Quantum Guarded Command Language (qGCL) was defined by P. Zuliani in his PhD thesis. It is based on Guarded Command Language created by Edsger Dijkstra. It can be described as a language of quantum programs specification. QMASM. Quantum Macro Assembler (QMASM) is a low-level language specific to quantum annealers such as the D-Wave Global Young Scientists Summit Jan Call for applications. Published: — The EPFL is looking for twenty outstanding and highly motivated PhD students or post-docs to represent the School at the 10th anniversary edition of the Global Young Scientists Summit (GYSS), taking place online from 18 to 21 January A DPhil (PhD) in Atomic and Laser Physics involves some of the most rapidly developing areas of physics ranging from the fundamental physics of quantum systems to high energy density plasmas and other interdisciplinary applications of lasers
Phase-space formulation - Wikipedia
The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. In contrast, the Schrödinger picture uses the position or momentum representations see also position and momentum space. The two density functional theory phd thesis features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution instead of a wave functionstate vectoror density matrix and operator multiplication is replaced by a star product.
The theory was fully developed by Hilbrand Groenewold in in his PhD thesis, [1] and independently by Joe Moyal[2] each building on earlier ideas by Hermann Weyl [3] and Eugene Wigner, density functional theory phd thesis.
The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar density functional theory phd thesis Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space ". Quantum mechanics in phase space is often favored in certain quantum optics applications see optical phase spaceor in the study of decoherence and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.
The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as Kontsevich's deformation-quantization see Kontsevich quantization formula and noncommutative geometry. The phase-space distribution f xp of a quantum state is a quasiprobability distribution.
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. There are several different ways to represent the distribution, all interrelated. Since the Wigner representation is the most common, this article will usually stick to it, unless otherwise specified. The phase-space distribution possesses properties akin density functional theory phd thesis the probability density in a 2 n -dimensional phase space.
For example, it is real-valuedunlike the generally complex-valued wave function. We can understand the probability of lying within a position interval, for example, density functional theory phd thesis, by integrating the Wigner function over all momenta and over the position interval:. If  xp is an density functional theory phd thesis representing an observable, it may be mapped to phase space as A xp through the Wigner transform.
Conversely, this operator may be recovered by the Weyl transform. The expectation value of the observable with respect to the phase-space distribution is [2] [15]. A point of caution, however: despite the similarity in appearance, W xp is not a genuine joint probability distributionbecause regions under it do not represent mutually exclusive states, as required in the third axiom of probability theory.
Moreover, it can, in general, take negative values even for pure states, with the unique exception of optionally squeezed coherent statesin violation of the first axiom. Regions of such negative value are provable to be "small": they cannot extend to compact regions larger than a few ħand hence disappear in the classical limit. They are shielded by the uncertainty principlewhich does not allow precise localization within phase-space regions smaller than ħdensity functional theory phd thesis, and thus renders such "negative probabilities" less paradoxical.
If the left side of the equation is to be interpreted as an expectation value in the Hilbert space with respect to an operator, then in the context of quantum optics this equation is known as the optical equivalence theorem. For details on the properties and interpretation of the Wigner function, see its main article. An alternative phase-space approach to quantum mechanics seeks to define a wave function not just a quasiprobability density on phase space, typically by means of the Segal—Bargmann transform, density functional theory phd thesis.
To be compatible with the uncertainty principle, the phase-space wave function cannot be an arbitrary function, or else it could be localized into an arbitrarily small region of phase space. There is a quasiprobability density associated to the phase-space wave function; it is the Husimi Q representation of the position wave function.
For concreteness, we restrict this discussion to the star product relevant to the Wigner-Weyl representation. For notational convenience, we introduce the notion of left and right derivatives. For a pair of functions f and gdensity functional theory phd thesis, the left and right derivatives are defined as.
The differential definition of the star product is. where the argument of the exponential function can be interpreted as a power series. Additional differential relations allow this to be written in terms of a change in the arguments of f and g :. Thus, e. where H is the Hamiltonian, a plain phase-space function, most often identical to the classical Hamiltonian. The time evolution of the phase space distribution is given by a quantum modification of Liouville flow.
where {{}} is the Moyal bracketthe Wigner transform of the quantum commutator, while {} is the classical Poisson bracket. In the quantum extension of the flow, however, the density of points in phase space is not conserved ; the probability fluid appears "diffusive" and compressible.
Given the restrictions placed by the uncertainty principle on localization, Niels Bohr vigorously denied the physical existence of such trajectories on the microscopic scale.
By means of formal phase-space trajectories, the time evolution problem of the Wigner function can be rigorously solved using the path-integral method [21] and the method of quantum characteristics[22] although there are severe practical obstacles in both cases. The Hamiltonian for the simple harmonic oscillator in one spatial dimension in the Wigner-Weyl representation is.
For the harmonic oscillator, the time evolution of an arbitrary Wigner distribution is simple. Suppose a particle is initially in a minimally uncertain Gaussian statewith the expectation values of position and momentum both centered at the origin in phase space. The Wigner function for such a state propagating freely is. Initially, the position and momenta are uncorrelated. Thus, in 3 dimensions, we expect the position and momentum vectors to be twice as likely to be perpendicular to each other as parallel.
However, the position and momentum become increasingly correlated as the state evolves, because portions of the distribution farther from the origin in position require a larger momentum to be reached: asymptotically, density functional theory phd thesis. This relative "squeezing" reflects the spreading of the free wave packet in coordinate space.
Indeed, it is possible to show that the kinetic energy of the particle becomes asymptotically radial only, in agreement with the standard quantum-mechanical notion of the ground-state nonzero angular momentum specifying orientation independence: [24]. The Morse potential is used to approximate the vibrational structure of a diatomic molecule.
Tunneling is a hallmark quantum effect where a quantum particle, not having sufficient energy to fly above, still goes through a barrier. This effect does not exist in classical mechanics.
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Main articles: Wigner quasiprobability distributionQuasiprobability distributionand Wigner—Weyl transform.
Main article: Moyal product. Main article: quantum harmonic oscillator. Time evolution of combined ground and 1st excited state Wigner function for the simple harmonic oscillator.
Note the rigid motion in phase space corresponding to the conventional oscillations in coordinate space. Wigner function for the harmonic oscillator ground state, displaced from the origin of phase space, i. Note the rigid rotation, identical to classical motion: this is a special feature of the SHO, illustrating the correspondence principle.
From the general pedagogy density functional theory phd thesis. Play media. Bibcode : Phy doi : Mathematical Proceedings of the Cambridge Philosophical Society. Bibcode : PCPS Zeitschrift für Physik.
Bibcode : ZPhy S2CID Physical Review. Bibcode : PhRv hdl : Twareque; Engliš, Miroslav Reviews in Mathematical Physics. Asia Pacific Physics Newsletter. arXiv : ZachosD. Fairlieand T. Curtright"Quantum Mechanics in Phase Space" World Scientific, Singapore, ISBN Journal of Mathematical Physics, density functional theory phd thesis.
Bibcode : JMP Quantum Mechanics in Phase Space". Physical Review D. Bibcode : PhRvD Physical Review Letters. Bibcode : PhRvL. Bibcode : PhRv.
Density Functional Theory, Part 1: Fundamentals
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