Quantum Physics (UCSD Physics 130)
... 11.1 Operators in a Vector Space . . . . . . . . . . . . . . 11.1.1 Review of Operators . . . . . . . . . . . . . . 11.1.2 Projection Operators |jihj| and Completeness 11.1.3 Unitary Operators . . . . . . . . . . . . . . . 11.2 A Complete Set of Mutually Commuting Operators . 11.3 Uncertainty Princi ...
... 11.1 Operators in a Vector Space . . . . . . . . . . . . . . 11.1.1 Review of Operators . . . . . . . . . . . . . . 11.1.2 Projection Operators |jihj| and Completeness 11.1.3 Unitary Operators . . . . . . . . . . . . . . . 11.2 A Complete Set of Mutually Commuting Operators . 11.3 Uncertainty Princi ...
Quantum Mechanics of Many-Particle Systems: Atoms, Molecules
... depends on simple qualitative ideas about how the AOs overlap – which depends in turn on their sizes and shapes. So even without doing a big SCF calculation it is often possible to make progress using only pictorial arguments. Once you have an idea of the probable order of the MO energies, you can s ...
... depends on simple qualitative ideas about how the AOs overlap – which depends in turn on their sizes and shapes. So even without doing a big SCF calculation it is often possible to make progress using only pictorial arguments. Once you have an idea of the probable order of the MO energies, you can s ...
Quantum Mechanics for Pedestrians 1: Fundamentals
... Certainly, there may be different answers to this question. After all, quantum mechanics is such a broad field that a single textbook cannot cover all the relevant topics. A selection or prioritization of subjects is necessary per se and, moreover, the physical and mathematical foreknowledge of the ...
... Certainly, there may be different answers to this question. After all, quantum mechanics is such a broad field that a single textbook cannot cover all the relevant topics. A selection or prioritization of subjects is necessary per se and, moreover, the physical and mathematical foreknowledge of the ...
Paired states of fermions in two dimensions with breaking of parity
... In this paper we will make extensive use of the methods for BCS paired states, and consider the transitions between the weak and strong coupling regimes in two dimensions. In the weak-coupling regime, exotic phenomena are possible when parity and time reversal are broken. The results are applied to ...
... In this paper we will make extensive use of the methods for BCS paired states, and consider the transitions between the weak and strong coupling regimes in two dimensions. In the weak-coupling regime, exotic phenomena are possible when parity and time reversal are broken. The results are applied to ...
Gravitational wave detection with advanced ground based detectors
... more “active stages”, with acceleration sensors and feedback loops, and four or more passive stages which use the isolation provided by mechanical inertia. In the end, it is the rejection of high frequency motion by the inertia of large masses that keeps the vibrations of the world from masking a gr ...
... more “active stages”, with acceleration sensors and feedback loops, and four or more passive stages which use the isolation provided by mechanical inertia. In the end, it is the rejection of high frequency motion by the inertia of large masses that keeps the vibrations of the world from masking a gr ...
Quantum Mechanics and Solid State Physics for Electric
... Chapter 3 we introduce the stationary wave function (or state function) of a microscopic particle (e.g electron) and solve a handful of problems that help to understand the concepts. In Chapter 4 we discuss the problem of time dependent phenomena and introduce the time dependent Schrödinger equatio ...
... Chapter 3 we introduce the stationary wave function (or state function) of a microscopic particle (e.g electron) and solve a handful of problems that help to understand the concepts. In Chapter 4 we discuss the problem of time dependent phenomena and introduce the time dependent Schrödinger equatio ...
The reduced Hamiltonian for next-to-leading-order spin
... both in harmonic gauge. Later, within the ADM canonical formalism, a Hamiltonian presentation was achieved [14] (see also [15]). The NLO spin(1)-spin(2) dynamics was found in [16, 15] and confirmed by [17, 18]. Higher PN orders linear in spin were tackled recently in [19, 20, 5]. In particular, Ref. ...
... both in harmonic gauge. Later, within the ADM canonical formalism, a Hamiltonian presentation was achieved [14] (see also [15]). The NLO spin(1)-spin(2) dynamics was found in [16, 15] and confirmed by [17, 18]. Higher PN orders linear in spin were tackled recently in [19, 20, 5]. In particular, Ref. ...
Topics in Ultracold Atomic Gases: Strong Interactions and Quantum
... in weakly interacting Bose gases in the condensate phase, even in the absence of a single realistic system of which the models are exactly suitable. After many years of effort in cooling down the dilute gases of neutral alkali atoms, in 1995, scientists from JILA[1] realized the first weakly interac ...
... in weakly interacting Bose gases in the condensate phase, even in the absence of a single realistic system of which the models are exactly suitable. After many years of effort in cooling down the dilute gases of neutral alkali atoms, in 1995, scientists from JILA[1] realized the first weakly interac ...
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.