Scattering of waves in the phase space, quantum mechanics. Thus, by expressing quantum mechanics in phase space the same ambit as for classical mechanics, the weyl map facilitates recognition of quantum mechanics as a deformation generalization of classical mechanics, with deformation parameter hs, where s is the action of the relevant process. Quantum optics in phase space has been successful in preparing students and researchers for their quest of unravelling the mysteries of nature. Hilbert space or by a quasiprobability distribution in the phase space. The second quantum number is often called the azimuthal quantum number l. Landsman, title quantum mechanics on phase space, year 1999. Quantum experiment in space confirms that reality is what. Quantum mechanics in phase space asia pacific physics. While the space qmps thereby gives a third point of view which formalism. Wigner functions density matrices in a special weyl. Fairlie skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Quantum experiment in space confirms that reality is what you. In classical mechanics, any choice of generalized coordinates q i for the position i.
This other quantum framework is equivalent to both the hilbert space approach and the path integral formulation. The formulation of quantum mechanics in terms of phase space functions volume 60 issue 3 d. Inner products, dual space, orthonormal bases, operators, operators as matrices in a given basis, adjoint operators, operator examples. The three pictures of quantum mechanics heisenberg in the heisenberg picture, it is the operators which change in time while the basis of the space remains fixed.
Klauder y departments of physics and mathematics university of florida gainesville, fl 32611 abstract phase space is the state space of classical mechanics, and this manifold is normally endowed only with a symplectic form. Adding a constant to the potential introduces a phase factor. Quantum mechanics professor allan adams massachusetts institute of technology 20 march 5. Quantum like classi96 cal mechanics in noncommutative phase space 2 mechanics with the principle of superposition of states 5, although experiments and theory show that superpositions of classical fields have the same phase space signature as quantum cat states 6. It is written at an introductory level, drawing on the remarkable history of the subject for inspiration and motivation. Quantum mechanics in coherent algebra on phase space. The operator h, which is an observable in this sense, generates time translations of ya2i. This we achieve by studying more thoroughly the structure of the space that underlies our physical objects, which as so often, is a vector space, the hilbert space. One might preclude that for a continuous phase space, any finite volume would contain an infinite number of states. One finds that classical volume of phase space qm number of states h or in other words, each quantum state corresponds to a classical. It is shown that the quantization naturally arises as an. These lecture notes were prepared by andrew turner, who was the teaching assistant ta for the class. A reformulation of quantum mechanics for a finite system is given using twisted multiplication of functions on phase space and tomitas theory of generalized.
The phase space rep is di erent in character not equivalent to the above prominent in recent years for applications to quantum optics, quantum information theory, quantum tomography, also for questions re foundations of qm and classical mechanics cm qm as a deformation of cm, the nature of the qmcm interface. The phase space distribution fx, p of a quantum state is a quasiprobability distribution. The value of l describes the shape of the region of space occupied by the electron. Classical phase space is defined as a space in which all possible states are represented. A function of the phase space coordinates of the system and time has total time derivative. And also gives you the possibility to get an associated operator, and a coordinateindependent sympleticgeometrical interpretation. Using classical statistics, schrodinger equation in quantum mechanics is derived from complex space model. Thus, teaching quantum mechanics in phase space may have some advantages when moving from classical to quantum. 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. Darooneh department of physics, university of zanjan, znu, 45378791, zanjan, iran. Hamiltonian mechanics gives nice phase space unified solutions for the equations of motion.
Chapter 3 mathematical formalism of quantum mechanics. Phase space geometry in classical and quantum mechanics john r. Oct 27, 2017 quantum experiment in space confirms that reality is what you make it. Trajectories in phase space classical and quantum trajectories p q. Tsallis entropy in phasespace quantum mechanics parvin sadeghi, siamak khademi, and amir h.
It has been useful in describing transport in quantum optics. Since quantum mechanics in phase space 369 v,w is generated by 1a2i, we will also consider a quantum observable any selfadjoint operator on h, affiliated with yaw. Time evolution of energy eigenfunctions through complex phase e. An odd space experiment has confirmed that, as quantum mechanics says, reality is what. The allowed values of l depend on the value of n and can range. Lecture notes quantum theory i physics mit opencourseware.
The state of a quantum particle can only be described by a vector on the. Pdf quantum mechanics in phase space cosmas zachos and. The phase space rep is di erent in character not equivalent to the above prominent in recent years for applications to quantum optics, quantum information theory, quantum tomography, also for questions re foundations of qm and classical mechanics cm qm as. Quantum mechanics is formulated on a quantum mechanical phase space. The basics of the wigner formulation of quantum mechanics and few related interpretational issues are presented in a simple language.
The state of a system of particles corresponds to a certain distribution of points in phase space. 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. This book gives a comprehensive introduction to modern quantum mechanics, emphasising the underlying hilbert space theory and generalised function theory. Phase space is a fundamental concept in hamiltonian mechanics and phase space formulations of quantum mechanics are therefore of great interest in order to compare quantum and hamiltonian mechanics, for. Quantum experiment in space confirms that reality is what you make it. Phasespace distribution functions for a nonrelativistic quantum particle are defined as the mean value of certain operators, whose most general forms are. These are based on the effective local field due to fixed neighbors in classical phase space, while the averages account for the noncommutativity of the position and momentum operators. Postal 20364, 0 mexico, df, mexico received 30 august 1983, in final form 29 may 1985 abstract. Quantum mechanics formulated in terms of wave functions over phase space is shown to have numerous advantages over the standard approach. All the major modern techniques and approaches used in quantum mechanics are introduced, such as berry phase, coherent and squeezed states, quantum computing, solitons and quantum mechanics. The aim here is to use the example of the sterngerlach experiment to develop this idea further, and to show that the. This is a text on quantum mechanics formulated simultaneously in terms of position and momentum, i. N 1 and n 2 form the heisenberg group in phase space, i.
Phase space geometry in classical and quantum mechanics. Schrodinger equation and phase space in quantum mechanics. We give an example of a mathematical model describing quantum mechanical processes interacting with medium. Advantages of quantum mechanics on phase space springerlink.
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. In relativistic scattering phase space generalizes to multiple particles in the final state and has a delta function for 4momentum conservation. In quantum mechanics however the state of a system is represented not by a point in 2n dimensional phase space, but by a state vector in the complex hilbert space of square integrable functions over i7n. Phase space probability amplitude, that can be defined on classical point of view, has connections to probability amplitude in internal space and to wave function in quantum mechanics. Visualizing quantum mechanics in phase space arxiv. For scattering processes, we assume that we have a free particle in the initial state before scatter. Moyal, quantum mechanics as a statistical theory, proc. Quantum mechanics in phase hilbrand groenewold and jos. Theory and examples of quantum mechanics on phase space.
Thomas l curtright, cosmas k zachos submitted on 27 apr. It is shown that the quantization naturally arises as an appropriate deformation of the classical hamiltonian mechanics. 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. Phase space in physics, phase space is a concept which uni. The time evolution of this state vector, or wavefunction, is generated by a selfadjoint. More abstractly, in classical mechanics phase space is the cotangent bundle of configuration space, and in this interpretation the procedure above expresses that a choice of. 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. Pdf quantum mechanics in hilbert space download full. The phase space quantum mechanics is such a natural formulation of quantum theory. Quantum effects lead to an entanglement of the ensemble.
Poisson brackets and constants of the motion the poisson bracket. Oneparticle energy eigenfunctions are used to obtain quantum averages in many particle systems. This formulation has extensive applications in quantum optics and in mixed quantum classical formulations. Wigners quasiprobability distribution function in phase space is a special weylwigner representation of the density matrix. A weird example in quantum mechanics, the fundamental postulates of quantum mechanics, hilbert spaces. Pdf on jan 1, 2005, cosmas zachos and others published quantum mechanics in phase space. Basic quantum mechanics in coordinate, momentum and. A concise treatise on quantum mechanics in phase space.
Niels bohr used to say that if you are not confused by quantum mechanics, then you really do not understand it. The counting of the number of states available to a particle amounts to determining the available volume in phase space. If, for a phase space function that is, no explicit time dependence then is a constant of the motion. As a model, we consider the process of heat scattering. However, due to the uncertainty principle caution is required when one wants to consider position and. The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. Scattering of waves in the phase space, quantum mechanics, and irreversibility. To explain this, let us first consider a scalar non spinorial density. Phase space is a fundamental concept in hamiltonian mechanics and phasespace formulations of quantum mechanics are therefore of great interest in order to compare quantum and hamiltonian mechanics, for. Pdf tsallis entropy in phasespace quantum mechanics. These advantages arise in the contexts of discussion of the theoretical framework and of descriptions of laboratory experiments. If each point in the phase space is considered as a random quantity with a particular probability ascribed to every possible state i.
The former is crucial in quantum mechanics, the later is. We can write the quantum phase space generators 11 as. The formulation of quantum mechanics in terms of phase. The quantum kinetic phase space description of the system is based on the wigner formulation of quantum mechanics 9. The phasespace formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. 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. Such a set of phase points is called a phase space ensemble. Thomas l curtright, cosmas k zachos submitted on 27 apr 2011, last revised 26 dec 2011 this version, v2. Chapter 8 vector spaces in quantum mechanics we have seen in the previous chapter that there is a sense in which the state of a quantum system can be thought of as being made up of other possible states.
Taati 1,3 1 institute for advanced studies in basic sciences, zanjan, 451951159, iran 2department of physics, zanjan university zanjan iran 3department of physics, university of kurdistan, d78457 sanadaj, iran. The mathematical formalism of quantum mechanics in phase space in terms of the wigner function resembles classical hamiltonian mechanics of noninteracting classical particles. The two key features of the phasespace formulation are that the quantum state is described by a quasiprobability distribution instead of a. An odd space experiment has confirmed that, as quantum mechanics says, reality is. On coherentstate representations of quantum mechanics. Ideally suited to a oneyear graduate course, this textbook is also a useful reference for researchers. The idea behind the phase space quantum mechanics relies on a deformation, with respect to some parameter h the planck constant, of a geometrical structure of a classical phase space m. Quantum statistical mechanics in classical phase space. As each member of the ensemble moves through phase space along a trajectory specified by hamiltons equations of motion. Adding two spin12 systems product and totals bases. Institute for quantum studies chapman university quantum stud math.
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