Quantum Mechanical Transition Probabilities Research Articles

Quantum work statistics in regular and classical-chaotic dynamical billiard systems.

In the thermodynamics of nanoscopic systems the relation between classical and quantum mechanical description is of particular importance. To scrutinize this correspondence we have chosen two two-dimensional billiard systems. Both systems are studied in the classical and the quantum mechanical settings. The classical conditional probability density p(E,L|E_{0},L_{0}) as well as the quantum mechanical transition probability P(n,l|n_{0},l_{0}) are calculated, which build the basis for the statistical analysis. We calculate the work distribution for one particle. The results in the quantum case in particular are of special interest since a suitable definition of mechanical work in small quantum systems is already controversial. Furthermore, we analyze the probability of both zero work and zero angular momentum difference. Using connections to an exactly solvable system analytical formulas are given for both systems. In the quantum case we get numerical results with some interesting relations to the classical case.

A generic approach to the quantum mechanical transition probability

In quantum theory, the modulus-square of the inner product of two normalized Hilbert space elements is to be interpreted as the transition probability between the pure states represented by these elements. A probabilistically motivated and more general definition of this transition probability was introduced in a preceding paper and is extended here to a general type of quantum logics: the orthomodular partially ordered sets. A very general version of the quantum no-cloning theorem, creating promising new opportunities for quantum cryptography, is presented and an interesting relationship between the transition probability and Jordan algebras is highlighted.

Quantum Probability's Algebraic Origin.

Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg’s and others’ uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.

Fermi\u2019s Golden Rule and the Second Law of Thermodynamics

We present a Gedankenexperiment that leads to a violation of detailed balance if quantum mechanical transition probabilities are treated in the usual way by applying Fermi’s “golden rule”. This Gedankenexperiment introduces a collection of two-level systems that absorb and emit radiation randomly through non-reciprocal coupling to a waveguide, as realized in specific chiral quantum optical systems. The non-reciprocal coupling is modeled by a hermitean Hamiltonian and is compatible with the time-reversal invariance of unitary quantum dynamics. Surprisingly, the combination of non-reciprocity with probabilistic radiation processes entails negative entropy production. Although the considered system appears to fulfill all conditions for Markovian stochastic dynamics, such a dynamics violates the Clausius inequality, a formulation of the second law of thermodynamics. Several implications concerning the interpretation of the quantum mechanical formalism are discussed.

Emission Intensity in the Hydrogen Atom Calculated from a Non-Probabilistic Approach to the Electron Transitions

Quantum aspects of the Joule-Lenz law for the transmission of energy allowed us to calculate the time rate of energy transitions between the quantum states of the hydrogen atom in a fully non-probabilistic way. The calculation has been extended to all transitions between p and s states having main quantum numbers not exceeding 6. An evident similarity between the intensity pattern obtained from the Joule-Lenz law and the corresponding quantum-mechanical transition pro-babilities has been shown.

Dynamics of disentanglement, density matrix, and coherence in neutrino oscillations

In charged current weak interaction processes, neutrinos are produced in an entangled state with the charged lepton. This correlated state is disentangled by the measurement of the charged lepton in a detector at the production site. We study the dynamical aspects of disentanglement, propagation, and detection, in particular, the conditions under which the disentangled state is a coherent superposition of mass eigenstates. The appearance and disappearance far-detection processes are described from the time evolution of this disentangled ``collapsed'' state. The familiar quantum mechanical interpretation and factorization of the detection rate emerges when the quantum state is disentangled on time scales much shorter than the inverse oscillation frequency, in which case the final detection rate factorizes in terms of the usual quantum mechanical transition probability provided the final density of states is insensitive to the neutrino energy difference. We suggest possible corrections for short-baseline experiments. If the charged lepton is unobserved, neutrino oscillations and coherence are described in terms of a reduced density matrix obtained by tracing out an unobserved charged lepton. The diagonal elements in the mass basis describe the production of mass eigenstates whereas the off-diagonal ones provide a measure of coherence. It is shown that coherences are of the same order of the diagonal terms on time scales up to the inverse oscillation frequency, beyond which the coherences oscillate as a result of the interference between mass eigenstates.

Algebraic description of the inelastic collision between an atom and a Morse oscillator in one dimension

The 1D collision between an atom and a diatomic molecule is investigated using an algebraic approach. Closed expressions for the quantum mechanical excitation transition probabilities are obtained. A Morse potential for the diatomic molecule is used. The classical trajectories are obtained in the united atom limit by taking an average over a finite number of states using the matrix density formalism, which allows the problem to be reformulated in terms of a time-dependent Hamiltonian. The system is studied in the interaction picture, which permits us to carry out an algebraic description through the approximation of the interaction potential in terms of a linear combination of the generators of the su(2) algebra. Comparisons with exact quantum mechanical results for transition probabilities in different systems are presented.

Topology of optimally controlled quantum mechanical transition probability landscapes

An optimally controlled quantum system possesses a search landscape defined by the physical objective as a functional of the control field. This paper particularly explores the topological structure of quantum mechanical transition probability landscapes. The quantum system is assumed to be controllable and the analysis is based on the Euler-Lagrange variational equations derived from a cost function only requiring extremizing the transition probability. It is shown that the latter variational equations are automatically satisfied as a mathematical identity for control fields that either produce transition probabilities of zero or unit value. Similarly, the variational equations are shown to be inconsistent (i.e., they have no solution) for any control field that produces a transition probability different from either of these two extreme values. An upper bound is shown to exist on the norm of the functional derivative of the transition probability with respect to the control field anywhere over the landscape. The trace of the Hessian, evaluated for a control field producing a transition probability of a unit value, is shown to be bounded from below. Furthermore, the Hessian at a transition probability of unit value is shown to have an extensive null space and only a finite number of negativemore » eigenvalues. Collectively, these findings show that (a) the transition probability landscape extrema consists of values corresponding to no control or full control, (b) approaching full control involves climbing a gentle slope with no false traps in the control space and (c) an inherent degree of robustness exists around any full control solution. Although full controllability may not exist in some applications, the analysis provides a basis to understand the evident ease of finding controls that produce excellent yields in simulations and in the laboratory.« less

Coherent dynamics on hierarchical systems

We study the coherent transport modeled by continuous-time quantum walks, focussing on hierarchical structures. For these we use Husimi cacti, lattices dual to the dendrimers. We find that the transport depends strongly on the initial site of the excitation. For systems of sizes N ⩽ 21 , we find that processes which start at central sites are nearly recurrent. Furthermore, we compare the classical limiting probability distribution to the long time average of the quantum mechanical transition probability which shows characteristic patterns. We succeed in finding a good lower bound for the (space) average of the quantum mechanical probability to be still or again at the initial site.

Coherent exciton transport in dendrimers and continuous-time quantum walks

We model coherent exciton transport in dendrimers by continuous-time quantum walks. For dendrimers up to the second generation the coherent transport shows perfect recurrences when the initial excitation starts at the central node. For larger dendrimers, the recurrence ceases to be perfect, a fact which resembles results for discrete quantum carpets. Moreover, depending on the initial excitation site, we find that the coherent transport to certain nodes of the dendrimer has a very low probability. When the initial excitation starts from the central node, the problem can be mapped onto a line which simplifies the computational effort. Furthermore, the long time average of the quantum mechanical transition probabilities between pairs of nodes shows characteristic patterns and allows us to classify the nodes into clusters with identical limiting probabilities. For the (space) average of the quantum mechanical probability to be still or to be again at the initial site, we obtain, based on the Cauchy-Schwarz inequality, a simple lower bound which depends only on the eigenvalue spectrum of the Hamiltonian.

Numerical study of the accuracy and efficiency of various approaches for Monte Carlo surface hopping calculations

A one-dimensional, two-state model problem with two well-separated avoided crossing points is employed to test the efficiency and accuracy of a semiclassical surface hopping technique. The use of a one-dimensional model allows for the accurate numerical evaluation of both fully quantum-mechanical and semiclassical transition probabilities. The calculations demonstrate that the surface hopping procedure employed accounts for the interference between different hopping trajectories very well and provides highly accurate transition probabilities. It is, in general, not computationally feasible to completely sum over all hopping trajectories in the semiclassical calculations for multidimensional problems. In this case, a Monte Carlo procedure for selecting important trajectories can be employed. However, the cancellation due to the different phases associated with different trajectories limits the accuracy and efficiency of the Monte Carlo procedure. Various approaches for improving the accuracy and efficiency of Monte Carlo surface hopping procedures are investigated. These methods are found to significantly reduce the statistical sampling errors in the calculations, thereby increasing the accuracy of the transition probabilities obtained with a fixed number of trajectories sampled.

A three-step model applied to free electron attachment to C60, Buckminsterfullerene

After a comparison of several experimental data sets, an analytical absolute cross-section for negative ion formation by low energy electron impact to C60 is derived. This cross-section is treated using a three-step model for electron attachment in which the electron is initially attracted by a polarizability potential with a Langevin capture rate, then it binds to the molecule after quantum mechanical energy transfer, and finally, following energy equilibration into vibrational modes, part of the electrons are lost due to auto-detachment. After removal of the Langevin and auto-detachment part from the experimental cross-section the energy dependent transition probability can be modelled by a series of Gaussian functions that are located at the allowed transition energies within the fullerene anion, thereby yielding the quantum mechanical transition probabilities. The transition probability can be compared to the density of valence states, if the electron affinity is correctly taken into account. This yields the energy dependence of the involved transition matrix elements. As a result, in C60 the transition matrix elements appear to be mostly independent of the transition, probably because Franck–Condon factors are of similar value over the entire energy range.

Theoretical model of photostructural changes in glassy semiconductors

A theoretical model is proposed which is able to describe the mechanism of photostructural changes in glassy semiconductors, involving negative-U centers as basic charge carriers, of which the electronic states are excited by gap light. This answers the old question of why such changes are only found in glassy semiconductors: negative-U centers are present only in these materials. The theory permits the calculation of the related quantum-mechanical transition probabilities.

Effects of the environment on nonadiabatic magnetization process in uniaxial molecular magnets at very low temperatures

We discuss the effect of the thermal environment on the low-temperature response of the magnetization of uniaxial magnets to a time-dependent applied magnetic field. At very low temperatures the stepwise magnetization curves observed in molecular magnets such as ${\mathrm{Mn}}_{12}$ and ${\mathrm{Fe}}_{8}$ display little temperature dependence where the apparent thermal assisted process are suppressed. We show that the changes of the magnetization at each step cannot be analyzed directly in terms of a quantum-mechanical nonadiabatic transition. In order to explain this nonadiabatic behavior, we study the quantum dynamics of the system weakly coupled to a thermal environment and propose a relation between the observed magnetization steps and the quantum-mechanical transition probability due to the nonadiabatic transition.

Doubly stochastic matrices in quantum mechanics

The general set of doubly stochastic matrices of order n corresponding to ordinary nonrelativistic quantum mechanical transition probability matrices is given. Lande's discussion of the nonquantal origin of such matrices is noted. Several concrete examples are presented for elementary and composite angular momentum systems with the focus on the unitary symmetry associated with such systems in the spirit of the recent work of Bohr and Ulfbeck. Birkhoff's theorem on doubly stochastic matrices of order n is reformulated in a geometrical language suitable for application to the subset of quantum mechanical doubly stochastic matrices. Specifically, it is shown that the set of points on the unit sphere in cartesian n!-space is surjective with the set of doubly stochastic matrices of order n. The question is raised, but not answered, as to what is the subset of points of this unit sphere that correspond to the quantum mechanical transition probability matrices, and what is the symmetry group of this subset of matrices.

The lifetimes of gas phase CO2˙− and N2O˙− calculated from the transition probability of the autodetachment process A− → A + e−

AbstractA procedure to calculate the quantum mechanical transition probability of a unimolecular primary chemical process, A− → A + e− is investigated for the circumstance where A− and A have different numbers of vibrational and rotational degrees of freedom (one is linear, the other not). A procedure is introduced to deal with the coupling between the vibrational and rotational motions. The proposed method was applied to calculating the lifetimes of CO2˙− and N2O˙− in the gas phase. The geometry optimizations and frequency calculations for CO2, CO2˙−, N2O, and N2O˙− are performed at HF, MP2, and QCISD(T) levels with 6‐31G* or 6–31+G* basis sets, in order to obtain reliable geometric and spectroscopic information on these systems. Lifetimes are calculated for several of the lower vibrational–rotational states of the anions, as well as for the Boltzmann distribution of states at 298 K. The lifetime of the lowest vibrational–rotational state of CO2˙−, is 1.03 × 10−4 s, and of the lowest vibrational state with rotational levels weighted by Boltzmann distribution at 298 K, 1.50 × 10−4 s. These values are in good agreement with the experimental number, 9.0 ± 2.0 × 10−5 s, and support the experimental evidence that CO2˙− was formed in its ground vibrational level by the techniques used. The lifetime of CO2˙− calculated with Boltzmann distribution over its vibrational and rotational levels at 298 K, is 1.51 × 10−5 s. There are no direct measurements of the lifetime of N2O˙−, but it was estimated to be greater than 10−4 s from experimental evidence. The predicted lifetimes of N2O˙−, at its lowest vibrational–rotational state (0 K) and lowest vibrational state with rotational levels weighted by the Boltzmann distribution at 298 K, are 238 and 19.1 s, respectively. The lifetime of N2O˙− at thermal equilibrium at 298 K is 6.66 × 10−2 s, indicating that electron loss from the excited vibrational states of N2O˙− is significant. This study represents the first theoretical investigation of CO2˙− and N2O˙− lifetimes. © 1994 John Wiley & Sons, Inc.

The classical limit of ionization in fast ion-atom collisions

The authors study the classical-quantum correspondence for ionization in three-body fast ion-atom collisions. The existence of a classical limit of the quantum mechanical transition probabilities as a function of the momentum transferred to the electron during the collision, the impact parameter, the energy and angle of the emitted electron, and the initial quantum level of the target is investigated. A well behaved classical limit is shown to exist for large momentum transfers whereas ionization by small momentum transfers or at large impact parameters is shown to be classically suppressed. It is found that ionization at small impact parameters does not possess a well-defined classical limit. Modifications of classical trajectory Monte Carlo (CTMC) methods to account for non-classical ionization are suggested. Applications related to recent data for electron ejection into large angles are presented.

Rotational-vibrational rainbows in impulsive electron ? diatomic molecule collisions: I. state-to-state transitions

Collisional induced combined rotational-vibrational excitation of diatomic molecules is discussed in a simple quantum mechanical spectator model with applications to electron-molecule collisions at intermediate collision energies (≈ 102 eV) within the rigid rotor/harmonic oscillator approximation. Quantum mechanical transition probabilities of the rotational-vibrational excitation, which show typical vibrational and rotational rainbow patterns, are calculated and compared with the structure of classical rainbow singularities.

Converged quantum-mechanical calculations of electronic-to-vibrational, rotational energy transfer probabilities in a system with a conical intersection

We present quantum-mechanical transition probabilities for the process Na(3p 2P) + H 2( v 0 = 0, j 0 = 0,2) → Na(3s 2S) + H 2( v′, j′) with zero total angular momentum where v 0, j 0, v′, and j′are initial and final vibrational and rotational quantum numbers. These calculations involve two coupled potential energy surfaces at a total energy of 2.431 eV, and the excited state has energetically accesible geometric configurations with HH internuclear distances 2.7 times larger than the H 2 equilibrium internuclear distance - which makes converged quantum-mechanical calculations very difficult. Covergence is demonstrated by obtaining the same results using two entirely different methods - R matrix propagation and the outgoing wave variational principle.

The transition probability of electron loss from anions in the gas phase: The lifetime of O2 ⋅−

The unimolecular primary chemical process, A−→A+e−, is investigated from a theoretical and computational point of view. A ‘‘standard’’ procedure for calculating the lifetime of A− directly is proposed based on the quantum mechanical transition probability. Specific application of the standard procedure is made to electron loss from O ⋅−2. The lifetime of O ⋅−2(ν′=4) calculated with the proposed method is 36×10−12 s, in good agreement with experimental data. An improvement to the standard procedure yields the best theoretical estimate for the lifetime, 125×10−12 s. The lifetimes of O ⋅−2 at higher vibrationally excited states are also calculated and agree well with those of previous calculations. The dependence of the calculated lifetime on the geometric and vibrational parameters of O2 and O ⋅−2 is discussed.