Quantum Mechanics
... has been brought up to date in the light of recent experimental and theoretical work on the conceptual basis of the subject and, in response to a number of requests from students, I have provided hints to the solution of the problems at the ends of the chapters. I should like to thank everyone who d ...
... has been brought up to date in the light of recent experimental and theoretical work on the conceptual basis of the subject and, in response to a number of requests from students, I have provided hints to the solution of the problems at the ends of the chapters. I should like to thank everyone who d ...
Resonances, dissipation and decoherence in exotic and artificial atoms
... to compute quantum dot spectra. Among the most common examples, one can mention density functional theory [29–31], configuration interaction [32–34], Hartree-Fock calculations [35–37], quantum Monte Carlo methods [38–40], variational techniques [41–43] or many-body perturbation theory [44]. The nece ...
... to compute quantum dot spectra. Among the most common examples, one can mention density functional theory [29–31], configuration interaction [32–34], Hartree-Fock calculations [35–37], quantum Monte Carlo methods [38–40], variational techniques [41–43] or many-body perturbation theory [44]. The nece ...
Algebraic approach to interacting quantum systems
... parameter and reduce its components according to the different subgroups of SUðNÞ that can be used to generate an equivalent language (for instance SUð2Þ). As an illustration, we show that the local order parameter for an S ¼ 1 SUð2Þ-spin Hamiltonian can be either the usual magnetization or a spin-ne ...
... parameter and reduce its components according to the different subgroups of SUðNÞ that can be used to generate an equivalent language (for instance SUð2Þ). As an illustration, we show that the local order parameter for an S ¼ 1 SUð2Þ-spin Hamiltonian can be either the usual magnetization or a spin-ne ...
NON-RELATIVISTIC QUANTUM MECHANICS - Philsci
... 1.2.3.2.c. Wavefunctions. Wavefunctions are just a specific way of representing statevectors. It is often convenient to take the Hilbert space for a quantum system to be the elements of L2 (R3 ), in which case statevectors are (equivalence classes of) complex-valued functions on R3 . The equation of ...
... 1.2.3.2.c. Wavefunctions. Wavefunctions are just a specific way of representing statevectors. It is often convenient to take the Hilbert space for a quantum system to be the elements of L2 (R3 ), in which case statevectors are (equivalence classes of) complex-valued functions on R3 . The equation of ...
Review - Sociedade Brasileira de Química
... Moreover, the addition of the relevant Wess-Zumino term to the quantum NL S model, equation (3), changes its properties dramatically since the critical dynamical exponent assumes z = 2 (nonrelativistic feature), in contrast with the value z = 1 found in the relativistic quantum NL S model, associate ...
... Moreover, the addition of the relevant Wess-Zumino term to the quantum NL S model, equation (3), changes its properties dramatically since the critical dynamical exponent assumes z = 2 (nonrelativistic feature), in contrast with the value z = 1 found in the relativistic quantum NL S model, associate ...
Lectures on Quantum Mechanics (nonlinear PDE point of view)
... We explain all details of the calculations and mathematical tools: Lagrangian and Hamiltonian formalism for the systems with finite degree of freedom and for fields, Geometric Optics, the HamiltonJacobi equation and WKB approximation, Noether theory of invariants including the theorem on currents, fou ...
... We explain all details of the calculations and mathematical tools: Lagrangian and Hamiltonian formalism for the systems with finite degree of freedom and for fields, Geometric Optics, the HamiltonJacobi equation and WKB approximation, Noether theory of invariants including the theorem on currents, fou ...
Non-perturbative Quantum Electrodynamics in low
... dimensional versions of QED can still excite the curiosity of theoreticians, as well as condensed matter physicists. Although interesting for their own sake, these theories provide also valuable playgrounds to study more realistic quantum field theories, as for example quantum chromodynamics. Beside ...
... dimensional versions of QED can still excite the curiosity of theoreticians, as well as condensed matter physicists. Although interesting for their own sake, these theories provide also valuable playgrounds to study more realistic quantum field theories, as for example quantum chromodynamics. Beside ...
1. QUARK MODEL
... is ℓ, then the parity P is (−1)ℓ+1 . The meson spin J is given by the usual relation |ℓ − s| ≤ J ≤ |ℓ + s|, where s is 0 (antiparallel quark spins) or 1 (parallel quark spins). The charge conjugation, or C-parity C = (−1)ℓ+s , is defined only for the qq̄ states made of quarks and their own antiquark ...
... is ℓ, then the parity P is (−1)ℓ+1 . The meson spin J is given by the usual relation |ℓ − s| ≤ J ≤ |ℓ + s|, where s is 0 (antiparallel quark spins) or 1 (parallel quark spins). The charge conjugation, or C-parity C = (−1)ℓ+s , is defined only for the qq̄ states made of quarks and their own antiquark ...
3rd Set of Note for Chem 655
... ½, 1, … (integral or half-integral) 2. The component on an arbitrary z-axis is limited to the 2l+1 values mlħ with ml = l, l-1, … -l Note that the half integral quantum numbers do not necessarily apply to a particular physical situation. For cyclical boundary conditions only integral values are admi ...
... ½, 1, … (integral or half-integral) 2. The component on an arbitrary z-axis is limited to the 2l+1 values mlħ with ml = l, l-1, … -l Note that the half integral quantum numbers do not necessarily apply to a particular physical situation. For cyclical boundary conditions only integral values are admi ...
Chapter 14
... In this equation, k is the spring constant, which depends on the stiffness and other properties of the spring, and x is the distance that the spring is stretched from its equilibrium position. Not all springs obey Hooke’s law, but many do. Those that do are called elastic. Potential energy When a fo ...
... In this equation, k is the spring constant, which depends on the stiffness and other properties of the spring, and x is the distance that the spring is stretched from its equilibrium position. Not all springs obey Hooke’s law, but many do. Those that do are called elastic. Potential energy When a fo ...
spin squeezing and quantum entanglement in interaction
... than the wavelength of the radiation. The simplest case of N = 1 is known under the name of the Jaynes-Cummings model (JCM) (Jaynes and Cummings, 1963). The interaction of a group of two-level atoms with a single mode cavity field has also been considered by Tavis and Cummings (Tavis and Cummings, 1 ...
... than the wavelength of the radiation. The simplest case of N = 1 is known under the name of the Jaynes-Cummings model (JCM) (Jaynes and Cummings, 1963). The interaction of a group of two-level atoms with a single mode cavity field has also been considered by Tavis and Cummings (Tavis and Cummings, 1 ...
quantum transition-state theory. II. Recovery of the exact quantum
... time-correlation function containing a ring-polymer dividing surface. This t → 0+ limit appears to be unique in giving positive-definite Boltzmann statistics, and is identical to ring-polymer molecular dynamics (RPMD) TST. Here, we show that quantum TST (i.e., RPMD-TST) is exact if there is no recro ...
... time-correlation function containing a ring-polymer dividing surface. This t → 0+ limit appears to be unique in giving positive-definite Boltzmann statistics, and is identical to ring-polymer molecular dynamics (RPMD) TST. Here, we show that quantum TST (i.e., RPMD-TST) is exact if there is no recro ...
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.