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... of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with non-constant coefficients of two confluent hypergeometric functions. Discussing the bound-state wave functions vanishing both at infinity and in the origin, we derive the exact equ ...
... of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with non-constant coefficients of two confluent hypergeometric functions. Discussing the bound-state wave functions vanishing both at infinity and in the origin, we derive the exact equ ...
Lagrange`s and Hamilton`s Equations
... We now apply the notion of the Legendre transform to the classical Lagrangian. In our previous developments, we have taken L to be a function of all the generalized coordinates and their respective time derivatives; i.e. L = L({qi }, {q̇i }, t). For generality, we have also included the possibility ...
... We now apply the notion of the Legendre transform to the classical Lagrangian. In our previous developments, we have taken L to be a function of all the generalized coordinates and their respective time derivatives; i.e. L = L({qi }, {q̇i }, t). For generality, we have also included the possibility ...
Quantum Mechanics
... The “State” of a system is described by a function Y of the coordinates and the time. This function, called the state function or wave function, contains all the information that can be determined about the system. We further postulate that Y is singlevalued, continuous, and quadratically integrable ...
... The “State” of a system is described by a function Y of the coordinates and the time. This function, called the state function or wave function, contains all the information that can be determined about the system. We further postulate that Y is singlevalued, continuous, and quadratically integrable ...
Elements of Particle Physics - The Institute of Mathematical Sciences
... total cross-section is in general a function of Lorentz invariant variables, which in this case is the square of the sum of the two four momenta in the initial state (or final state). At very high energies we may neglect the masses of the particles, and purely by dimensional considerations the cross ...
... total cross-section is in general a function of Lorentz invariant variables, which in this case is the square of the sum of the two four momenta in the initial state (or final state). At very high energies we may neglect the masses of the particles, and purely by dimensional considerations the cross ...
QUANTUM FIELD THEORY
... concept of a wave function becomes a meaningful (albeit approximate) concept. For a massless particle one gets ∆x ≫ ~/p = λ/2π, i.e. the coordinates of a photon only become meaningful in cases where the typical dimensions are much larger than the wavelength. For the momentum or energy of a particle ...
... concept of a wave function becomes a meaningful (albeit approximate) concept. For a massless particle one gets ∆x ≫ ~/p = λ/2π, i.e. the coordinates of a photon only become meaningful in cases where the typical dimensions are much larger than the wavelength. For the momentum or energy of a particle ...
Coupling and Dissociation in Artificial Molecules
... lower in energy than the P = 2 polarized one, namely the two states are practically degenerate, which implies that for the set of parameters employed here the QD molecule is located well in the dissociation regime. Although the symmetry breaking within the HF theory allows one to correct for the art ...
... lower in energy than the P = 2 polarized one, namely the two states are practically degenerate, which implies that for the set of parameters employed here the QD molecule is located well in the dissociation regime. Although the symmetry breaking within the HF theory allows one to correct for the art ...
Historical pseudo simplified solution of the Dirac
... in a paper[9] published in Physical Review 22 years ago is a pseudo solution. For the said secondorder Dirac-Coulomb equations in which two equations were written in the same form by using sign “±”, two eigenvalues set should be given and they are actually different. It is well known that two differ ...
... in a paper[9] published in Physical Review 22 years ago is a pseudo solution. For the said secondorder Dirac-Coulomb equations in which two equations were written in the same form by using sign “±”, two eigenvalues set should be given and they are actually different. It is well known that two differ ...
Exact solutions of a Dirac equation with a varying CP
... Modern particle physics studies the fundamental interactions and properties of the known subatomic particles. The most notable particle physics theory is the Standard Model, which very successfully describes a major part of observed particle phenomena. Despite its success, the Standard model also le ...
... Modern particle physics studies the fundamental interactions and properties of the known subatomic particles. The most notable particle physics theory is the Standard Model, which very successfully describes a major part of observed particle phenomena. Despite its success, the Standard model also le ...
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.