I Complex Ion Formation
... when the labels of electrons one and two are interchanged and hence the electrons appear distinguishable. However, the linear combinations (sums) $,(1)9J2(2) $1(2)$2(l) and h(l)$2(2) - W W ( 1 ) conserve the indistinguishability since the values of their squares are unchanged when the labels are int ...
... when the labels of electrons one and two are interchanged and hence the electrons appear distinguishable. However, the linear combinations (sums) $,(1)9J2(2) $1(2)$2(l) and h(l)$2(2) - W W ( 1 ) conserve the indistinguishability since the values of their squares are unchanged when the labels are int ...
Lecture 6: 3D Rigid Rotor, Spherical Harmonics, Angular Momentum
... electron via an effect known as the Zeeman effect. The number of discrete states observed in the Zeeman effect is related to the orbital angular momentum quantum number l. In a famous experiment by Stern and Gerlach in 1921, where they passed Ag atoms in a magnetic field, they observed that the spli ...
... electron via an effect known as the Zeeman effect. The number of discrete states observed in the Zeeman effect is related to the orbital angular momentum quantum number l. In a famous experiment by Stern and Gerlach in 1921, where they passed Ag atoms in a magnetic field, they observed that the spli ...
detailed technical description
... In many materials, the strongly interacting electrons can be understood using Fermi-liquid theory. The idea is that the main effect of interactions can be coded in a few phenomenological parameters, the most important one being the effective electron mass. Taking these so called renormalizations int ...
... In many materials, the strongly interacting electrons can be understood using Fermi-liquid theory. The idea is that the main effect of interactions can be coded in a few phenomenological parameters, the most important one being the effective electron mass. Taking these so called renormalizations int ...
The Klein-Gordon Equation as a time-symmetric
... the free parameters of the Schrödinger Equation solution But 80 years of experiments say we can’t learn any more information (at one time) than can be encoded by . (Deeply connected with the Uncertainty Principle) Therefore, if we start with the KGE as the master Equation, one gets the axiomatic ...
... the free parameters of the Schrödinger Equation solution But 80 years of experiments say we can’t learn any more information (at one time) than can be encoded by . (Deeply connected with the Uncertainty Principle) Therefore, if we start with the KGE as the master Equation, one gets the axiomatic ...
Document
... In 1924, Louis de Broglie suggested a relationship between momentum and wavelength for light applying to particles - any particle, not only photons, travelling with a linear momentum p should have (in some sense) a wavelength given by the de Broglie relation: ...
... In 1924, Louis de Broglie suggested a relationship between momentum and wavelength for light applying to particles - any particle, not only photons, travelling with a linear momentum p should have (in some sense) a wavelength given by the de Broglie relation: ...
Stationary Solutions of the Klein-Gordon Equation in a Potential Field
... We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r, t) = ψ(r)e−iEt/h̄ , ...
... We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r, t) = ψ(r)e−iEt/h̄ , ...
Lecture 9 - Scattering and tunneling for a delta
... particle can cross the potential barrier. This is the effect known as tunnelling. It can also be seen that if the particle energy is greater than there is still a chance that the particle will be reflected from the potential barrier. Neither of these phenomena exist in classical mechanics. The wav ...
... particle can cross the potential barrier. This is the effect known as tunnelling. It can also be seen that if the particle energy is greater than there is still a chance that the particle will be reflected from the potential barrier. Neither of these phenomena exist in classical mechanics. The wav ...
The strange (hi)story of particles and waves*
... that was to replace the concept of deterministic trajectories by formal probabilistic rules. Together with Born and Jordan, Heisenberg then constructed an elegant algebraic framework that could be used to “quantize” all mechanical systems. This mathematical abstraction perfectly matched Heisenberg’s ...
... that was to replace the concept of deterministic trajectories by formal probabilistic rules. Together with Born and Jordan, Heisenberg then constructed an elegant algebraic framework that could be used to “quantize” all mechanical systems. This mathematical abstraction perfectly matched Heisenberg’s ...
Chapter 7: Quantum Mechanical Model of Atom
... • Werner Heisenberg - showed that it is impossible to know (or measure) precisely both the position and velocity (or the momentum) at the same time. • The simple act of “seeing” an electron would change ...
... • Werner Heisenberg - showed that it is impossible to know (or measure) precisely both the position and velocity (or the momentum) at the same time. • The simple act of “seeing” an electron would change ...
Quantum Mechanics: Particles in Potentials
... As the quantum number increases to large values, probability of particle position approaches uniform distribution in the region [0,a]. This is the classical limit. Quantum mechanics approaches classical mechanics in the limit of large quantum numbers. As the quantum number increases to large values ...
... As the quantum number increases to large values, probability of particle position approaches uniform distribution in the region [0,a]. This is the classical limit. Quantum mechanics approaches classical mechanics in the limit of large quantum numbers. As the quantum number increases to large values ...
the original file
... are like the macroscopic version of stationary states. Classical normal modes can be seen in molecular vibrations. Imagine for a moment, that a molecule represents our quantum mechanical operator. Then each oscillatory degree of freedom for the molecule (asymmetric and symmetric flexing, stretching, ...
... are like the macroscopic version of stationary states. Classical normal modes can be seen in molecular vibrations. Imagine for a moment, that a molecule represents our quantum mechanical operator. Then each oscillatory degree of freedom for the molecule (asymmetric and symmetric flexing, stretching, ...
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.