Commutation relations for functions of operators
... 共6兲 and 共7兲 is a typical and almost obligatory exercise in a modern text on quantum mechanics. The standard way of proceeding is to consider the commutator of x with increasing powers of p, to use induction, and to develop a Taylor expansion of the function f. It may come as a surprise therefore tha ...
... 共6兲 and 共7兲 is a typical and almost obligatory exercise in a modern text on quantum mechanics. The standard way of proceeding is to consider the commutator of x with increasing powers of p, to use induction, and to develop a Taylor expansion of the function f. It may come as a surprise therefore tha ...
Path-memory induced quantization of classical orbits
... longer as hysteresis increases. Limiting ourselves to the observed plateaus we find that the radii are located on the same steps when they are rescaled by the Faraday wavelength Rn ∕λF and plotted as a function of ðV W ∕2ΩλF Þ1∕2 (see Fig. 2C). Let us now look for an analogy between the discretizati ...
... longer as hysteresis increases. Limiting ourselves to the observed plateaus we find that the radii are located on the same steps when they are rescaled by the Faraday wavelength Rn ∕λF and plotted as a function of ðV W ∕2ΩλF Þ1∕2 (see Fig. 2C). Let us now look for an analogy between the discretizati ...
Nonparametric estimation of the purity of a quantum state in
... probabilities of its measurable properties, or "observables" (examples of observables include energy, position, momentum and angular momentum). Generally, quantum mechanics does not assign deterministic values to observables. Instead, it makes predictions about probability distributions; that is, th ...
... probabilities of its measurable properties, or "observables" (examples of observables include energy, position, momentum and angular momentum). Generally, quantum mechanics does not assign deterministic values to observables. Instead, it makes predictions about probability distributions; that is, th ...
Theory and simulations of quantum glass forming liquids
... Understanding the fundamental causes of the dramatic slowdown of dynamics when a liquid transforms into a glass is still a subject of great debate.1–4 Essentially all discussion of the glass transition has focused on the strictly classical regime of liquid state behavior, namely where the de Broglie ...
... Understanding the fundamental causes of the dramatic slowdown of dynamics when a liquid transforms into a glass is still a subject of great debate.1–4 Essentially all discussion of the glass transition has focused on the strictly classical regime of liquid state behavior, namely where the de Broglie ...
A Spin Chain Primer - University of Miami Physics
... and chains with spins in representations of higher-rank algebras (su(N ) , . . .). Using also so-called K matrices, one can construct integrable open spin chains [11]. Bethe Ansatz equations have been obtained for many such models. 11 Obtaining Bethe Ansatz equations (BAE) for an integrable model is ...
... and chains with spins in representations of higher-rank algebras (su(N ) , . . .). Using also so-called K matrices, one can construct integrable open spin chains [11]. Bethe Ansatz equations have been obtained for many such models. 11 Obtaining Bethe Ansatz equations (BAE) for an integrable model is ...
Quantum Field Theory I, Lecture Notes
... Usually, excitations of the quantum field will be described by “particles”. In QFT the number of these particles is not conserved, they are created and annihilated when they interact. It is precisely what we observe in elementary particle physics, hence QFT has become the mathematical framework for ...
... Usually, excitations of the quantum field will be described by “particles”. In QFT the number of these particles is not conserved, they are created and annihilated when they interact. It is precisely what we observe in elementary particle physics, hence QFT has become the mathematical framework for ...
Electronic structure of rectangular quantum dots
... shown that the improvement gained with the new parametrization is directly proportional to the electron density and the polarization of the system. In our recent paper,26 we compare different LSD functionals in small 2D quantum dots. We show that in comparison with the variational quantum Monte Carl ...
... shown that the improvement gained with the new parametrization is directly proportional to the electron density and the polarization of the system. In our recent paper,26 we compare different LSD functionals in small 2D quantum dots. We show that in comparison with the variational quantum Monte Carl ...
Computer simulation by quantum mechanical time dependent wave
... model to describe the physical phenomena. Classical and semi-classical models are also applied in the literature to characterise TEAS and MBS. Since, the interaction of the atom (e.g. He) – solid surface system changes abruptly near the classical turning point the classical trajectories are washed o ...
... model to describe the physical phenomena. Classical and semi-classical models are also applied in the literature to characterise TEAS and MBS. Since, the interaction of the atom (e.g. He) – solid surface system changes abruptly near the classical turning point the classical trajectories are washed o ...
LeCtURe Notes QUANTUM STATISTICAL FIELD THEORY
... for all allowed values αi , αi0 and 0 < λ < 1. As a consequence, Φ cannot support a discontinuity in the interior of the range of its arguments. However, it can have a discontinuous derivative. This implies that the necessary and sufficient condition for the uniqueness of the equilibrium state is th ...
... for all allowed values αi , αi0 and 0 < λ < 1. As a consequence, Φ cannot support a discontinuity in the interior of the range of its arguments. However, it can have a discontinuous derivative. This implies that the necessary and sufficient condition for the uniqueness of the equilibrium state is th ...
Quantum Imaging beyond the Diffraction Limit by
... local oscillator field / xA ðxÞ. The theory presented in this Letter is in fact more general; it shows that repeated ^ without the linearization, can not measurements of X, only produce a Gaussian spot, but also complex quantum interference patterns like those in quantum lithography at the de Brogl ...
... local oscillator field / xA ðxÞ. The theory presented in this Letter is in fact more general; it shows that repeated ^ without the linearization, can not measurements of X, only produce a Gaussian spot, but also complex quantum interference patterns like those in quantum lithography at the de Brogl ...
Atomic Structure Institute of Lifelong Learning, University of Delhi
... an electron has a certain set of quantum numbers, then no other electron in that atom can have the same set of quantum numbers. Physicists call this "Pauli's exclusion principle." ...
... an electron has a certain set of quantum numbers, then no other electron in that atom can have the same set of quantum numbers. Physicists call this "Pauli's exclusion principle." ...
ON THE GENERAL FORM OF QUANTUM STOCHASTIC
... for quantum states, giving the dynamical solution for the well-known quantum measurement problem. Some particular types of such equations have been considered recently in the phenomenological theories of quantum permanent reduction [4, 5], continuous measurement collapse [6, 7], spontaneous jumps [8 ...
... for quantum states, giving the dynamical solution for the well-known quantum measurement problem. Some particular types of such equations have been considered recently in the phenomenological theories of quantum permanent reduction [4, 5], continuous measurement collapse [6, 7], spontaneous jumps [8 ...
msc_f_p1b2 - Bhoj University
... Important Note-In all the above text and text that follows we are using only h line or h cross, which is equal to h (Planck’s constant) divided by 2. It is quite likely that print does not show it distinctly. In some cases the size of h line is not same due to unavoidable reasons but it conveys th ...
... Important Note-In all the above text and text that follows we are using only h line or h cross, which is equal to h (Planck’s constant) divided by 2. It is quite likely that print does not show it distinctly. In some cases the size of h line is not same due to unavoidable reasons but it conveys th ...
coherent states in quantum mechanics
... In classical physics the properties of a certain system can be described using its position x and mass m. With these variables it is possible to determine the velocity v(=dx/dt), the momentum p(=mv) and any other dynamical variable of interest. Quantum mechanics describes the time evolution of physi ...
... In classical physics the properties of a certain system can be described using its position x and mass m. With these variables it is possible to determine the velocity v(=dx/dt), the momentum p(=mv) and any other dynamical variable of interest. Quantum mechanics describes the time evolution of physi ...
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.