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 wave equation is of the same form as that derived in a recent paper by Scott and Milburn for the specific case of joint Gaussian states.
Exact solutions of effective radial Schrödinger equation are obtained for some inverse potentials by using the point canonical transformation. The energy eigenvalues and the corresponding wave functions are calculated by using a set of mass distributions
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.
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.
We investigate two methods of constructing a solution of the Schrödinger equation from the canonical transformation in classical mechanics. One method shows that we can formulate the solution of the Schrödinger equation from linear canonical transformations, the other focuses on the generating function which satisfies the Hamilton-Jacobi equation in classical mechanics. We also show that these two methods lead to the same solution of the Schrödinger equation.
The representation of a Schrodinger equations as a classic Hamiltonian system allows to construct a unified perturbation theory both in classic, and in a quantum mechanics grounded on the theory of canonical transformations, and also to receive asymptotic estimations of affinity of the precisian approximated solutions of Schrodinger equations
Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over Grassmannian and its relation with Berry phase is considered …
The Canonical Function Method (CFM) is a powerful accurate and fast method that solves the Schrödinger equation for the eigenvalues directly without having to evaluate the eigenfunctions. Its versatility allows to solve several types of problems and in this work it is applied to the solution of several 1D potential problems, the 3D Hydrogen atom and the Morse potential.
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.
Noise sequences of infinite matrices associated with covariant phase and box localization observables are defined and determined. The canonical observables are characterized within the relevant classes of observables as those with asymptotically minimal or minimal noise, i.e., the noise tending to 0 or having the value 0.
Measurements of single-mode phase observables are studied in the spirit of the quantum theory of measurement. We determine the minimal measurement models of phase observables and consider methods of measuring such observables by using a double homodyne detector. We show that, in principle, the canonical phase distribution of the signal state can be measured via double homodyne detection by first processing the state using a two-mode unitary channel.
Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg-type uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state …
Uncertainties in successive measurements of general canonically conjugate variables are examined. Such operators are approached within a limiting procedure of the Pegg-Barnett type. Dealing with unbounded observables, we should take into account a finiteness of detector resolution. An appropriate reformulation of two scenarios of successive measurements is proposed and motivated. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. The Rényi and Tsallis formulations of uncertainty relations are obtained for both the scenarios of successive meas…
Canonically conjugated observables such as position-momentum and phase-number are found to play a 3-fold role in the drama of the quantum teleportation. Firstly, the common eigenstate of two commuting canonical observables like phase-difference and number-sum provides the quantum channel between two systems. Secondly, a similar pair of canonical observables from another two systems is measured in the Bell operator measurements.
Investigated idea was actuated by the old opinion that a measurement of a quantum observable should be regarded a as a single deterministic sampling. But, according to the last decades studies, such observables are veritable random variables and their measurements must imply significant sets of statistical samplings. So one finds the indubitable caducity of the approached idea. Contiguously the respective finding allows to put into a new light the controversial questions like the Schrodinger's cat thought experiment or description of quantum measurements.
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 …
The canonical function method (CFM) is a powerful means for solving the Radial Schrodinger Equation. The mathematical difficulty of the RSE lies in the fact it is a singular boundary value problem. The CFM turns it into a regular initial value problem and allows the full determination of the spectrum of the Schrodinger operator without calculating the eigenfunctions. Following the parametrisation suggested by Klapisch and Green, Sellin and Zachor we develop a CFM to optimise the potential parameters in order to reproduce the experimental Quantum Defect results for various Rydberg series of He,…
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.
It is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be Borel summable to the solutions themselves appear also to be the unique solutions to the 1D Schrödinger equation having this property. Namely, it is shown in this paper that for the polynomial potentials the Borel function defined by the fundamental solutions can be considered as the canonical one. The latter means that any Borel summable solution can be obtained by the Borel transformation of this unique canonical Borel function multiplied by some $\hbar$-dependent and Bo…
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.
Weak measurement of a subset of noncommuting observables of a quantum system can be modeled by the open-system evolution, governed by the master equation in the Lindblad form. The open-system density operator can be represented as statistical mixture over non unitarily evolving pure states, driven by the stochastic Nonlinear Schrodinger equation (sNLSE). The globally stable solution of the sNLSE is obtained in the case where the measured subset of observables comprises the spectrum-generating algebra of the system. This solution is a generalized coherent state (GCS), associated with the algebr…
We regard the real and imaginary parts of the Schrodinger wave function as canonical conjugate variables.With this pair of conjugate variables and some other 2n pairs, we construct a quadratic Hamiltonian density. We then show that the Schrodinger Equation follows from the Hamilton's Equation of motion when the Planck frequency is much larger than the characteristic frequencies of the Hamiltonian system. The Hamiltonian and the normal mode solutions coincide, respectively, with the energy expectation value and the energy eigenstates of the corresponding quantum mechanical system.
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…
For angular observables pairs (angular momentum-angle and number-phase) the adequate reference element of normality is not the Robertson-Schrödinger uncertainty relation but a Schwarz formula regarding the quantum fluctuations. Beyond such a fact the traditional interpretation of the uncertainty relations appears as an unjustified doctrine.
We show that the stochastic Schrodinger equation for the filtered state of a system, with linear free dynamics, undergoing continual non-demolition measurement or either position or momentum, or both together, can be solved explicitly within a class of Gaussian states which we call extended coherent states. The asymptotic limit yields a class of relaxed states which we describe explicitly. Bellman's principle is then applied directly to optimal feedback control of such dynamical systems and the Hamilton Jacobi Bellman equation for the minimum cost is derived. The situation of quadratic perform…
We consider the problem of designing a variety of "system guided" basis sets for quantum mechanical anharmonic oscillators. Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to generate generalized "Cartesian-like positions, W and momenta, p_W" with unit Poisson brackets. These are quantized following Dirac, leading to an infinite family of potential "operator observables". The fundamental issue is that all but one of the operators are not Hermitian (formally self-adjoint) in the original position representation. We sh…
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…
The non-Hermitian Schrödinger equation is re-expressed generally in the form of Hamilton's canonical equation without any approximation. Its quantization called non-Hermitian quantum field theory is discussed. By virtue of the canonical equation, the theory of non-Hermitian quantum mechanics is totally reformulated, including the probability amplitudes of states, the expectations of operators, as well as the expressions of interaction terms. The conventional difficulties in non-Hermitian quantum mechanics are totally overcome by the reformulation. Specifically, the imaginary parts the non-Herm…
Nonlocal nonlinear Schrodinger model is quantised and exactly solved using the canonical framework. It is found that the usual canonical quantisation of the model leads to a theory with pathological inner product. This problem is resolved by constructing another inner product over the vector space of the theory. The resultant theory is found to be identical to that of nonrelativistic bosons with delta function interaction potential, devoid of any nonlocality. The exact eigenstates are found using the Bethe ansatz technique.
We address measurement schemes where certain observables are chosen at random within a set of non-degenerate isospectral observables and then measured on repeated preparations of a physical system. Each observable has a given probability to be measured, and the statistics of this generalized measurement is described by a positive operator-valued measure (POVM). This kind of schemes are referred to as quantum roulettes since each observable is chosen at random, e.g. according to the fluctuating value of an external parameter. Here we focus on quantum roulettes for qubits involving the measureme…
We propose a non-Gaussian continuous variable (CV) gate which is able to conditionally produce superposition of two "copies" of an arbitrary input state well separated in the coordinate and momentum plane - a Schrödinger cate state. The gate uses cubic phase state of an ancillary oscillator as a non-Gaussian resource, an entangling Gaussian gate, and homodyne measurement which provides nonunique information about the target system canonical variables, which is a key feature of the scheme. We show that this nonuniqueness manifests problems which may arise by extension of the Heisenberg picture …
We consider systems, which conserve the particle number and are described by Schrödinger equations containing complex nonlinearities. In the case of canonical systems, we study their main symmetries and conservation laws. We introduce a Cole-Hopf like transformation both for canonical and noncanonical systems, which changes the evolution equation into another one containing purely real nonlinearities, and reduces the continuity equation to the standard form of the linear theory. This approach allows us to treat, in a unifying scheme, a wide variety of canonical and noncanonical nonlinear syste…
A canonical quantization scheme is represented for a quantum system interacting with a nonlinear absorbing environment. The environment is taken anisotropic and the main system is coupled to its environment through some coupling tensors of various ranks. The nonlinear response equation of the environment against the motion of the main system is obtained. The nonlinear Langevin-schrödinger equation is concluded as the macroscopic equation of motion of the dissipative system. The effect of nonlinearity of the environment is investigated on the spontaneous emission of an initially excited two lev…
$p$-Mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how non-linear canonical transformations affect $p$-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of $p$-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state…
Extending the point canonical transformation approach in a manner distinct from the previous ones, we propose a unified approach of generating potentials of all classes having non-constant masses.
Algebra of quantum mechanics via classical phonons. I: The Schrodinger equation as the Newtonian equation of motion and quantum observables as classical averages
In a recent paper by Jafarov, Nagiyev, Oste and Van der Jeugt (2020 {\sl J.\ Phys.\ A} {\bf 53} 485301), a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position, was constructed and it was shown that the confinement parameter gets quantized. By using a point canonical transformation starting from the constant-mass Schrödinger equation for the Rosen-Morse II potential, it is shown here that similar results can be easily obtained without quantizing the confinement parameter. In addition, an extension t…
A field state containing photons propagating in different directions has a non vanishing mass which is a quantum observable. We interpret the shift of this mass under transformations to accelerated frames as defining space-time observables canonically conjugated to energy-momentum observables. Shifts of quantum observables differ from the predictions of classical relativity theory in the presence of a non vanishing spin. In particular, quantum redshift of energy-momentum is affected by spin. Shifts of position and energy-momentum observables however obey simple universal rules derived from inv…
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.
Based on his extension of the classical argument of Einstein, Podolsky and Rosen, Schrödinger observed that, in certain quantum states associated with pairs of particles that can be far away from one another, the result of the measurement of an observable associated with one particle is perfectly correlated with the result of the measurement of another observable associated with the other particle. Combining this with the assumption of locality and some "no hidden variables" theorems, we showed in a previous paper [11] that this yields a contradiction. This means that the assumption of localit…
Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrödinger equation. A simple generalization of this …
Applying ideas from monadic dynamics to the well-established framework of categorical quantum mechanics, we provide a novel toolbox for the simulation of finite-dimensional quantum dynamics. We use strongly complementary structures to give a graphical characterisation of quantum clocks, their action on systems and the relevant energy observables, and we proceed to formalise the connection between unitary dynamics and projection-valued spectra. We identify the Weyl canonical commutation relations in the axioms of strong complementarity, and conclude the existence of a dual pair of time/energy o…
Lagrange and Hamilton equations for thermodynamic evolution near equilibrium as well as Schrodinger-like equation for the non-equilibrium case are obtained extending the CPDQ Principle (Constancy of the product momentum-oordinate uncertainty) to a new conjugate pair where the momentum f is the extension of the mechanical f=p.δq product for the case the coordinate q is not directly observable. Applied to the case of a particle in a one dimension box, the results are in agreement with reported calculated and measured thermal conductance.
Newtonian and Schrodinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrodinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed thr…
Students in a quantum mechanics course are often introduced to the Schrödinger equation as the standard mathematical tool. However, rarely do students develop an understanding as to why the equation is the choice for modeling quantum phenomena or where it came from. While many books do have a heuristic derivation, they sometimes differ in their approach and often fail to give a satisfying explanation to the so-called canonical substitution of momentum. In this paper, we use de Broglie's hypothesis along with experimental results and theories to provide a heuristic derivation of the Schrödinger…
We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one but it is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it a pure state of some quantum system is described by a state section (along paths) of a (Hilbert) fibre bundle. Its evolution is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and the density operator are described via bundle morphisms (along paths). The menti…
During recent years, exact solutions of position-dependent mass Schrödinger equations have inspired intense research activities, based on the use of point canonical transformations, Lie algebraic methods or supersymmetric quantum mechanical techniques. Here we highlight the interest of another approach to such problems, relying on quadratic algebras. We illustrate this point by constructing spectrum generating algebras for a class of $d$-dimensional radial harmonic oscillators with $d\ge2$ (including the one-dimensional oscillator on the line via some minor changes) and a specific mass choice.…
