Manipulation and detection of electron charge/spin qubits
... strong classical fields. This interaction, is shown to yield a single, versatile process that provides induces an effective coupling between the quantum modes that can be e.g. ⇒ addresses all DiVincenzo Criteria a full set of photonic quantum processing tools. This set satisfies the tuned and enhanc ...
... strong classical fields. This interaction, is shown to yield a single, versatile process that provides induces an effective coupling between the quantum modes that can be e.g. ⇒ addresses all DiVincenzo Criteria a full set of photonic quantum processing tools. This set satisfies the tuned and enhanc ...
Regular and irregular semiclassical wavefunctions
... (I, have the de Broglie wavelength h / [ 2 m ( E- V(q))]1’2. Now, however, the oscillations near q are statistically isotropic, even close to the anticaustics. The autocorrelation function is not by itself sufficient to determine all statistical properties of cc/. However it is likely that for stoch ...
... (I, have the de Broglie wavelength h / [ 2 m ( E- V(q))]1’2. Now, however, the oscillations near q are statistically isotropic, even close to the anticaustics. The autocorrelation function is not by itself sufficient to determine all statistical properties of cc/. However it is likely that for stoch ...
applied theta functions
... We were led from the top statement (wave representation) to the bottom statement (particle representation) by appeal to Jacobi’s identity, and know on those grounds that the two representations are “Appell equivalent” in the sense that the Appell transformation sends either into the other. I draw at ...
... We were led from the top statement (wave representation) to the bottom statement (particle representation) by appeal to Jacobi’s identity, and know on those grounds that the two representations are “Appell equivalent” in the sense that the Appell transformation sends either into the other. I draw at ...
Quantum Numbers and Rules
... momentum vectors lie on cones as illustrated. This behavior is not observed on the large scale. To see how the correspondence principle holds here, consider that the smallest angle (θ1 in the example) is for the maximum value of ml = 0, namely ml = l. For that smallest angle, ...
... momentum vectors lie on cones as illustrated. This behavior is not observed on the large scale. To see how the correspondence principle holds here, consider that the smallest angle (θ1 in the example) is for the maximum value of ml = 0, namely ml = l. For that smallest angle, ...
Generation of macroscopic pair-correlated atomic beams by four
... state wavefunction [1]. Once BEC had been achieved, the initial well-defined quantum state can be transformed into other more complex states by manipulating it with magnetic and optical fields. This can result in a variety of time-dependent macroscopic wavefunctions [1], including oscillating conden ...
... state wavefunction [1]. Once BEC had been achieved, the initial well-defined quantum state can be transformed into other more complex states by manipulating it with magnetic and optical fields. This can result in a variety of time-dependent macroscopic wavefunctions [1], including oscillating conden ...
On the Derivation of the Time-Dependent Equation of Schrodinger
... Eq. (2) is redundant? However, there is a more serious objection to the solution Eq. (2) and its FeynmanHibbs interpretation. An oscillation frequency is, by definition, positive definite. However, the quantity (E S ) is not absolutely defined and can take on any value, negative or positive, arbi ...
... Eq. (2) is redundant? However, there is a more serious objection to the solution Eq. (2) and its FeynmanHibbs interpretation. An oscillation frequency is, by definition, positive definite. However, the quantity (E S ) is not absolutely defined and can take on any value, negative or positive, arbi ...
Post-Markov master equation for the dynamics of open quantum
... Ž11. to preserve the positivity of r t if applied to an arbitrary initial density operator. Our derivation shows that for the asymptotic Eq. Ž11. to be meaningful, one has to use an effectiÕe initial condition w4x obtained by fully determining the initial slip arising from the time dependent coeffic ...
... Ž11. to preserve the positivity of r t if applied to an arbitrary initial density operator. Our derivation shows that for the asymptotic Eq. Ž11. to be meaningful, one has to use an effectiÕe initial condition w4x obtained by fully determining the initial slip arising from the time dependent coeffic ...
Quantum Contributions to Cosmological Correlations
... two different kinds of wave number: the fixed wave numbers q associated with external lines, and the internal wave numbers p circulating in loops, over which we must integrate. It is only if the integrals over internal wave numbers p are dominated by values of order p ≈ q that we can speak of a defi ...
... two different kinds of wave number: the fixed wave numbers q associated with external lines, and the internal wave numbers p circulating in loops, over which we must integrate. It is only if the integrals over internal wave numbers p are dominated by values of order p ≈ q that we can speak of a defi ...
Atomic wave packet dynamics in finite time
... from Vin . Since we are interested in the dynamics over time scales which are long compared to the early interaction-driven expansion, we can, without loss of generality, assume that each particle behaves independently. The Hamiltonian describing the dynamics along the z-axis therefore adopts the si ...
... from Vin . Since we are interested in the dynamics over time scales which are long compared to the early interaction-driven expansion, we can, without loss of generality, assume that each particle behaves independently. The Hamiltonian describing the dynamics along the z-axis therefore adopts the si ...
Wave function
A wave function in quantum mechanics describes the quantum state of an isolated system of one or more particles. There is one wave function containing all the information about the entire system, not a separate wave function for each particle in the system. Its interpretation is that of a probability amplitude. Quantities associated with measurements, such as the average momentum of a particle, can be derived from the wave function. It is a central entity in quantum mechanics and is important in all modern theories, like quantum field theory incorporating quantum mechanics, while its interpretation may differ. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi).For a given system, once a representation corresponding to a maximal set of commuting observables and a suitable coordinate system is chosen, the wave function is a complex-valued function of the system's degrees of freedom corresponding to the chosen representation and coordinate system, continuous as well as discrete. Such a set of observables, by a postulate of quantum mechanics, are Hermitian linear operators on the space of states representing a set of physical observables, like position, momentum and spin that can, in principle, be simultaneously measured with arbitrary precision. Wave functions can be added together and multiplied by complex numbers to form new wave functions, and hence are elements of a vector space. This is the superposition principle of quantum mechanics. This vector space is endowed with an inner product such that it is a complete metric topological space with respect to the metric induced by the inner product. In this way the set of wave functions for a system form a function space that is a Hilbert space. The inner product is a measure of the overlap between physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The actual space depends on the system's degrees of freedom (hence on the chosen representation and coordinate system) and the exact form of the Hamiltonian entering the equation governing the dynamical behavior. In the non-relativistic case, disregarding spin, this is the Schrödinger equation.The Schrödinger equation determines the allowed wave functions for the system and how they evolve over time. A wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name ""wave function"", and gives rise to wave–particle duality. The wave of the wave function, however, is not a wave in physical space; it is a wave in an abstract mathematical ""space"", and in this respect it differs fundamentally from water waves or waves on a string.For a given system, the choice of which relevant degrees of freedom to use are not unique, and correspondingly the domain of the wave function is not unique. It may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space, the two are related by a Fourier transform. These descriptions are the most important, but they are not the only possibilities. Just like in classical mechanics, canonical transformations may be used in the description of a quantum system. Some particles, like electrons and photons, have nonzero spin, and the wave function must include this fundamental property as an intrinsic discrete degree of freedom. In general, for a particle with half-integer spin the wave function is a spinor, for a particle with integer spin the wave function is a tensor. Particles with spin zero are called scalar particles, those with spin 1 vector particles, and more generally for higher integer spin, tensor particles. The terminology derives from how the wave functions transform under a rotation of the coordinate system. No elementary particle with spin 3⁄2 or higher is known, except for the hypothesized spin 2 graviton. Other discrete variables can be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g. a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g. z-component of spin). These values are often displayed in a column matrix (e.g. a 2 × 1 column vector for a non-relativistic electron with spin 1⁄2).In the Copenhagen interpretation, an interpretation of quantum mechanics, the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle as being at a given place at a given time or having a definite momentum, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation, this general requirement a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured. Its value does not in isolation tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.The unit of measurement for ψ depends on the system, and can be found by dimensional analysis of the normalization condition for the system. For one particle in three dimensions, its units are [length]−3/2, because an integral of |ψ|2 over a region of three-dimensional space is a dimensionless probability.