The Hydrogen Atom Revisited
... the solution which represents the electron, a suitable boundary condition must be imposed. Schrödinger postulated a boundary condition: Ψ → 0 as r → ∞ , which leads to a purely mathematical model of the electron. This equation is not based on first principles, has no validity as such, and should not ...
... the solution which represents the electron, a suitable boundary condition must be imposed. Schrödinger postulated a boundary condition: Ψ → 0 as r → ∞ , which leads to a purely mathematical model of the electron. This equation is not based on first principles, has no validity as such, and should not ...
Brief history of the atom
... If an object is hotter than its surroundings it will cool by giving off light. In order to study this effect scientist had to eliminate the other modes of cooling. Blocks of graphite were hollowed and a small hole was drilled into the carbon. Although the outside of the carbon block would cool by co ...
... If an object is hotter than its surroundings it will cool by giving off light. In order to study this effect scientist had to eliminate the other modes of cooling. Blocks of graphite were hollowed and a small hole was drilled into the carbon. Although the outside of the carbon block would cool by co ...
A Quantum Mechanical Supertask
... is undefined. Thus it will turn out that the time evolution governed by the differential SchroÈdinger equation need not conserve the norm. We will see cases in which a normed state develops to an unnormalizable state. In Sec. 1 below, I review some classical supertasks, which will be the models for ...
... is undefined. Thus it will turn out that the time evolution governed by the differential SchroÈdinger equation need not conserve the norm. We will see cases in which a normed state develops to an unnormalizable state. In Sec. 1 below, I review some classical supertasks, which will be the models for ...
Chapter_9 - Experimental Elementary Particle Physics Group
... and the empty set) which is closed under unions and finite intersections, and such that a point p is a limit point of a subset A of S if and only if every open set containing p also contains a point of A distinct from p. For example, if we define the collection of open spherical regions in Euclidean ...
... and the empty set) which is closed under unions and finite intersections, and such that a point p is a limit point of a subset A of S if and only if every open set containing p also contains a point of A distinct from p. For example, if we define the collection of open spherical regions in Euclidean ...
Spin and Pauli`s Principle
... allowed values of s = 0, 12 , 1, 32 ... (note difference with angular momentum!) For a given s, ms can take values from −s, ..., s The main point is for each type of quantum particle s is fixed....is an intrinsic property of the particle...for electrons s = 12 VBS/MRC ...
... allowed values of s = 0, 12 , 1, 32 ... (note difference with angular momentum!) For a given s, ms can take values from −s, ..., s The main point is for each type of quantum particle s is fixed....is an intrinsic property of the particle...for electrons s = 12 VBS/MRC ...
7.1 Electronic states of helium atom 7.2 The Variation Method
... eigenvalues ms=1/2 and -1/2, respectively. By taking a spin function for each electron we can form 22=4 products as shown in the first row of Table 7.4. The second and third rows of the table give the eigenvalues of ̂ and ̂ when the product spin function is an eigenfunction of the corresponding oper ...
... eigenvalues ms=1/2 and -1/2, respectively. By taking a spin function for each electron we can form 22=4 products as shown in the first row of Table 7.4. The second and third rows of the table give the eigenvalues of ̂ and ̂ when the product spin function is an eigenfunction of the corresponding oper ...
One-Dimensional Mass-Spring Chains Supporting Elastic Waves
... and supports elastic waves with conventional topology in wave vector space. Perturbing the 1D harmonic chain through linear or nonlinear coupling may create resonant phonon modes that are topology in wave vector space. Perturbing the 1D harmonic chain through linear or nonlinear coupling dispersive, ...
... and supports elastic waves with conventional topology in wave vector space. Perturbing the 1D harmonic chain through linear or nonlinear coupling may create resonant phonon modes that are topology in wave vector space. Perturbing the 1D harmonic chain through linear or nonlinear coupling dispersive, ...
Quantum Statistical Mechanics - Physics Department
... from the world of macroscopic phenomena using some general principles or from the microscopic world of molecules, atoms, elementary particles. A macroscopic approach is used in thermodynamics, while the microscopic approach constitutes the framework of statistical mechanics. Depending on whether the ...
... from the world of macroscopic phenomena using some general principles or from the microscopic world of molecules, atoms, elementary particles. A macroscopic approach is used in thermodynamics, while the microscopic approach constitutes the framework of statistical mechanics. Depending on whether the ...
Nonabelions in the fractional quantum hall effect
... The fractional quantum Hall effect (FQHE) [3], i.e. a plateau in the Hall resistance, is observed in two-dimensional electron gases in high magnetic fields only when the mobile charged excitations have a gap in their excitation spectrum, so the system is incompressible (in the absence of disorder). ...
... The fractional quantum Hall effect (FQHE) [3], i.e. a plateau in the Hall resistance, is observed in two-dimensional electron gases in high magnetic fields only when the mobile charged excitations have a gap in their excitation spectrum, so the system is incompressible (in the absence of disorder). ...
Emergence of exponentially small reflected waves
... Since φright (x, E) is essentially of order e−ξc (E)/ε , the leading order contribution to Ψright (x, t) comes from energies E near the maximum E ∗ (ε) of |Q(E, ε)|e−ξc (E)/ε . In order to see nice asymptotics in the semiclassical limit ε → 0 we impose conditions on Q(E, ε) that guarantee that E ∗ ...
... Since φright (x, E) is essentially of order e−ξc (E)/ε , the leading order contribution to Ψright (x, t) comes from energies E near the maximum E ∗ (ε) of |Q(E, ε)|e−ξc (E)/ε . In order to see nice asymptotics in the semiclassical limit ε → 0 we impose conditions on Q(E, ε) that guarantee that E ∗ ...
Review: Castro-Neto et al, Rev. Mod Phys. Abanin, Lee and Levitov
... reversal T. Sign of mass the same for K and K’. 3. Second way is to have complex t’ as suggested by Haldane. Sign of mass is opposite for K and K’ and T is broken. ...
... reversal T. Sign of mass the same for K and K’. 3. Second way is to have complex t’ as suggested by Haldane. Sign of mass is opposite for K and K’ and T is broken. ...
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.