Copyright c 2016 by Robert G. Littlejohn Physics 221A Fall 2016
... As we have already noted, H⊥ and Hk commute with each other and with H = H⊥ + Hk , so we can take (H⊥ , Hk ), or, better, (H⊥ , P̂3 ) as a pair of commuting operators. Recall that Hk = P̂32 /2m. But these operators do not form a complete set because the first degree of freedom (Q̂1 , P̂1 ), or, equi ...
... As we have already noted, H⊥ and Hk commute with each other and with H = H⊥ + Hk , so we can take (H⊥ , Hk ), or, better, (H⊥ , P̂3 ) as a pair of commuting operators. Recall that Hk = P̂32 /2m. But these operators do not form a complete set because the first degree of freedom (Q̂1 , P̂1 ), or, equi ...
chap3
... Suppose that f(x) and g(x) are two eigenfunctions of an operator Q, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfuntion of Q, with eigenvalue q. ...
... Suppose that f(x) and g(x) are two eigenfunctions of an operator Q, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfuntion of Q, with eigenvalue q. ...
P410M: Relativistic Quantum Fields
... For example, the position-space wavefunction is hxjÃi = Ãx(x) and jhxjÃij2 = jÃx(x)j2 is the probability of finding the particle at position x. Sincejxi and jpi are not aligned bases, the state cannot be an eigenvector of position and momentum simultaneously. Also, since the measurements change the ...
... For example, the position-space wavefunction is hxjÃi = Ãx(x) and jhxjÃij2 = jÃx(x)j2 is the probability of finding the particle at position x. Sincejxi and jpi are not aligned bases, the state cannot be an eigenvector of position and momentum simultaneously. Also, since the measurements change the ...
Solving Schrödinger`s equation around a desired energy
... both to the bulk band structure and to the surface work function (4.9 eV)Y We passivate the surface dangling bonds by hydrogen atoms and model the surface relaxations of the chemisorbed layer according to data for hydrogen-covered (001), (111),16 and (110) 17 Si surfaces. The hydrogen empirical pseu ...
... both to the bulk band structure and to the surface work function (4.9 eV)Y We passivate the surface dangling bonds by hydrogen atoms and model the surface relaxations of the chemisorbed layer according to data for hydrogen-covered (001), (111),16 and (110) 17 Si surfaces. The hydrogen empirical pseu ...
Many-body systems
... so that we have to perform N integrals over real space (all operators we deal with are diagonal in positions); in general 2N sums over spin indexes (which reduce to N if the operator is diagonal in spin-space, as I assumed here) ... and this is out of a combination of a product of order (N !) 2 term ...
... so that we have to perform N integrals over real space (all operators we deal with are diagonal in positions); in general 2N sums over spin indexes (which reduce to N if the operator is diagonal in spin-space, as I assumed here) ... and this is out of a combination of a product of order (N !) 2 term ...
7 WZW term in quantum mechanics: single spin
... In this simplified treatment we just found some classical action which reproduces the classical limit of operator equations of motion (7.4). One can proceed more formally starting with commutation relations (7.1) and quantum Hamiltonian (7.3) and derive the classical action (7.7) using, e.g., cohere ...
... In this simplified treatment we just found some classical action which reproduces the classical limit of operator equations of motion (7.4). One can proceed more formally starting with commutation relations (7.1) and quantum Hamiltonian (7.3) and derive the classical action (7.7) using, e.g., cohere ...
Quantum Mechanics and Common Sense
... A wave has a phase, so it is necessary to ascribe phases also to these bra or ket corpuscles. One may imagine them as the ”messengers with clocks” used in [12,13 ] for the numerical simulation of quantum phenomena. Actually are these bra or ket waves or corpuscles is the detail of secondary importan ...
... A wave has a phase, so it is necessary to ascribe phases also to these bra or ket corpuscles. One may imagine them as the ”messengers with clocks” used in [12,13 ] for the numerical simulation of quantum phenomena. Actually are these bra or ket waves or corpuscles is the detail of secondary importan ...
Quantum states in phase space • classical vs. quantum statistics
... look formally the same. The expectation value of both classical and quantum observables are obtained by averaging the values of the observable in phase space with a distribution function. The important difference is that the in the quantum case the Wigner function is in general not a well-defined prob ...
... look formally the same. The expectation value of both classical and quantum observables are obtained by averaging the values of the observable in phase space with a distribution function. The important difference is that the in the quantum case the Wigner function is in general not a well-defined prob ...
Chapter 1. The Basics of Quantum Mechanics
... So, now we have some idea about why Newton’s equations fail to account for the dynamical motions of light and small particles such as electrons and nuclei. We see that extra conditions (e.g., the Bragg condition or constraints on the de Broglie wavelength) could be imposed to achieve some degree of ...
... So, now we have some idea about why Newton’s equations fail to account for the dynamical motions of light and small particles such as electrons and nuclei. We see that extra conditions (e.g., the Bragg condition or constraints on the de Broglie wavelength) could be imposed to achieve some degree of ...
11. Scattering from a Barrier
... incoming quantum particle is not precisely defined. Technically that’s always true in the real world, but real-world wavepackets are often much wider in space, and hence narrower in momentum space and energy resolution, than in the simulation. The more common approach to analyzing barrier scattering ...
... incoming quantum particle is not precisely defined. Technically that’s always true in the real world, but real-world wavepackets are often much wider in space, and hence narrower in momentum space and energy resolution, than in the simulation. The more common approach to analyzing barrier scattering ...
Chapter 1
... So, now we have some idea about why Newton’s equations fail to account for the dynamical motions of light and small particles such as electrons and nuclei. We see that extra conditions (e.g., the Bragg condition or constraints on the de Broglie wavelength) could be imposed to achieve some degree of ...
... So, now we have some idea about why Newton’s equations fail to account for the dynamical motions of light and small particles such as electrons and nuclei. We see that extra conditions (e.g., the Bragg condition or constraints on the de Broglie wavelength) could be imposed to achieve some degree of ...
PPT
... y(x) must be single-valued, and finite. Finite to avoid infinite probability density. y(x) must be continuous, with finite dy/dx. dy/dx is related to the momentum. ...
... y(x) must be single-valued, and finite. Finite to avoid infinite probability density. y(x) must be continuous, with finite dy/dx. dy/dx is related to the momentum. ...
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.