Chapter 3 Foundations II: Measurement and Evolution 3.1
... finite-dimensional context; it becomes more apt when the index a can be continually varying.) In our discussion we have arrived at the special case of a POVM by one-dimensional operators (operators with one nonvanishing eigenvalue). In the generalized measurement theory, each outcome has a probabili ...
... finite-dimensional context; it becomes more apt when the index a can be continually varying.) In our discussion we have arrived at the special case of a POVM by one-dimensional operators (operators with one nonvanishing eigenvalue). In the generalized measurement theory, each outcome has a probabili ...
Physics Formulary
... prints, publishing by electronic or other means, implies full agreement to the above non-profit-use clause, unless upon explicit prior written permission of the author. This document is provided by the author “as is”, with all its faults. Any express or implied warranties, including, but not limited ...
... prints, publishing by electronic or other means, implies full agreement to the above non-profit-use clause, unless upon explicit prior written permission of the author. This document is provided by the author “as is”, with all its faults. Any express or implied warranties, including, but not limited ...
Lecture Notes in Quantum Mechanics Doron Cohen
... The major attempt in this set of lectures was to give a self contained presentation of quantum mechanics, which is not based on the historical ”quantization” approach. The main inspiration comes from Ref.[3] and Ref.[1]. The challenge was to find a compromise between the over-heuristic approach of R ...
... The major attempt in this set of lectures was to give a self contained presentation of quantum mechanics, which is not based on the historical ”quantization” approach. The main inspiration comes from Ref.[3] and Ref.[1]. The challenge was to find a compromise between the over-heuristic approach of R ...
What is matter? The fundamental ontology of atomism and structural
... admits only the quantum state of the universe in its ontology (as represented by the universal wave function); this state always undergoes a linear evolution (as represented e.g. by the Schrödinger equation) (see Wallace (2012) for the most detailed contemporary account). However, although it is a ...
... admits only the quantum state of the universe in its ontology (as represented by the universal wave function); this state always undergoes a linear evolution (as represented e.g. by the Schrödinger equation) (see Wallace (2012) for the most detailed contemporary account). However, although it is a ...
III. Contact-ing Schrödinger
... But let me put off talking about the NEGF model till the next Lecture, and use subsequent lectures to illustrate its application to interesting problems in quantum transport. As indicated in Fig.18.1b the NEGF method requires two types of inputs: the Hamiltonian, [H] describing the dynamics of an el ...
... But let me put off talking about the NEGF model till the next Lecture, and use subsequent lectures to illustrate its application to interesting problems in quantum transport. As indicated in Fig.18.1b the NEGF method requires two types of inputs: the Hamiltonian, [H] describing the dynamics of an el ...
Mean spin direction and spin squeezing in superpositions of spin
... integer N = 2j, Sn⊥ = S ◦ n⊥ , and j is the spin number. The inequality ξ 2 < 1 indicates that the system is spin squeezed. In this paper, we study the MSD and spin squeezing in a general superpositions of two spin coherent states (SCSs) [24], the study of the MSD is important at least for the follo ...
... integer N = 2j, Sn⊥ = S ◦ n⊥ , and j is the spin number. The inequality ξ 2 < 1 indicates that the system is spin squeezed. In this paper, we study the MSD and spin squeezing in a general superpositions of two spin coherent states (SCSs) [24], the study of the MSD is important at least for the follo ...
Fock Matrix Construction for Large Systems
... polynomials P are chosen as the real solid harmonics[5]. The exponents α and expansion ...
... polynomials P are chosen as the real solid harmonics[5]. The exponents α and expansion ...
Quantum Theory of Molecular Magnetism - cond
... (“ferric wheels”) are most popular [8, 41, 53–60]. Iron-6 rings for instance can host alkali ions such as lithium or sodium which allows to modify the parameters of the spin Hamiltonian within some range [61, 62]. Another realization of rings is possible using chromium ions as paramagnetic centers [ ...
... (“ferric wheels”) are most popular [8, 41, 53–60]. Iron-6 rings for instance can host alkali ions such as lithium or sodium which allows to modify the parameters of the spin Hamiltonian within some range [61, 62]. Another realization of rings is possible using chromium ions as paramagnetic centers [ ...
Coherent Decay of Bose-Einstein Condensates
... As the coldest form of matter known to exist, atomic Bose-Einstein condensates are unique forms of matter where the constituent atoms lose their individual identities, becoming absorbed into the cloud as a whole. Effectively, these gases become a single macroscopic object that inherits its propertie ...
... As the coldest form of matter known to exist, atomic Bose-Einstein condensates are unique forms of matter where the constituent atoms lose their individual identities, becoming absorbed into the cloud as a whole. Effectively, these gases become a single macroscopic object that inherits its propertie ...
Photoemission studies of quantum well states in thin films
... The ®rst photoemission observation of quantum size effects was reported in 1986 [18]. The evidence was clear but the quantum well peaks were very broad, again due to ®lm roughness. Later work, however, clearly established the importance of quantum size effects in ®lms [19±25]. The argument that phot ...
... The ®rst photoemission observation of quantum size effects was reported in 1986 [18]. The evidence was clear but the quantum well peaks were very broad, again due to ®lm roughness. Later work, however, clearly established the importance of quantum size effects in ®lms [19±25]. The argument that phot ...
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.