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... It is the probability of finding the particle that waves. A matter wave is described by the wave function, Ψ. The square of the wave function |Ψ| 2 (probability distribution) at any point is proportional to the number of particles expected to be found there. For a single particle, the wave function ...
... It is the probability of finding the particle that waves. A matter wave is described by the wave function, Ψ. The square of the wave function |Ψ| 2 (probability distribution) at any point is proportional to the number of particles expected to be found there. For a single particle, the wave function ...
Homework 2
... (b) According to the Bohr model, electrons move on circular orbits and the angular momentum L can assume the values L = n~, n ∈ {1, 2, . . .}. Determine the possible energies En , orbital radii rn , and velocity ratios vn /c, where c is the speed of light. (c) Bohr assumed that only radiation of fre ...
... (b) According to the Bohr model, electrons move on circular orbits and the angular momentum L can assume the values L = n~, n ∈ {1, 2, . . .}. Determine the possible energies En , orbital radii rn , and velocity ratios vn /c, where c is the speed of light. (c) Bohr assumed that only radiation of fre ...
Physical Chemistry Postulates of quantum mechanics Origins of
... corresponding to a b dV 0 if a b different eigenvalues all space are orthogonal. The set of all eigenfunctions of a hermitian operator is complete. Any function f ( x, y, z ) ck k ( x, y , z ) of the coordinates on k which it depends can be expressed as a linear combination of the membe ...
... corresponding to a b dV 0 if a b different eigenvalues all space are orthogonal. The set of all eigenfunctions of a hermitian operator is complete. Any function f ( x, y, z ) ck k ( x, y , z ) of the coordinates on k which it depends can be expressed as a linear combination of the membe ...
Statistical laws
... molecules. This is obviously not possible and also unnecessary. The reason is that the presence of a large number of particles causes new types of regularity (statistical law). These statistical laws cannot in any ...
... molecules. This is obviously not possible and also unnecessary. The reason is that the presence of a large number of particles causes new types of regularity (statistical law). These statistical laws cannot in any ...
Periodic boundary physics etc
... In physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics. In the standard interpretation of quantum mechanics, the q ...
... In physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics. In the standard interpretation of quantum mechanics, the q ...
Lecture 16
... Completely solving the problem is simply a matter of equating the wave function and first derivative of the wave function at the potential boundary. More interesting is comparing the probability for a wave to exist as a right moving, transmitted wave, or left moving, reflected, wave after encounteri ...
... Completely solving the problem is simply a matter of equating the wave function and first derivative of the wave function at the potential boundary. More interesting is comparing the probability for a wave to exist as a right moving, transmitted wave, or left moving, reflected, wave after encounteri ...
Quantum Mechanics Lecture Course for 4 Semester Students by W.B. von Schlippe
... The history of optical theories shows that the scientific view has for long oscillated between a mechanical and an undulatory conception of light; however, these two views are perhaps less opposed to one another than was previously thought, and the development of quantum theory, in particular, appea ...
... The history of optical theories shows that the scientific view has for long oscillated between a mechanical and an undulatory conception of light; however, these two views are perhaps less opposed to one another than was previously thought, and the development of quantum theory, in particular, appea ...
Lecture-XXIII Quantum Mechanics
... Therefore, one can't impose an arbitrary condition on ψ without checking that the two are consistent. Interestingly, if ψ(x, t) is a solution, Aψ(x, t) is also a solution where A is any (complex) constant. Therefore, one must pick a undetermined multiplicative factor in such a way that the Schroding ...
... Therefore, one can't impose an arbitrary condition on ψ without checking that the two are consistent. Interestingly, if ψ(x, t) is a solution, Aψ(x, t) is also a solution where A is any (complex) constant. Therefore, one must pick a undetermined multiplicative factor in such a way that the Schroding ...
WAVE MECHANICS (Schrödinger, 1926)
... * The orbitals are named by giving the n value followed by a letter symbol for l: l= 0,1, 2, 3, 4, 5, ... s p d f g h ... * All orbitals with the same n are called a “shell”. All orbitals with the same n and l are called a “subshell”. ...
... * The orbitals are named by giving the n value followed by a letter symbol for l: l= 0,1, 2, 3, 4, 5, ... s p d f g h ... * All orbitals with the same n are called a “shell”. All orbitals with the same n and l are called a “subshell”. ...
BasicQuantumMechanics18And20January2017
... • Inconsistency with classical light theory According to the classical wave theory, maximum kinetic energy of the photoelectron is only dependent on the incident intensity of the light, and independent on the light frequency; however, experimental results show that the kinetic energy of the photoele ...
... • Inconsistency with classical light theory According to the classical wave theory, maximum kinetic energy of the photoelectron is only dependent on the incident intensity of the light, and independent on the light frequency; however, experimental results show that the kinetic energy of the photoele ...
PHYS-2100 Introduction to Methods of Theoretical Physics Fall 1998 1) a)
... instead of k to avoid confusion with the wave number. Here, m is the mass of the particle as usual, and ω just turns out to be the classical angular frequency of the oscillator.) The solution involves the series approach to solving differential equations (see Landshoff, Appendix A) but in this probl ...
... instead of k to avoid confusion with the wave number. Here, m is the mass of the particle as usual, and ω just turns out to be the classical angular frequency of the oscillator.) The solution involves the series approach to solving differential equations (see Landshoff, Appendix A) but in this probl ...
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.