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.
In the present article, we discuss one of the basic relations of Quantum Mechanics - the Uncertainty Relation (UR). In 1930, few years after Heisenberg, Erwin Schrodinger generalized the famous Uncertainty Relation in Quantum Mechanics, making it more precise than the original. The present study discusses recent generalizations of Schrodinger's work and explains why his paper remains almost forgotten in the last century.
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.
The original Schrodinger's paper is translated and annotated in honour of the 70-th anniversary of his Uncertainty Relation [published also in: Bulg. Journal of Physics,vol.26,no.5/6 (1999) pp.193-203]. In the annotation it is shown that the Uncertainty Relation can be written in a complete compact canonical form.
It is explicitly shown that there exist physical states (normalized to 1) in which the Robertson- Schrödinger and Heisenberg uncertainty relations are invalid, namely, the mean values of the physical operators are infinite. Consequently, these relations cannot imply a general physical principle. The explanation by the theory of functional analysis is given : for these states even the definition of the uncertainty notion through the dispersion notion in the probability theory is irrelevant.
We consider quantum steering by non-Gaussian entangled states. The Reid steering criterion based on the Heisenberg uncertainty relation fails to detect steerability for many categories of such states. Here, we derive a tighter steering criterion using the Robertson-Schrödinger uncertainty relation. We show that our steering condition is able to detect steerability of several classes of non-Gaussian states such as entangled eigenstates of the two-dimensional harmonic oscillator, the photon subtracted squeezed vacuum state and the NOON state.
Inspired by the generalized uncertainty principle (GUP), which adds gravitational effects to the standard description of quantum uncertainty, we extend the exact uncertainty principle (EUP) approach by Hall and Reginatto [J. Phys. A: Math. Gen. (2002) 35 3289], and obtain a (quasi)nonlinear Schrödinger equation. This quantum evolution equation of unusual form, enjoys several desired properties like separation of non-interacting subsystems or planewave solutions for free particles. Starting with the harmonic oscillator example, we show that every solution of this equation respects the gravitati…
Resonance (quasinormal) states correspond to non-Hermitian solutions to the Schrödinger equation obeying outgoing boundary conditions which lead to complex energy eigenvalues and momenta. Following the normalization rule for resonance states obtained from the residue at a complex pole of the outgoing Green's function to the problem, we propose a definition of expectation value for these states and use it to investigate the extent of validity of the Heisenberg uncertainty relations for potentials that vanish after a distance. We derive analytical expressions for the expectation values involving…
The local phase-invariance of the momentum-space Schrödinger equation for free-particle has been used to construct quantum kinematics that describes a motion of the particle in external U(1) background gauge field. The gauge structure over the momentum space of the particle is interpreted in terms of helicity and spin carried by the particle. As a by-product an effective one-particle Schrödinger equation of motion for the helicity-carrying particle in external potential field is derived. An effect of screening of the external potential is predicted, that can affect quantization and splittings …
We formulate an inseparability criterion based on the recently derived generalized Schrödinger-Robertson uncertainty relation (SRUR) [Ivan {\it et al.} J. Phys. A :Math. Theor. {\bf 45}, 195305 (2012)] together with the negativity of partial transpose (PT). This generalized SRUR systematically deals with two orthogonal quadrature amplitudes to higher-orders, so is relevant to characterize non-Gaussian quantum statistics. We first present a method that relies on the single-mode marginal distribution of two-mode fields under PT followed by beam-splitting operation. We then extend the SRUR to two…
A more general measurement disturbance uncertainty principle is presented in a Robertson-Schrödinger formulation. It is shown that it is stronger and having nicer properties than Ozawa's uncertainty relations. In particular is invariant under symplectic transformations. One shows also that there are states of the probe (measuring device) that saturate the matrix formulation of measurement disturbance uncertainty principle.
Recently, the Asymptotic Iteration Method (AIM) was used to calculate the energy spectrum for a short rang three parameter central potential which was introduced by H. Bahlouli and A. D. Alhaidari. The S-orbital wave solution of the Schrödinger equation was obtained for different parameters of the potential. In this work a non-zero angular momentum term were introduced to the problem and the energy eigenvalues were obtained for different potential parameters. Our results show very good agreements compared with other methods such as potential parameter spectrum method (PPSM) and the complex sca…
One of the advantages of a reconstruction of quantum mechanics based on transparent physical axioms is that it may offer insight to naturally generalize quantum mechanics by relaxing the axioms. Here, we discuss possible extensions of quantum mechanics within a general epistemic framework based on an operational scheme of estimation of momentum given positions under epistemic restriction. The epistemic restriction is parameterized by a global-nonseparable random variable on the order of Planck constant, an ontic extension to the separable classical phase space variables. Within the estimation …
Analytical solution of the Schrodinger equation for the neutral helium atom in the ground state considering the uncertainty principle, vibrational modes and quantum-electrodynamical effects
In the recent paper, Ref. 1, the l-waves Schrödinger equation for the Cornell's potential is solved in quantum mechanics with a generalized uncertainty principle by following Ref. 2. It is showed here that the approach of Ref. 2 can only be used for the s-waves, and then the solution given in Ref. 1 would be true only in the special case l=0. Furthermore, it is highlighted that the abstract and the conclusion of Ref. 1 do not accurately reflect the results of the paper.
We investigate properties of the covariance matrix in the framework of non-commutative quantum mechanics for an one-parameter family of transformations between the familiar Heisenberg-Weyl algebra and a particular extension of it. Employing as a measure of the Robertson-Schrödinger uncertainty principle the linear symplectic capacity of the Weyl ellipsoid (and its dual), we determine its corresponding bounds. Inequalities between the capacities for non-commutative phase-spaces are established. We also present a constructive example based on a simple model to justify our theoretical predictions…
We study and generalize the momentum operator satisfying the extended uncertainty principle relation (EUP). This generalized extended momentum operator (GEMO) consists of an arbitrary auxiliary function of position operator, $μ\left( x\right) $, in such a combination that not only GEMO satisfies the EUP relation but also it is Hermitian. Next, we apply the GEMO to construct the generalized one-dimensional Schrödinger equation. Upon using the so called point canonical transformation (PCT), we transform the generalized Schrödinger equation from $x$-space to $z$-space where in terms of the transf…
This paper presents the uncertainty related to position and momentum localization of a quantum state in terms of entropic uncertainty relations. We slightly improve the inequality given in [Phys. Rev. A 74, 052101 (2006)] and introduce a new entropic measure with corresponding uncertainty relation.
Commutator-based entropic uncertainty relations in multidimensional position and momentum spaces are derived, twofold generalizing previous entropic uncertainty relations for one-mode states. They provide optimal lower bounds and imply the multidimensional variance-based uncertainty principle. The article concludes with an open conjecture.
In order to know if the Schrodinger's cat is alive or dead without opening the box, observers have to play a game with nature. The observers have to "guess" (with degrees of belief) the state of the cat due to incomplete information; in other words, the observers' decision has to be made under uncertainty. We propose a quantum expected value theory for decision-making under uncertainty. Value operator is proposed to guide observers to choose corresponding actions based on their subjective beliefs through objective quantum world by maximizing the value from the measured historical results. The …
Analytical solutions of the Schrodinger equation are obtained for some diatomic molecular potentials with any angular momentum. The energy eigenvalues and wave functions are calculated exactly. The asymptotic form of the equation is also considered. Algebraic method is used in the calculations.
Propagation effects are analyzed for electromagnetic (EM) waves which satisfy the one-soliton non-linear Schrodinger (NLS) equation in a dispersive wave guide. The coupling between momentum and frequencies due to dispersion relation is treated by a coupled Hamiltonian-Momentum operator with equal-space commutation relations(CR). Kerr interactions in the soliton are treated. The classical solution of the one-soliton with possible quantum effects are analyzed. The integrability of nonlinear equations is related to compatibility-condition for scattering matrices and such relation is applied to th…
We have obtained the uncertainty relations for arbitrary states of the hydrogen atom. It is shown that the minimal value of the uncertainty relation is attained for the circular Rydberg states.
The exact solutions of Schrodinger equation are obtained for a noncentral potential which is a ring-shaped potential. The energy eigenvalues and corresponding eigenfunctions are obtained for any angular momentum l. Nikiforov-Uvarov method is used in the computations.
The equality in the uncertainty principle for linear momentum and position is obtained for states which also minimize the uncertainty product. However, in the uncertainty relation for angular momentum and angular position both sides of the inequality are state dependent and therefore the intelligent states, which satisfy the equality, do not necessarily give a minimum for the uncertainty product. In this paper, we highlight the difference between intelligent states and minimum uncertainty states by investigating a class of intelligent states which obey the equality in the angular uncertainty r…
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.
Uncertainty relations between a bounded coordinate operator and a conjugate momentum operator frequently appear in quantum mechanics. We prove that physically reasonable minimum-uncertainty solutions to such relations have quantized expectation values of the conjugate momentum. This implies, for example, that the mean angular momentum is quantized for any minimum-uncertainty state obtained from any uncertainty relation involving the angular-momentum operator and a conjugate coordinate. Experiments specifically seeking to create minimum-uncertainty states localized in angular coordinates theref…
The position-momentum uncertainty-like inequality based on moments of arbitrary order for d-dimensional quantum systems, which is a generalization of the celebrated Heisenberg formulation of the uncertainty principle, is improved here by use of the Renyi-entropy-based uncertainty relation. The accuracy of the resulting lower bound is physico-computationally analyzed for the two main prototypes in d-dimensional physics: the hydrogenic and oscillator-like systems.
In this study, we explore the validity of the original Heisenberg position- momentum uncertainty relation for a macroscopic harmonic oscillator interacting with environmental micro particles. Our results show that, in the quasi-classical situation, the original uncertainty relation does not hold when the number of particles in the environment is small. Nonetheless, increasing the environmental degrees of freedom resolves the violation in the region of our investigation.
We point out two interesting features of position-momentum uncertainty product: $U=Δx Δp$. We show that two special (non-differentiable) eigenstates of the Schr{ö}dinger operator with the Dirac Delta potential $[V(x)=-V_0 δ(x)],V_0>0$, also satisfy the Heisenberg's uncertainty principle by yielding $U> \frac{\hbar}{2}$. One of these eigenstates is a zero-energy and zero-curvature bound state.
It is suggested that a measurement of the products of photoemission by alkali atoms excited after extraction from a trap, might, using the EPR strategy, show a significant violation of the momentum-position uncertainty relation. If this failed, as is quite likely, possible causes, such as retroactive propagation of influences and retrodiction failure, could be tested on the proposed apparatus.
The uncertainty principle is considered to be one of the most striking features in quantum mechanics. In the textbook literature, uncertainty relations usually refer to the preparation uncertainty which imposes a limitation on the spread of measurement outcomes for a pair of non-commuting observables. In this work, we study the preparation uncertainty for the angular momentum, especially in the spin-1/2 representation. We derive uncertainty relations encompassing the triple components of angular momentum, and show that compared with the relations involving only two components, a triple constan…
For a quantum particle with a single degree of freedom, we derive preparational sum and product uncertainty relations satisfied by $N$ linear combinations of position and momentum observables. The state-independent bounds depend on their degree of incompatibility defined by the area of a parallelogram in an $N$-dimensional coefficient space. Maximal incompatibility occurs if the observables give rise to regular polygons in phase space. We also conjecture a Hirschman-type uncertainty relation for N observables linear in position and momentum, generalizing the original relation which lower-bound…
The uncertainty relations for angle and angular momentum are revisited. We use the exponential of the angle instead of the angle itself and adopt dispersion as a natural measure of resolution. We find states that minimize the uncertainty product under the constraint of a given uncertainty in angle or in angular momentum. These states are described in terms of Mathieu wave functions and may be approximated by a von Mises distribution, which is the closest analogous of the Gaussian on the unit circle. We report experimental results using beam optics that confirm our predictions.
A weakly nonlocal extension of ideal fluid dynamics is derived from the Second Law of thermodynamics. It is proved that in the reversible limit the additional pressure term can be derived from a potential. The requirement of the additivity of the specific entropy function determines the quantum potential uniquely. The relation to other known derivations of Schrödinger equation (stochastic, Fisher information, exact uncertainty) is clarified.
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 consider particles prepared by the von Neumann-Lüders projection. For those particles the standard deviation of the momentum is discussed. We show that infinite standard deviations are not exceptions but rather typical. A necessary and sufficient condition for finite standard deviations is given. Finally, a new uncertainty relation is derived and it is shown that the latter cannot be improved.
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…
We consider two entropic uncertainty relations of position and momentum recently discussed in literature. By a suitable rescaling of one of them, we obtain a smooth interpolation of both for high-resolution and low-resolution measurements respectively. Because our interpolation has never been mentioned in literature before, we propose it as a candidate for an improved entropic uncertainty relation of position and momentum. Up to now, the author has neither been able to falsify nor prove the new inequality. In our opinion it is a challenge to do either one.
In the Entropic Dynamics (ED) approach to quantum theory the particles have well-defined positions but since they follow non differentiable Brownian trajectories they cannot be assigned an instantaneous momentum. Nevertheless, four different notions of momentum can be usefully introduced. We derive relations among them and the corresponding uncertainty relations. The main conclusion is that momentum is a statistical concept: in ED the momenta are not properties of the particles; they are attributes of the probability distributions.
Recently it was shown in [New J. Phys. 8, 330 (2006)] that the three dimensional Heisenberg uncertainty principle might be signifficantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a new lower bound which involves the mean value and the variance of the $\hat{L}^2$ operator.
This note shows that Heisenberg's choice for a wave function in his original paper on the uncertainty principle is simply a renormalized characteristic function of a stable distribution with certain restrictions on the parameters. Relaxing Heisenberg's restrictions leads to a more general formulation of the uncertainty principle. This reformulation shows quantum uncertainty can exist at a macroscopic level. These modifications also give rise to a new form of Schrodinger's wave equation as the equation of a vibrating string. Although a heat equation version can also be given, the latter shows t…
The uncertainty relation between angle and orbital angular momentum had not been formulated in a similar form as the uncertainty relation between position and linear momentum because the angle variable is not represented by a quantum mechanical self-adjoint operator. Instead of the angle variable operator, we introduce the complex position operator $ \hat{Z} = \hat{x}+i \hat{y} $ and interpret the order parameter $ μ= \langle \hat{Z} \rangle / \sqrt{ \langle \hat{Z}^\dagger \hat{Z} \rangle} $ as a measure of certainty of the angle distribution. We prove the relation between the uncertainty of …
In this work, we consider the relativistic Duffin-Kemmer-Petiau equation for spin-one particles with a nonminimal vector interaction in the presence of minimal uncertainty in momentum. By using the position space representation we exactly determine the bound-states spectrum and the corresponding eigenfunctions. We discuss the effects of the deformation and nonminimal vector coupling parameters on the energy spectrum analytically and numerically.
We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form $(ΔP)(ΔQ)\geq C\hbar$, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision $ΔQ$ of the position measurement, and the perturbation $ΔP$ of the conjugate variable introduced by such a measurement. We also determine the best constant $C$, which is attained for a unique p…
The Schrödinger-Coulomb Sturmian problem in $\mathbb{R}^{N}$, $N\geqslant2$, is considered in the momentum representation. An integral formula for the Gegenbauer polynomials, found recently by Cohl [arXiv:1105.2735], is used to separate out angular variables and reduce an integral Sturmian eigenvalue equation in $\mathbb{R}^{N}$ to a Fredholm one on $\mathbb{R}_{+}$. A kernel of the latter equation contains the Legendre function of the second kind. A symmetric Poisson-type series expansion of that function into products of the Gegenbauer polynomials, established by Ossicini [Boll. Un. Mat. Ita…
Uncertainty in the determination of emission angles of signal and idler photons is investigated. The role of divergence of the pump beam, width of the pump spectrum, and size of the crystal is elucidated. Experimental data obtained by an iCCD camera are in a good agreement with a numerical model that provides angular and spectral characteristics of the signal and idler fields.
Exact solution of Schrodinger equation for the pseudoharmonic potential is obtained for an arbitrary angular momentum. The energy eigenvalues and corresponding eigenfunctions are calculated by Nikiforov-Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The energy eigenvalues are calculated numerically for some values of l and n with n<5 for some diatomic molecules.
In this work, we derive a generalization of the so-called Schrödinger-Langevin or Kostin equation for a Brownian particle interacting with a heat bath. This generalization is based on a nonlinear interaction model providing a state-dependent dissipation process exhibiting multiplicative noise. Two straightforward applications to the measurement process are then analyzed, continuous and weak measurements in terms of the quantum Bohmian trajectory formalism. Finally, it is also shown that the generalized uncertainty principle, which appears in some approaches to quantum gravity, can be expressed…
So-called quantum limits and their achievement are important themes in physics. Heisenberg's uncertainty relations are the most famous of them but are not universally valid and violated in general. In recent years, the reformulation of uncertainty relations is actively studied, and several universally valid uncertainty relations are derived. On the other hand, several measuring models, in particular, spin-1/2 measurements, are constructed and quantitatively examined. However, there are not so many studies on simultaneous measurements of position and momentum despite their importance. Here we s…
The state-extended uncertainty relations generalizing the Robertson Schrodinger inequality are presented in the form appropriate for the experimental check by homodyne photon state detection. The method of qubit portrait of qudit states identified with the tomographic probability distributions is discussed to analyze the entanglement of two-mode field.
The Schrodinger formalism of quantum mechanics is used to demonstrate the existence of the Aharonov-Bohm effect in momentum space and set-ups for experimentally demonstrating it are proposed for either free or ballistic electrons.
Uncertainty relations based on information theory for both discrete and continuous distribution functions are briefly reviewed. We extend these results to account for (differential) Rényi entropy and its related entropy power. This allows us to find a new class of information-theoretic uncertainty relations (ITURs). The potency of such uncertainty relations in quantum mechanics is illustrated with a simple two-energy-level model where they outperform both the usual Robertson-Schrödinger uncertainty relation and Kraus-Maassen Shannon entropy based uncertainty relation. In the continuous case th…
Using the quantum transition path time probability distribution we show that time averaging of weak values leads to unexpected results. We prove a weak value time energy uncertainty principle and time energy commutation relation. We also find that time averaging allows one to predict in advance the momentum of a particle at a post selected point in space with accuracy greater than the limit of $\hbar /2$ as dictated by the uncertainty principle. This comes at a cost - it is not possible at the same time to predict when the particle will arrive at the post selected point. A specific example is …
We define an uncertainty observable, acting on several replicas of a continuous-variable bosonic state, whose trivial uncertainty lower bound induces nontrivial phase-space uncertainty relations for a single copy of the state. By exploiting the Schwinger representation of angular momenta in terms of bosonic operators, we construct such an observable that is invariant under symplectic transformations (rotation and squeezing in phase space). We first design a two-copy uncertainty observable, which is a discrete-spectrum operator vanishing with certainty if and only if it is applied on (two copie…
At the recent QSCP XIX, the author claimed a procedure of using a scaled Fourier transform (the scaling being determined by the detailed interaction and particle mass for a harmonic oscillator) to achieve simultaneous resolution of position and momentum greater than the standard Heisenberg value of 1/2. The procedure is, in fact, invalid for quantum mechanics. The purpose of this paper is simply to give the correct analysis of the uncertainty product, thereby clarifying the error made.
