigenvalues and the corresponding eigenfunctions are calculated for any angular momentum for target potentials. Various forms of point canonical transformations are applied. PACS numbers: 03.65.-w; 03.65.Ge; 12.39.Fd Keywords: Morse potential, Kratzer potential, Position-dependent mass, Point canonical transformation, Effective mass Schrödinger equation.
The analytical solutions of the N-dimensional Schrodinger equation with position-dependent mass for a general class of central potentials is obtained via the series expansion method. The position-dependent mass is expanded in series about origin. As a special case, the analytical bound-state series solutions and the recursion relation of the linear-plus-Coulomb (Cornell) potential with the decaying position-dependent mass m=m_{0}e^{-λr} are also found.
A recursion technique of obtaining the asymptotical expansions for the bound-state energy eigenvalues of the radial Schrödinger equation with a position-dependent mass is presented. As an example of the application we calculate the energy eigenvalues for the Coulomb potential in the presence of position-dependent mass and we derive the inequalities regulating the shifts of the energy levels from their constant-mass positions.
An important and well established area of quantum optics is the theory of Markovian stochastic Schrödinger equations (or by another name quantum trajectory theory). Recently stochastic Schrödinger equations have been developed for non-Markovian systems. In this paper we extend the current known stochastic Schrödinger equations for non-Markovian systems to include the position unraveling. We also discuss and illustrate that this stochastic Schrödinger equation can have an interpretation under both the orthodox and the de Broglie-Bohm hidden variable interpretation of quantum mechanics. We concl…
Exact solutions of Schrodinger equation for PT-/non-PT-symmetric and non-Hermitian Morse and Poschl-Teller potentials are obtained with the position-dependent effective mass by applying a point canonical transformation method. Three kinds of mass distributions are used in order to construct exactly solvable target potentials and obtain energy spectrum and corresponding wave functions.
In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schrödinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schrödinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples in which the coherent state…
PT-symmetric solutions of Schrodinger equation are obtained for the Scarf and generalized harmonic oscillator potentials with the position-dependent mass. A general point canonical transformation is applied by using a free parameter. Three different forms of mass distributions are used. A set of the energy eigenvalues of the bound states and corresponding wave functions for target potentials are obtained as a function of the free parameter.
Exact solutions of the Schrodinger equation are obtained for the Rosen-Morse and Scarf potentials with the position-dependent effective mass by appliying a general point canonical transformation. The general form of the point canonical transformation is introduced by using a free parameter. Two different forms of mass distributions are used. A set of the energy eigenvalues of the bound states and corresponding wave functions for target potentials are obtained as a function of the free parameter.
Macroscopic objects appear to have definite positions. In a many-worlds interpretation of quantum theory, this appearance is an illusion; the correct view is the "view from outside" in which even macroscopic objects are in general in a superposition of different positions. In the Bohm model, objects really are in definite positions. This additional aspect of reality is accessible only from the "inside"; thus in the Bohm model the view from inside can be more correct than is the view from outside.
We consider non-relativistic position-dependent mass (PDM) Schrodinger particles moving in an elastic medium with space-like screw dislocation. Within the cylindrical coordinates, we study and report the effects of screw dislocation as well as PDM settings on the energy levels of some PDM-quantum mechanical systems. In so doing, we use a power-law type positive-valued dimensionless scalar multiplier f(r) (Which manifestly introduces the metaphoric notion of PDM-Schrodinger particles). Next, we subject such PDM particles to magnetic and Aharonov-Bohm flux fields. We report exact or conditionall…
Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass(PDM). The explicit expressions for the potentials, energy eigenvalues and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hemite and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term $V_m$ which produced from the dependence of mass on the coordinate, compared with that for the systems of constant mass. The properties of $V_m$ for several mass functions are discussed.
Recent studies have shown that the use of Dunkl derivatives instead of ordinary derivatives leads to deriving parity-dependent dynamic solutions. According to this motivation in this manuscript, we formulate the Dunkl-Schrödinger equation within the position-dependent mass formalism and derive an algebraic solution out of it. Our systematic approach lets us observe some new findings in addition to the earlier ones. For example, we find that the solution of the Dunkl-Schröodinger equation with position-dependent mass cannot be considered independent of the choice of parameters. Similarly, throu…
We study the generalized harmonic oscillator which has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and the its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.
With the consideration of spherical symmetry for the potential and mass function, one-dimensional solutions of non-relativistic Schrodinger equations with spatially varying effective mass are successfully extended to arbitrary dimensions within the frame of recently developed elegant non-perturbative technique, where the BenDaniel-Duke effective Hamiltonian in one-dimension is assumed like the unperturbed piece, leading to well-known solutions, whereas the modification term due to possible use of other effective Hamiltonians in one-dimension and, together with, the corrections coming from the …
We deal with the exact solutions of Schrodinger equation characterized by position-dependent effective mass via point canonical transformations. The Morse, Poschl-Teller and Hulthen type potentials are considered respectively. With the choice of position-dependent mass forms, exactly solvable target potentials are constructed. Their energy of the bound states and corresponding wavefunctions are determined exactly.
Ordering ambiguity associated with the von Roos position dependent mass (PDM) Hamiltonian is considered. An affine locally scaled first order differential introduced, in Eq.(9), as a PDM-pseudo-momentum operator. Upon intertwining our Hamiltonian, which is the sum of the square of this operator and the potential function, with the von Roos d-dimensional PDM-Hamiltonian, we observed that the so-called von Roos ambiguity parameters are strictly determined, but not necessarily unique. Our new ambiguity parameters' setting is subjected to Dutra's and Almeida's [11] reliability test and classified …
The effect of the built-in supersymmetric quantum mechanical language on the spectrum of the (1+1)-Dirac equation, with position-dependent mass (PDM) and complexified Lorentz scalar interactions, is re-emphasized. The signature of the "quasi-parity" on the Dirac particles' spectra is also studied. A Dirac particle with PDM and complexified scalar interactions of the form S(z)=S(x-ib) (an inversely linear plus linear, leading to a PT-symmetric oscillator model), and S(x)=S_{r}(x)+iS_{i}(x) (a PT-symmetric Scarf II model) are considered. Moreover, a first-order intertwining differential operator…
We derive the stochastic Schrodinger equation for the limit of continuous weak measurement where the observables monitored are canonical position and momentum. To this end we extend an argument due to Smolianov and Truman from the von Neumann model of indirect measurement of position to the Arthurs and Kelly model for simultaneous measurement of position and momentum. We only require unbiasedness of the detector states and an integrability condition sufficient to ensure a central limit effect. Despite taking a weak interaction, as opposed to weak measurement limit, the resulting stochastic wav…
It is shown the role of a scalar potential in the Schrödinger equation for a steady-state two-particle system is equivalent to an isometric entanglement of the position coordinates of the particles in space and time. The entangled coordinates of each particle are complex quantities related through the entangling transformation to the real positions of both particles. The transformation takes into account all of the states in the Hilbert space of the composite system. Transforming the Schrödinger equation into these entangled coordinates eliminates the scalar potential.
We show that the exact discrete analogue of Schrödinger equation can be derived naturally from the Hamiltonian operator of a Schrödinger field theory by using the discrete Fourier transform that transforms the operator from momentum representation into position representation. The standard central difference equation that is often used as the discretized Schrödinger equation actually describes a different theory since it is derived from a different Hamiltonian operator. The commutator relation between the position and momentum operators in discrete space is also derived and found to be differe…
The scattering solutions of the one-dimensional Schrodinger equation for the Woods-Saxon potential are obtained within the position-dependent mass formalism. The wave functions, transmission and reflection coefficients are calculated in terms of Heun's function. These results are also studied for the constant mass case in detail.
A new method for generating exactly solvable Schrödinger equations with a position-dependent mass is proposed. It is based on a relation with some deformed Schrödinger equations, which can be dealt with by using a supersymmetric quantum mechanical approach combined with a deformed shape-invariance condition. The solvability of the latter is shown to impose the form of both the deformed superpotential and the position-dependent mass. The conditions for the existence of bound states are determined. A lot of examples are provided and the corresponding bound-state spectrum and wavefunctions are re…
Using a recently developed technique to solve Schrödinger equation for constant mass, we studied the regime in which mass varies with position i.e position dependent mass Schrödinger equation(PDMSE). We obtained an analytical solution for the PDMSE and applied our approach to study a position dependent mass $m(x)$ particle scattered by a potential $\mathcal{V}(x)$. We also studied the structural analogy between PDMSE and two-level atomic system interacting with a classical field.
We show in this paper that the basic representations of position and momentum in a quantum mechanical system, that are guided by a generalized uncertainty principle and lead to a corresponding one-parameter eigenvalue problem, can be interpreted in terms of an extended Schrodinger equation embodying momentum-dependent mass. Some simple consequences are pointed out.
An approximate method is proposed to solve position dependent mass Schrödinger equation. The procedure suggested here leads to the solution of the PDM Schrödinger equation without transforming the potential function to the mass space or vice verse. The method based on asymptotic Taylor expansion of the function, produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schrödinger equation. The results show that PDM and constant mass Schrödinger equations are not isospectral. The calculations are carried out with the aid of a computer syst…
The study of the Schrödinger equation with the position-dependent effective mass has attracted a lot of attention, due to its applications in many fields of physics, including the properties of the semiconductors, semiconductor heterostructures, graded alloys, quantum liquids, Helium-3 clusters, quantum wells, wires and dots etc. In the present work we obtain several classes of solutions of the one-dimensional Schrödinger equation with position-dependent particle mass. As a first step the single particle Schrödinger equation with position-dependent mass is transformed into an equivalent Riccat…
In this paper, we study the time-independent Schrödinger equation within the formalism of position dependent effective mass. For a generalized decomposition of the non-central effective potential, the deformed Schrödinger equation can be easily solved analytically through separation of variables. The energy eigenvalues and the normalization constant of the radial wave functions are obtained, as well as the scattering phase shifts.
By using the point canonical transformation approach in a manner distinct from previous ones, we generate some new exactly solvable or quasi-exactly solvable potentials for the one-dimensional Schrödinger equation with a position-dependent effective mass. In the latter case, SUSYQM techniques provide us with some additional new potentials.
An exact uncertainty principle, formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrodinger equation. Thus there is an exact formulation of the uncertainty principle which precisely captures the essence of what is "quantum" about quantum mechanics.
We employ the point canonical transformation (PCT) to solve the D-dimensional Schrödinger equation with position-dependent effective mass (PDEM) function for two molecular pseudoharmonic and modified Kratzer (Mie-type) potentials. In mapping the transformed exactly solvable D-dimensional ($D\geq 2$) Schrödinger equation with constant mass into the effective mass equation by employing a proper transformation, the exact bound state solutions including the energy eigenvalues and corresponding wave functions are derived. The well-known pseudoharmonic and modified Kratzer exact eigenstates of vario…
A one-dimensional Schrödinger equation with position-dependent effective mass in the kinetic energy operator is studied in the framework of an $so(2,1)$ algebra. New mass-deformed versions of Scarf II, Morse and generalized Pöschl-Teller potentials are obtained. Consistency with an intertwining condition is pointed out.
We have obtained a set of coupled differential equations from the continuous limit of the transfer matrix method. Decoupling such a set of equations yields an extension to the Wentzel-Kramers-Brillouin (WKB) approximation for the Schrödinger equation with the position-dependent effective mass (PDEM). In the classically allowed region, the decoupling is to ignore the reflection resulting from the variations of both the potential and effective mass. By considering an infinite-well example with the PDEM, it is shown that the extended WKB approximation can provide not only an estimation to eigenen…
The effective mass one-dimensional Schrödinger equation for the generalized Morse potential is solved by using Nikiforov-Uvarov method. Energy eigenvalues and corresponding eigenfunctions are computed analytically. The results are also reduced to the case of constant mass. Energy eigenvalues are computed numerically for some diatomic molecules. The results are in agreement with the ones obtained before.
Recently developed simple approach for the exact/approximate solution of Schrodinger equations with constant/position-dependent mass, in which the potential is considered as in the perturbation theory, is shown to be equivalent to the one leading to the construction of exactly solvable potentials via the solution of second-order differential equations in terms of known special functions. The formalism in the former solves difficulties encountered in the latter in revealing the corrections explicitly to the unperturbed piece of the solutions whereas the other obviate cumbersome procedures used …
We consider a free particle,V(r)=0, with position-dependent mass m(r)=1/(1+zeta^2*r^2)^2 in the d-dimensional schrodinger equation. The effective potential turns out to be a generalized Poschl-Teller potential that admits exact solution.
Maximal kinematical invariance groups of $2d$ Schrödinger equation with a position dependent mass and arbitrary potential are classified. It is demonstrated that there exist seven classes of such equations possessing non-equivalent continuous symmetry group. Three of these classes include arbitrary functions while the remaining ones are defined up to arbitrary parameters. In particular, for the case of a constant mass the class missing in the Boyer classification (Boyer C P 1974 Helv. Phys. Acta{\bf 47}, 450) is indicated. A constructive test of (non)equivalence of a PDM system to a constant m…
We provide a squeeze-like transformation that allows one to remove a position dependent mass from the Hamiltonian. Methods to solve the Schrödinger equation may then be applied to find the respective eigenvalues and eigenfunctions. As an example, we consider a position-dependent-mass that leads to the integrable Morse potential and therefore to well-known solutions.
We discuss the relationship between exact solvability of the Schrödinger equation with a position-dependent mass and the ordering ambiguity in the Hamiltonian operator within the frame of supersymmetric quantum mechanics. The one-dimensional Schrödinger equation, derived from the general form of the effective mass Hamiltonian, is solved exactly for a system with exponentially changing mass in the presence of a potential with similar behaviour, and the corresponding supersymmetric partner Hamiltonians are related to the effective-mass Hamiltonians proposed in the literature.
An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical w…
Transmission probabilities of the scattering problem with a position dependent mass are studied. After sketching the basis of the theory, within the context of the Schrödinger equation for spatially varying effective mass, the simplest problem, namely, tranmission through a square well potential with a position dependent mass barrier is studied and its novel properties are obtained. The solutions presented here may be adventageous in the design of semiconductor devices.
This study presents the Fisher information for the position-dependent mass Schrödinger equation with hyperbolical potential {$V(x)=-V_0{\rm csch}^2(ax)$}. The analysis of the quantum-mechanical probability for the ground and exited states $(n=0, 1, 2)$ has been obtained via the Fisher's information. This controls both chemical and physical properties of some molecular systems. The Fisher information is considered only for $x>0$ due to the singular point at $x=0$. We found that Fisher-information-based uncertainty relation and the Cramer-Rao inequality holds. Some relevant numerical results are…
A logarithmic Schrödinger equation with time-dependent coupling to the non-linearity is presented as a model of collisional decoherence of the wavefunction of a quantum particle in position-space. The particular mathematical form of the logarithmic Schrödinger equation has been shown to follow from conditional wave theory, but the validity of the logarithmic Schrödinger equation has not yet been investigated numerically for general initial conditions. Using an operator-splitting approach, we solve the non-linear equation of motion for the wavefunction numerically and compare it it to the solut…
Two new methods for investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. 1)The first one - $SUSY-$ separation of variables - is based on the intertwining relations of Higher order SUSY Quantum Mechanics (HSUSY QM) with supercharges allowing for separation of variables. 2)The second one is a generalization of shape invariance. While in one dimension shape invariance allows to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been yet explored. Here we prov…
We investigate the Schrodinger equation for a particle with a nonuniform solitonic mass density. First, we discuss in extent the (nontrivial) position-dependent mass $V(x)=0$ case whose solutions are hypergeometric functions in $\tanh^2(x)$. Then, we consider an external hyperbolic-tangent potential. We show that the effective quantum mechanical problem is given by a Heun class equation and find analytically an eigenbasis for the space of solutions. We also compute the eigenstates for a potential of the form $V(x)=V_0 \sinh^2(x)$.
On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schrödinger equation and a deformation of the canonical commutation relations, a method based on deformed shape invariance has recently been devised for generating pairs of potential and PDM for which the Schrödinger equation is exactly solvable. This approach has provided the bound-state energy spectrum, as well as the ground-state and the first few excited-state wavefunctions. The general wavefunctions have however remained unknown in explicit form because for their determination one would need the …
We outline a general method for obtaining exact solutions of Schrödinger equations with a position dependent effective mass and compare the results with those obtained within the frame of supersymmetric quantum theory. We observe that the distinct effective mass Hamiltonians proposed in the literature in fact describe exactly equivalent systems having identical spectra and wave functions as far as exact solvability is concerned. This observation clarifies the Hamiltonian dependence of the band-offset ratio for quantum wells.
A novel exactly solvable Schrödinger equation with a position-dependent mass (PDM) describing a Coulomb problem in $D$ dimensions is obtained by extending the known duality relating the quantum $d$-dimensional oscillator and $D$-dimensional Coulomb problems in Euclidean spaces for $D = (d+2)/2$. As an intermediate step, a mapping between a quantum $d$-dimensional nonlinear oscillator of Mathews-Lakshmanan type (or oscillator in a space of constant curvature) and a quantum $D$-dimensional Coulomb-like problem in a space of nonconstant curvature is derived. It is finally reinterpreted in a PDM b…
