This paper suggests a new way to compute the path integral for simple quantum mechanical systems. The new algorithm originated from previous research in string theory. However, its essential simplicity is best illustrated in the case of a free non relativistic particle, discussed here, and can be appreciated by most students taking an introductory course in Quantum Mechanics. Indeed, the emphasis is on the role played by the {\it entire family of classical trajectories} in terms of which the path integral is computed exactly using a functional representation of the Dirac delta-distribution. We argue that the new algorithm leads to a deeper insight into the connection between classical and quantum systems, especially those encountered in high energy physics.
We present a derivation of the propagator for a particle in the presence of the step and delta function potentials. These propagators are known, but we present a direct path integral derivation, based on the path decomposition expansion and the Brownian motion definition of the path integral. The derivation exploits properties of the Catalan numbers, which enumerate certain classes of lattice paths.
We present a direct path integral derivation of the propagator in the presence of a step potential. The derivation makes use of the Path Decomposition Expansion (PDX), and also of the definition of the propagator as a limit of lattice paths.
This paper presents an application of white noise functional approach to derive the quantum propagator and the evolution of the reduced density matrix for an open quantum system consisting of coupled harmonic oscillators which are coupled to a bath consisting of a multimode harmonic oscillator. It is shown that the full quantum propagator is a product of three individual propagators. These propagators were obtained after two successive transformations of the coordinates for the system, and the coordinates for the coupling between the bath and one of the two harmonic oscillators in the system. …
A novel derivation of quantum propagator of a system described by a general quadratic Lagrangian is presented in the framework of Heisenberg equations of motion. The general corresponding density matrix is obtained for a derived quantum harmonic oscillator and a particle confined in a one dimensional Paul trap. Total mean energy, work and absorbed heat, Wigner function and excitation probabilities are found explicitly. The method presented here is based on the Heisenberg representation of position and momentum operators and can be generalized to a system consisting of a set of linearly interac…
Starting from a total Lagrangian describing an oscillator-bath system, a novel derivation of exact quantum propagator is presented. Having the quantum propagator, the exact density matrix, reduced density matrix of the main oscillator and thermal equilibrium fixed point are obtained. The problem is generalised to the cases where the main oscillator is under the influence of a classical external force. By introducing generalised auxiliary classical fields, the generalised quantum propagator or generating functional of position correlation functions is obtained.
We introduce a notion of isolated units, elementary particles or more general physical phenomena that do not significantly affect their surrounding environment, and we build a primitive ontology to describe their evolution and interaction. We further introduce a notion of indistinguishability of distinct spacetime paths of a unit, for which the evolution of the state variables of the unit is the same, and a generalization of the equivalence principle based on indistinguishability. Under a time invertibility condition on the whole set of indistinguishable paths of a unit, we show that the quant…
Quantum computing takes fully advantage of the superposition principle to increase greatly (even exponentially) the speed of calculations, relative to the classical approach. The Deutsch-Jozsa algorithm is the simplest quantum algorithm illustrating this power. Unfortunately, the standard derivation involves several ingenious steps which usually leave students feeling that they could never have figured out the algorithm by themselves. We present here a different formulation of the problem which allows students to derive the algorithm using only basic knowledge of quantum mechanics and linear a…
We present a detailed derivation of the semiclassical propagator in the SU(n) coherent state representation. In order to provide support for immediate physical applications, we restrict this work to the fully symmetric irreducible representations, which are suitable for the treatment of bosonic dynamics in n modes, considering systems with conservation of total particle number. The derivation described here can be easily extended to other classes of coherent states, thus representing an alternative approach to previously published methods.
When a quantum system undergoes unitary evolution in accordance with a prescribed Hamiltonian, there is a class of states |psi> such that, after the passage of a certain time, |psi> is transformed into a state orthogonal to itself. The shortest time for which this can occur, for a given system, is called the passage time. We provide an elementary derivation of the passage time, and demonstrate that the known lower bound, due to Fleming, is typically attained, except for special cases in which the energy spectra have particularly simple structures. It is also shown, using a geodesic argument, t…
There is no known exact expression for the propagator of a non-relativistic particle colliding with a hard sphere. De Prunelé (2008 {\it J.~Phys.~A:~Math.~Theor.} {\bf 41} 255305) derived a partial wave expansion of the propagator and compared it against some known approximations, including the semiclassical Van Vleck-Gutzwiller (VG) propagator; the VG propagator was evaluated entirely numerically. Here we point out that the VG propagator for the particle-sphere problem admits an analytic expression in terms of elementary functions.
We present a novel derivation of the original Levitov formula, for the statistics of charge transported between electron reservoirs. This is done by proving a trace formula, which relates certain traces in Fock space to single particle determinants. Using the present approach we find in addition several generalizations, such as a corresponding formula for Bosons.
We present a complete derivation of the semiclassical limit of the coherent state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial value representation for the semiclassical propagator, based on an initial gaussian wavepacket. It tu…
A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman. The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later. In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform. The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be ge…
A detailed derivation of the semiclassical propagator in the generalized coherent-state representation is performed by applying the saddle-point method to a path integral over the classical phase space. With the purpose of providing greater accessibility and applicability to the developed formalism, a brief review of the generalized concept of coherent states is presented, in which three examples of coherent-state sets are examined, namely, the canonical, spin, and SU(n) bosonic coherent states.
A quantum mechanical path integral derivation is given of a thermal propagator in non-static Gui spacetime. The thermal nature of the propagator is understood in terms of homotopically non-trivial paths in the configuration space appropriate to tortoise coordinates. The connection to thermal emission from collapsing black holes is discussed.
We present a quantum field theoretical derivation of the nondecay probability of an unstable particle with nonzero three-momentum $\mathbf{p}$. To this end, we use the (fully resummed) propagator of the unstable particle, denoted as $S,$ to obtain the energy probability distribution, called $d_{S}^{\mathbf{p} }(E)$, as the imaginary part of the propagator. The nondecay probability amplitude of the particle $S$ with momentum $\mathbf{p}$ turns out to be, as usual, its Fourier transform: $a_{S}^{\mathbf{p}}(t)=\int_{\sqrt{m_{th} ^{2}+\mathbf{p}^{2}}}^{\infty}dEd_{S}^{\mathbf{p}}(E)e^{-iEt}$ ($m_…
Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we re-formulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it…
Recently quantum walks have been considered as a possible fundamental description of the dynamics of relativistic quantum fields. Within this scenario we derive the analytical solution of the Weyl walk in 2+1 dimensions. We present a discrete path-integral formulation of the Feynman propagator based on the binary encoding of paths on the lattice. The derivation exploits a special feature of the Weyl walk, that occurs also in other dimensions, that is closure under multiplication of the set of the walk transition matrices. This result opens the perspective of a similar solution in the 3+1 case.
The path decomposition expansion is a path integral technique for decomposing sums over paths in configuration space into sums over paths in different spatial regions. It leads to a decomposition of the configuration space propagator across arbitrary surfaces in configuration space. It may be used, for example, in calculations of the distribution of first crossing times. The original proof relied heavily on the position representation and in particular on the properties of path integrals. In this paper, an elementary proof of the path decomposition expansion is given using projection operators…
We present a semiclassical approach to many-body quantum propagation in terms of coherent sums over quantum amplitudes associated with the solutions of corresponding classical nonlinear wave equations. This approach adequately describes interference effects in the many-body space of interacting bosonic systems. The main quantity of interest, the transition amplitude between Fock states when the dynamics is driven by both single-particlecontributions and many-body interactions of similar magnitude, is non-perturbatively constructed in the spirit of Gutzwiller's derivation of the van Vleck propa…
Master equations describe the quantum dynamics of open systems interacting with an environment. They play an increasingly important role in understanding the emergence of semiclassical behavior and the generation of entropy, both being related to quantum decoherence. Presently we derive the exact master equation for a homogeneous scalar Higgs or inflaton like field coupled to an environment field represented by an infinite set of harmonic oscillators. Our aim is to demonstrate a derivation directly from the path integral representation of the density matrix propagator. Applications and general…
We present a derivation of the Landau-Zener solution through the explicit evaluation of the time ordered propagator. The result is exact and does not involve the solution of the differential equation for the spin amplitudes.
The Lindblad equation is an evolution equation for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad equation are given. The derivation of the Lindblad equation presented here is "simple" in that all it uses is the expression of a hermitian matrix in terms of its orthonormal eigenvectors and real eigenvalues. Thus, it is appropriate for students who have learned the algebra of quantum theory. Where helpful, arguments are first giv…
Quantum theory can be derived from purely informational principles. Five elementary axioms-causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning-define a broad class of theories of information-processing that can be regarded as a standard. One postulate-purification-singles out quantum theory within this class. The main structures of quantum theory, such as the representation of mixed states as convex combinations of perfectly distinguishable pure states, are derived directly from the principles without using the Hilbert space framework.
