We explore a few advantages of studying the change in information entropy of a bound quantum state with energy. It is known that the property generally increases with the quantum number for stationary states, in spite of the concomitant gradual increase in the number of constraints for higher levels. A simple semiclassical proof of this observation is presented via the Wilson-Sommerfeld quantization scheme. In the small quantum number regime, we numerically demonstrate how far the semiclassical predictions are valid for a few systems, some of which are exactly solvable and some not so. Our findings appear to be significant in a number of ways. We observe that, for most problems, information entropy tends to a maximum as the quantum number tends to infinity. This sheds some light on the Bohr limit as a classical limit. Noting that the dependence of energy on the quantum number governs the rate of increase of information entropy with the degree of excitation, we extend our analysis to include the role of the kinetic energy. The endeavor yields a relation that possesses a universal character for any one-dimensional problem. Relevance of information entropy in studying the goodness of approximate stationary states obtained from finite-basis linear variational calculations is also delineated. Finally, we expound how this property behaves in situations where shape resonances show up. A typical variation is indeed observed in such cases when we proceed to detect Siegert states via the stabilization method.
We explore a few advantages of studying the change in information entropy of a bound quantum state with energy. It is known that the property generally increases with the quantum number for stationary states, in spite of the concomitant gradual increase in the number of constraints for higher levels. A simple semiclassical proof of this observation is presented via the Wilson-Sommerfeld quantization scheme. In the small quantum number regime, we numerically demonstrate how far the semiclassical predictions are valid for a few systems, some of which are exactly solvable and some not so. Our findings appear to be significant in a number of ways. We observe that, for most problems, information entropy tends to a maximum as the quantum number tends to infinity. This sheds some light on the Bohr limit as a classical limit. Noting that the dependence of energy on the quantum number governs the rate of increase of information entropy with the degree of excitation, we extend our analysis to include the role of the kinetic energy. The endeavor yields a relation that possesses a universal character for any one-dimensional problem. Relevance of information entropy in studying the goodness of approximate stationary states obtained from finite-basis linear variational calculations is also delineated. Finally, we expound how this property behaves in situations where shape resonances show up. A typical variation is indeed observed in such cases when we proceed to detect Siegert states via the stabilization method.
Historically, thermodynamics was formulated for macroscopic systems followed by the development of statistical mechanics which provided microscopic formulations for quantities such as entropy. Entropy is traditionally linked to the ln of the number states available or to the Shannon’s form -k Sum over i P(ei) ln(P(ei)) where P(ei) is the probability for the quantity ei. In addition a variable T (linked to average kinetic energy) appears in both theories. Later the Schrodinger equation was developed to describe bound electron states in an atom etc. These states were characterized by a math object W(x) called the wavefunction and a set of discrete energy levels was calculated which matched data of emitted photons. In a previous note (1) we argued that the average energy En for a quantum bound state plays the same role of characterizing equilibrium as T in classical thermodynamics/statistical mechanics. We further argued that En is linked to parameters in the system. For example, for a particle in a box with infinite potential walls, En is linked to L as well as to n the energy level. One may change En by either changing n or L separately (or both). Holding L constant and changing n leads to a certain type of energy change. We argue this is similar to holding the volume of a gas constant and increasing T. Alternatively, one may leave n unchanged (a quantum adiabatic process) and change L i.e. a mechanical change or work. We argue that one may already consider the concept of the first law of thermodynamics at the quantum bound state level i.e. dE = dQ - dWork where Work is linked to a change in a physical parameter, such as L the length of the box, whereas dQ is linked to energy changes not involving a physical parameter e.g. a change in n the energy level. The Schrodinger equation yields solutions for W(x)=Sum over p a(p)exp(ipx) the wavefunction as well as En’s. We have argued in previous notes that W(x) is part of a conditional probability P(p/x)= a(p)exp(ipx)/W(x) and that En is constant at each point x just as T is constant at each x for classical statistical equilibrium (1). Thus a kind of statistical mechanics exists at the wavefunction level we argue. This statistical mechanics makes use of a free particle wavefunction exp(ipx) and weights a(p), but does not contain the usual P(x), P(p) picture of the classical world. We have shown in (2) that one may obtain P(x)=W*(x)W(x) and P(p)=a*(p)a(p) from the conditional probability. The essential physics we argue, however, is at the level of the wavefunction together with its own unique statistical scenario of exp(ipx)’s interacting with exp(ikx)’s from V(x)=Sum over Vkexp(ikx) (see (1)). Given that P(x) and P(p) exist for a quantum bound state, it may be convenient to use them in formulas because they are linked to a classical statistical picture. We argue that one may use P(x) and P(p) to construct an object S such that T dS does not change when one changes L, the size of box with infinite potential walls. We argue that such an object follows from properties of the Schrodinger equation together with various assumptions. We arrive at S=Sp+Sx where Sp=-k Integral dp a(p)a(p) ln{|a(p)a(p)}| and Sx= -k Integral dx W(x)W(x) ln(W(x)W(x)) for a given energy level. We do not use arguments related to taking ln of the number of states present or Shannon’s entropy. Rather we look for a solution which does not depend on L and is additive. We note, however, that for both W(x)W(x) and a(p)a(p) the solution found is mathematically consistent with the entropy form found by considering independent probabilities. This, however, applies to P(x) and P(p), but we argue that the physical situation is based on W(x) and a(p) as solutions of the Schrodinger equation. Nevertheless P(x) and P(p) are physical observables and not simply math constructs and so may think of them in a somewhat classical manner in terms of probability. Thus we argue that thermodynamics is already present at the level of a quantum bound state and that one may obtain Shannon’s entropy at this level using P(x) and P(p), but without considering the overall probability of independent probabilities as in information theory or numbers of states. Rather obtaining a function which is additive in Sp and Sx and eliminates L is the main consideration. We also argue that S=Sp+Sx carries over into mixed quantum state wavefunctions even though it is usually ignored. Thus we try to argue that it is the bound quantum state which gives rise to the notion of thermodynamics and entropy.
Entropy and free energy are central concepts in both statistical physics and information theory, with quantum and classical facets. In mathematics these concepts appear quite often in different contexts (dynamical systems, probability theory, von Neumann algebras, etc.). In this work, we study the von Neumann and Wehrl entropies from the point of view of semiclassical analysis. We first prove the semiclassical convergence of the von Neumann to the Wehrl entropy for quantum Gibbs states (thermal equilibrium), after a suitable renormalization has been taken into account. Then, we show that, in the same limit, the free energy functional defined with the Wehrl entropy $ Γ-$converges to its classical counterpart, so implying convergence of the minima and the associated minimizers.
The Laplace method is considered in application to one-dimensional two-state eigenvalue problems in the example of simple models, including that of coupled parabolic terms and radial oscillators with quadratic, linear and constant coupling (basic models of Renner, pseudo-Jahn-Teller effects etc.). In all cases, the solution can be presented either as a two-state analogue of the Bohr-Sommerfeld quasiclassical quantization rule, or in more general form. Compared with the semiclassical picture of inelastic collisions in the momentum-space representation, the contour of integration of 'trajectory equations' for the case of bound states is finite, and any desired degree of accuracy can easily be achieved. Because no convergence problems arise, the need to compute the equation possessing 'physical sense' (e.g. for adiabatic amplitudes) is circumvented, suggesting that the method would be applicable also to complicated potentials and to many-state problems. Numerical tests for the energy level positions and calculations of eigenfunctions are presented; practical applications and generalizations are outlined.
In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.