Bringing quantum mechanics to life: from Schrödinger’s cat to Schrödinger’s microbe

The question whether quantum mechanics is complete and the nature of the transition between quantum mechanics and classical mechanics have intrigued physicists for decades. There have been many experimental breakthroughs in creating larger and larger quantum superposition and entangled states since Erwin Schrödinger proposed his famous thought experiment of putting a cat in a superposition of both alive and dead states in 1935. Remarkably, recent developments in quantum optomechanics and electromechanics may lead to the realisation of quantum superposition of living microbes soon. Recent evidence also suggests that quantum coherence may play an important role in several biological processes. In this review, we first give a brief introduction to basic concepts in quantum mechanics and the Schrödinger’s cat thought experiment. We then review developments in creating quantum superposition and entangled states and the realisation of quantum teleportation. Non-trivial quantum effects in photosynthetic light harvesting and avian magnetoreception are also discussed. At last, we review recent proposals to realise quantum superposition, entanglement and state teleportation of micro-organisms, such as viruses and bacteria.     

Topological Solitons of the Nonlinear Schrödinger’s Equation with Fourth Order Dispersion

This paper obtains the topological 1-soliton solution of the nonlinear Schrödinger’s equation, in a non-Kerr law media, with fourth order dispersion. An exact 1-soliton solution is obtained. The types of nonlinearity that are studied in this paper are Kerr law and power law.     

On Shtelen’s Solution of the Free Linear Schrödinger Equation

Abstract

The solution of the three-dimensional free Schrödinger equation due to W.M. Shtelen based on the invariance of this equation under the Lorentz Lie algebra so(1,3) of nonlocal transformations is considered. Various properties of this solution are examined, including its extension to n ≥ 3 spatial dimensions and its time decay; which is shown to be slower than that of the usual solution of this equation. These new solutions are then used to define certain mappings, F _n, on L ²(?ⁿ) and a number of their properties are studied; in particular, their global smoothing properties are considered. The differences between the behavior of F _n and that of analogous mappings constructed from usual solutions of the free Schrödinger equation are discussed.     

Gauge Transformations for a Family of Nonlinear Schrödinger Equations

Abstract

An enlarged gauge group acts nonlinearly on the class of nonlinear Schrödinger equations introduced by the author in joint work with Doebner. Here the equations and the group action are displayed in the presence of an external electromagnetic field. All the gauge-invariants are listed for the coupled nonlinear “Schrödinger-Maxwell” theory. Time-dependent gauge parameters result in additional terms of the type introduced by Kostin and Bialynicki-Birula and Mycielski, but Maxwell’s equations for the (non-quantized) gauge-invariant electric and magnetic fields remain linear.     

Derivation of Nonlinear Schrödinger Equation

We propose some nonlinear Schrödinger equations by adding some higher order terms to the Lagrangian density of Schrödinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrödinger equation is weak.     

Exact Solutions of the Schrödinger Equation with Inverse-Power Potential

The Schrödinger equation for stationary states is studied in a central potential V(r) proportional to r ^– in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrödinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case = 4 is elucidated. In general, whenever the parameter is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.     

Exact Solutions of the Schrödinger Equation with Irregular Singularity

The 1D nonrelativistic Schrödinger equation possessing an irregular singularpoint is investigated. We apply a general theorem about existence and structureof solutions of linear ordinary differential equations to the Schrödinger equationand obtain suitable ansatz functions and their asymptotic representations for alarge class of singular potentials. Using these ansatz functions, we work out allpotentials for which the irregular singularity can be removed and replaced by aregular one. We obtain exact solutions for these potentials and present sourcecode for the computer algebra system Mathematica to compute the solutions. Forall cases in which the singularity cannot be weakened, we calculate the mostgeneral potential for which the Schrödinger equation is solved by the ansatzfunctions obtained and develop a method for finding exact solutions.     

Quantum Monodromy in the Spectrum of Schrödinger Equation with a Decatic Potential

In this study the spectral problem of the two-dimensional Schrödinger equation with the cylindrically symmetrical decatic potential is carried out. The concept of quantum monodromy is introduced to give insight into the energy levels of system with this potential. It is shown that quantum monodromy occurs at = 0 in the distribution of eigenstates around a critical point on the spectrum at E = 0 with zero angular momentum, such that there can be no smoothly valid assignment of quantum number. Cases with the three-well and four-well potentials are presented to give rise to the double degeneracies with respect to energy except for the angular momentum m = 0.     

Dilaton Quantum Cosmology with a Schrödinger-like Equation

A quantum cosmological model with radiation and a dilaton scalar field is analyzed. The Wheeler–DeWitt equation in the minisuperspace induces a Schrödinger equation, which can be solved. An explicit wavepacket is constructed for a particular choice of the ordering factor. A consistent solution is possible only when the scalar field is a phantom field. Moreover, although the wavepacket is time-dependent, a Bohmian analysis allows to extract a bouncing behavior for the scale factor.     

Lorentz Transformations for the Schrödinger Equation

Abstract

We show that the free Schrödinger equation admits Lorentz space-time transformations when corresponding transformations of the ψ-function are nonlocal. Some consequences of this symmetry are discussed.

Dedicated to Wilhelm Fushchych – Inspirer, Mentor, Friend and Pioneer in non–Lie symmetry methods – on the occasion of his sixtieth birthday     

A Flea on Schrödinger’s Cat

We propose a technical reformulation of the measurement problem of quantum mechanics, which is based on the postulate that the final state of a measurement is classical; this accords with experimental practice as well as with Bohr’s views. Unlike the usual formulation (in which the post-measurement state is a unit vector in Hilbert space), our version actually opens the possibility of admitting a purely technical solution within the confines of conventional quantum theory (as opposed to solutions that either modify this theory, or introduce unusual and controversial interpretative rules and/or ontologies). To that effect, we recall a remarkable phenomenon in the theory of Schrödinger operators (discovered in 1981 by Jona-Lasinio, Martinelli, and Scoppola), according to which the ground state of a symmetric double-well Hamiltonian (which is paradigmatically of Schrödinger’s Cat type) becomes exponentially sensitive to tiny perturbations of the potential as ?→0. We show that this instability emerges also from the textbook wkb approximation, extend it to time-dependent perturbations, and study the dynamical transition from the ground state of the double well to the perturbed ground state (in which the cat is typically either dead or alive, depending on the details of the perturbation). Numerical simulations show that adiabatically arising perturbations may (quite literally) cause the collapse of the wave-function in the classical limit. Thus, at least in the context of a simple mathematical model, we combine the technical and conceptual virtues of decoherence (which fails to solve the measurement problem but launches the key idea that perturbations may come from the environment) with those of dynamical collapse models à la grw (which do solve the measurement problem but are ad hoc), without sharing their drawbacks: single measurement outcomes are obtained (instead of merely diagonal reduced density matrices), and no modification of quantum mechanics is needed.     

Green’s Function of the Schrödinger Equation in the Potential Quantization Method

Journal of Experimental and Theoretical Physics - The problem for determining Green’s function G(r, r') for the time-independent Schrödinger equation is considered using the...     

Symmetries of the Free Schrödinger Equation

An algorithm is proposed for studying the symmetry properties of equations used in theoretical and mathematical physics. The application of this algorithm to the free Schrödinger equation permits one to establish that, in addition to the known Galilei symmetry, the free Schrödinger equation possesses also relativistic symmetry in some generalized sense. This property of the free Schrödinger equation provides an extension of the equation into the relativistic domain of the free particle motion under study.     

Thermodynamic Gravity and the Schrödinger Equation

We adopt a formulation of the Mach principle that the rest mass of a particle is a measure of it’s long-range collective interactions with all other particles inside the horizon. As a consequence, all particles in the universe form a ‘gravitationally entangled’ statistical ensemble and one can apply the approach of classical statistical mechanics to it. It is shown that both the Schrödinger equation and the Planck constant can be derived within this Machian model of the universe. The appearance of probabilities, complex wave functions, and quantization conditions is related to the discreetness and finiteness of the Machian ensemble.     

Iterative Solutions of the Schrödinger Equation

In the first example containing a long ranged potential, the long range part of the solution is obtained by an iterative Born-series type method. The convergence is illustrated for a case with the long range part of the potential given by C ₆/r ⁶. Accuracies of 1 : 10⁸ are achieved after 8 iterations. The second example iteratively calculates the solution of a non-linear Gross–Pitaevskii equation for condensed Bose atoms contained in a trap at low temperature.     

Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise

We construct generalized grand-canonical- and canonical Gibbs measures for a Hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear Schrödinger equation with a focusing nonlinearity of order p < 6. Key properties of these Gibbs measures, in particular absence of “phase transitions” and regularity properties of field samples, are established. We then study a time evolution of this system given by the Hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature. The noise is of Ornstein–Uhlenbeck type for the Fourier modes of the field, with the strength of the noise decaying to zero, as the frequency of the mode tends to ∞. We prove exponential approach of the state of the system to a grand-canonical Gibbs measure at a temperature and “chemical potential” determined by the stochastic noise term.     

Semiclassical Limit for the Schrödinger Equation¶with a Short Scale Periodic Potential

We consider the dynamics generated by the Schr?dinger operator H=−?Δ+V(x)+W(ɛx), where V is a lattice periodic potential and W an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit ɛ→ 0 the time dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined by the semiclassical dynamics. Received: 7 February 2000 / Accepted: 7 July 2000     

Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term

In this paper, we establish a family of symplectic integrators for a class of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization. Then we apply the symplectic Euler method to the Hamiltonian system. It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system, but also does not require to resolve coupled nonlinear algebraic equations which is different from the general implicit symplectic schemes. The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated. It shows that the semi-explicit scheme is conditionally stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones, such as backward Euler integrators.     

Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.