Unified view on multiconfigurational time propagation for systems
... DHF for fermions and MCTDHB for bosons, can be formulated in a unified and compact form. In particular, the equations of motion for the orbitals depend explicitly and solely on the reduced one- and two-body density matrices of the system’s many-particle wave function. We also point out that represen ...
... DHF for fermions and MCTDHB for bosons, can be formulated in a unified and compact form. In particular, the equations of motion for the orbitals depend explicitly and solely on the reduced one- and two-body density matrices of the system’s many-particle wave function. We also point out that represen ...
Indistinguishable Particles in Quantum Mechanics: An Introduction
... off to zero asymptotically. We associate the spread of the wave function to this volume V , which can evolve in time. Finally, recall that because of the Heisenberg’s uncertainty relations we cannot simultaneously measure the particle’s position and its momentum with an arbitrary precision (see, for ...
... off to zero asymptotically. We associate the spread of the wave function to this volume V , which can evolve in time. Finally, recall that because of the Heisenberg’s uncertainty relations we cannot simultaneously measure the particle’s position and its momentum with an arbitrary precision (see, for ...
ESI Bose-Einstein Condensation as a Quantum Phase Transition in an Optical Lattice
... temperature. Units are chosen so that Boltzmann’s constant equals 1. The thermal expectation value of some observable O will be denoted by hOi = Z −1 Tr O exp(−βH). In the proof of BEC we focus on dimensions d ≥ 3, but, using the technique employed in [17], an extension to the ground state in two di ...
... temperature. Units are chosen so that Boltzmann’s constant equals 1. The thermal expectation value of some observable O will be denoted by hOi = Z −1 Tr O exp(−βH). In the proof of BEC we focus on dimensions d ≥ 3, but, using the technique employed in [17], an extension to the ground state in two di ...
Frustrated Magnetism in Vanadium Oxides
... state of LiV2 O4 . This approach was developed in Ref. [9] within random phase approximation (RPA) spin fluctuation theory based on ab initio LDA electronic structure. 3.1. Electronic structure and spin susceptibility The pyrochlore lattice has four atoms per unit cell. A next neighbor tight binding ...
... state of LiV2 O4 . This approach was developed in Ref. [9] within random phase approximation (RPA) spin fluctuation theory based on ab initio LDA electronic structure. 3.1. Electronic structure and spin susceptibility The pyrochlore lattice has four atoms per unit cell. A next neighbor tight binding ...
An Introduction to Quantum Field Theory, Mrinal Dasgupta
... found it to be equivalent to the Euler-Lagrange equation. The benefit is that the latter can be easily generalised to other systems in any number of dimensions, multi-particle systems, or systems with an infinite number of degrees of freedom, such as needed for field theory. For example, if we now cons ...
... found it to be equivalent to the Euler-Lagrange equation. The benefit is that the latter can be easily generalised to other systems in any number of dimensions, multi-particle systems, or systems with an infinite number of degrees of freedom, such as needed for field theory. For example, if we now cons ...
class notes
... d) The origin for any coordinate system is arbitrary. Since the value for a coordinate depends on the origin chosen, you should specify your origin in every problem. ...
... d) The origin for any coordinate system is arbitrary. Since the value for a coordinate depends on the origin chosen, you should specify your origin in every problem. ...
Simple examples of second quantization 4
... Excitation spectrum of the one-dimensional x-y ferromagnet, showing how the negative energy states are filled. The negative-energy dispersion curve is “folded over” to describe the positive hole excitation energy. so that, remarkably, there is no ground-state magnetization. We may interpret this loss ...
... Excitation spectrum of the one-dimensional x-y ferromagnet, showing how the negative energy states are filled. The negative-energy dispersion curve is “folded over” to describe the positive hole excitation energy. so that, remarkably, there is no ground-state magnetization. We may interpret this loss ...
Path integral approach to the heat kernel 1 Introduction
... The fixing of a “renormalization condition” in this context essentially means fixing which value of α one chooses for the quantum theory. In the absence of other requirements, one may fix α = 0 as “renormalization conditions” (if needed, one may always introduce an additional coupling to R by redefi ...
... The fixing of a “renormalization condition” in this context essentially means fixing which value of α one chooses for the quantum theory. In the absence of other requirements, one may fix α = 0 as “renormalization conditions” (if needed, one may always introduce an additional coupling to R by redefi ...
An Introduction to the Mathematical Aspects of Quantum Mechanics:
... Roughly speaking the quantities ∆x and ∆p can interpreted as the error obtained in measurements of position and momentum respectively. That is Heisenberg’s uncertainty principle, is a fundamental limit to the precision with which the pair (x, p) can be known simultaneously, or in another words it gi ...
... Roughly speaking the quantities ∆x and ∆p can interpreted as the error obtained in measurements of position and momentum respectively. That is Heisenberg’s uncertainty principle, is a fundamental limit to the precision with which the pair (x, p) can be known simultaneously, or in another words it gi ...
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... This equation is the reason the operator L2 plays a very important role in the study of central potentials. L2 will feature as one of the operators in complete sets of commuting observables. An operator, such as L2 , that commutes with all the angular momentum operators is called a “Casimir” of the ...
... This equation is the reason the operator L2 plays a very important role in the study of central potentials. L2 will feature as one of the operators in complete sets of commuting observables. An operator, such as L2 , that commutes with all the angular momentum operators is called a “Casimir” of the ...
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.