Ever since werner heisenbergs 1927 paper on uncertainty, there has been considerable hesitancy in simultaneously considering. In particular, we should only expect that classical statements agree. After all, there is more to quantum mechanics than classical phase space can tell us. Quantum mechanics in phase space world scientific series. Pdf continuous phase spaces and the time evolution of spins. As a consequence, this is called the moyal formalism or sometimes the weylwignermoyal formalism of q. Negative values in these distributions imply nonclassical behavior in the sense that the system necessarily exhibits coherent superpositions of states corresponding to different measurement outcomes. Phase space in quantum mechanics and heisenberg uncertainty principle. Thus, we will present in this contribution methods to visualize quantum dynamics in phase space. It represents an expansion of a previous overview with selected papers collected by the authors, and includes a historical narrative account due the subject. An overview with selected papers world scientific series in 20th century physics 20051212 on. On the representation of quantum mechanics on phase space.
Quantum mechanics in phase hilbrand groenewold and jos. In classical mechanics, the phase space is the space of all possible states of a physical. A mixture of quantum states is again a quantum state. Overview of phasespace quantization quantum mechanics in.
This invaluable book is a collection of the seminal papers on the formulation, with an introductory overview which provides a trail map for those papers. Pdf continuous phase spaces and the time evolution of. These advantages arise in the contexts of discussion of the theoretical framework and of descriptions of laboratory experiments. An overview with selected papers find, read and cite all the research you need on researchgate. Mar 01, 2009 i am not quite sure i agree with that description. An overview with selected papers, world scien tific, 2005. While the formalism for quantum mechanics in phase space was wholly cast at that time, it was not completely understood nor widely known much less generally accepted until the late 20th. As any generalization o f a concept is nonunique, a whole collection of different proposals appeared in the literature see for example. Owing to its peculiar properties, the wigner quasidistribution attracted attention of numerous researches. View quantum mechanics research papers on academia. Moshinsky atom and density functional theory a phase space. Geometry and motion of the wigner distribution by surya ganguli submitted to the department of electrical engineering and computer science and departments of physics and mathematics on may 26, 1998, in partial fulfillment of the requirements for the degrees of master of engineering and. It has been useful in describing transport in quantum optics.
The formulation of quantum mechanics in terms of phase. It is shown that the quantization naturally arises as an appropriate deformation of the classical hamiltonian mechanics. Quantum mechanics on phase space and the coulomb potential. Phase space formulation of quantum mechanics tony bracken centre for mathematical physics and department of mathematics university of queensland smft07, melbourne, januaryfebruary 2007. Quantum optics in phase space has been successful in preparing students and researchers for their quest of unravelling the mysteries of nature. Advantages of quantum mechanics on phase space springerlink. The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. If the address matches an existing account you will receive an email with instructions to reset your password.
In einstein submitted an essay for publication in scientific papers. The formulation of quantum mechanics in terms of phase space functionsthe third equation. The curious quantum mechanics of pre and postselected. Ever since werner heisenbergs 1927 paper on uncer tainty, there has been considerable hesitancy in simultaneously considering positions and momenta in quantum contexts, since these are incompatible observables. The problem of two particles in a common harmonic oscillator potential interacting through harmonic oscillator forces is discussed in the weylwigner phasespace representation. Apr 27, 2011 while the formalism for quantum mechanics in phase space was wholly cast at that time, it was not completely understood nor widely known much less generally accepted until the late 20th century.
Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Pdf on jan 1, 2005, cosmas zachos and others published quantum mechanics in phase space. World scientific series in 20th century physics quantum mechanics in phase space, pp. Advanced undergraduates, beginning graduate students and researchers in physics, quantum computing. On the other hand, if they feel completely lost in all the di. Selecting contours as indicated in the preceding figure, one obtains. It works in full phase space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. The map between function in the phase space formulation of quantum mechanics and the operators in the corresponding hilbert space is known as wignerweyl transform. Symplectic quantum mechanics sqm considers a noncommutative algebra of functions on a phase space. An overview with selected papers world scientific on free shipping on qualified orders. It represents an expansion of a previous overview with selected papers collected by the authors, and includes a. Wigner s quasiprobability distribution function in phase space is a special weyl representation of the density matrix.
The formulation of quantum mechanics in terms of phase space functions volume 60 issue 3 d. Familiar quantities such as energy, momentum, electric charge, mass possibly even time and space are not continuous. Its account of quantum jumps supplanted the bohr models electron orbits. Chapter 6 also demonstrates that thermodynamics is a straightforward consequence of quantum mechanics and that we no longer need to derive the laws of thermodynamics through the traditional, rather subtle, arguments about heat engines. An overview with selected papers world scientific series in 20th century physics 20051212. Wigners quasiprobability distribution function in phase space is a special weylwigner representation of the density matrix. Phase space in quantum mechanics and heisenberg uncertainty.
Quantum mechanics in phase space by thomas l curtright. We then argue that this is a consequence of the noncommutative geometry. Quantum mechanics in phase space postgraduate study. In addition, there is a very useful overview, consisting of an introduction of 30 pages with an emphasis on the wigner quasiprobability. In the phase space formulation, the phase space distribution may be treated as the fundamental, primitive description of the quantum system, without any reference to wave functions or density matrices. Unlike parameter space such as phase space, hilbert space is an in.
Ever since werner heisenbergs 1927 paper on uncertainty, there has been considerable hesitancy in simultaneously considering positions and momenta in quantum contexts, since these are incompatible observables. In the phasespace formulation, the phase space distribution may be treated as the fundamental, primitive description of the quantum system, without. The phase space formulation of quantum mechanics radiates from a definition. Classical phase space is defined as a space in which all possible states are represented. Matrix mechanics is a formulation of quantum mechanics created by werner heisenberg, max born, and pascual jordan in 1925. If momentum space is the space where both the cartans hypothesis on the spinorial origin of euclidean geometry and the spinorial origin of quantum mechanics might be naturally realized, configuration space, and in particular minkowski space m ir31, remains the space where the. Curtright wigners quasiprobability distribution function in phase space is a special weyl representation of the density matrix. Pre and post selected ensembles in quantum mechanics 453 placed gaussian wavepackets re2 out of phase.
A third way is the formulation of quantum mechanics on phase space also known as the moyal quantization or the deformation quantization which is grounded on wigners quasidistribution function and weyls correspondence between ordinary cnumber functions in phase space and quantum mechanical operators in hilbert space. Summary of important ideas in quantum physics 1 the universe is quantized. One finds that classical volume of phase space qm number of states h or in other words, each quantum state corresponds to a classical. We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time. Quantum mechanics formulated in terms of wave functions over phase space is shown to have numerous advantages over the standard approach. We develop a general framework to describe the fluctutions of arbitrary noncommuting observables by quasiprobability distributions. A quantum mechanical representation in phase space go. Phase space in physics, phase space is a concept which uni. In quantum mechanics in phase space, by zachos, fairlie and curtright, the bulk of the book of 551 pages consists of reprints of a selection of 23 papers dealing with the phase space formalism. The origin of a formulation of quantum mechanics qm on phase space lies on a paper by moyal 1 written in the forties.
It is shown that hilbert space quantum mechanics can be represented on phase space in the sense that the density operators can be identified with phase space densities and the observables can be described by functions on phase space. In order to establish a relation between quantum dynamics and classical dynamics, the phase space seems to be more appropriate because classical hamiltonian mechanics is formulated in phase space. Phase space formulation of the quantum mechanical particleinabox problem nicholas wheeler, reed college physics department december 2000 introduction. However, with the advent of geometric quantum mechanics.
The wigner function, was introduced as a fourier transform of the density matrix. Their combined citations are counted only for the first. Phase space quantum mechanics the physics travel guide. Mathematically, a pure quantum state can be represented by a ray in a hilbert space over the complex numbers. An overview with selected papers isbn 9789812383846 hos adlibris. It has been useful in describing quantum transport in quantum optics nuclear physics decoherence, quantum computing, and quantum.
The remainder of this paper is organized as follows. This approach naturally relates phase spaces of increasing spin number to their quantum optical limit and allows for efficient approximations of the time evolution for large spin numbers. But this persistent discomfort with addressing positions and momenta jointly in the quantum world is not really warranted, as was. Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics in phase space, but finite dimensional systems.
I think it is better to say that there is no such thing as absolute phase in qm or in any other field in physics meaning we are free to choose whatever value we want as a starting point. The formulations of quantum mechanics on phase space are based on wigner quasidistribution function and weyls correspondence between ordinary cnumber functions in phase space and quantummechanical operators in hilbert space. Visualizing quantum mechanics in phase space arxiv. While the formalism for quantum mechanics in phase space was wholly cast at that time.
An overview with selected papers world scientific at. But this persistent discomfort with addressing positions and momenta jointly in the quantum world is not really warranted, as was first fully appreciated by hilbrand groenewold and. More precisely, the deformation of the pointwise product of observables to an appropriate noncommutative. Quasiprobability distributions for observables in dynamic. This allows us to associate with s an ordinary quantum system q such that the values of the quantized energy coincide for the fields in the phase space of s and for q. The phase space distribution fx, p of a quantum state is a quasiprobability distribution. Wave mechanics in classical phase space, brownian motion. The density functional description of the system is touched upon.
Since the early years of the development of quantum mechanics, its formulation in phase space has been troublesome. Pdf quantum mechanics in phase space cosmas zachos and. Pdf ever since werner heisenbergs 1927 paper on uncertainty, there has been considerable. A promising technology is the quantum computers, and this paper gives a general overview about this subject. Overview of phasespace quantization quantum mechanics. Niels bohr used to say that if you are not confused by quantum mechanics, then you really do not understand it. In the phase space formulation, the transition from classical mechanics to quantum mechanics is known as deformation quantization.
Rather than abandoning all hope of constructing such a phasespace for quantum phenomena, we take aspects from both the wignermoyal and bohm approaches and show that although there is no unique phase space we can form shadow phase spaces. Thomas curtright wigners quasiprobability distribution function in phase space is a special weyl representation of the density matrix. It is no surprise that we obtain destructive interference resulting in a measured state with a large component of momentum in the z direction. The wigner function of the system is an ordinary function of the phasespace constants of the motion. The seminal paper by wigner in 1932 addressed such a problem in an attempt to deal with the superfluidity of helium. This can often b e done b y analogy with the corresp onding classical problem, thereb y pro viding a direct route for the quan tum classical. Ouantum mechanics in phase space an overview with selected papers world scientific series in 20th century physics published vol. It has been useful in describing quantum transport in quantum optics. Quantum states that cannot be written as a mixture of other states are called pure quantum states, all other states are called mixed quantum states. Quantum mechanics in phase space world scientific series in. Calculate the volume of phase space in this case, the area of the twodimensional phase space, and compare this classical system to the corresponding qm system, in a semiclassical limit. Every state corresponds to a unique point in the phase space on the other hand, in quantum mechanics every quantum mechanical observable corresponds to a unique function distribution on phase space. It might seem that with the construction of the local observables def. An overview with selected papers world scientific on.