2010 midterm exam solutions
... probability of obtaining the nth eigenvalue is given by |an |2 . Here then we have p(E91 , t = t1 ) = |a91 |2 . The probability of a particular energy does not change with time. More precisely, the wavefunction at a later time would be given by ψ(x, t) = a1 w1 (x)e−iE1 (t−t1 )/n + a91 w91 (x)e−iE91 ...
... probability of obtaining the nth eigenvalue is given by |an |2 . Here then we have p(E91 , t = t1 ) = |a91 |2 . The probability of a particular energy does not change with time. More precisely, the wavefunction at a later time would be given by ψ(x, t) = a1 w1 (x)e−iE1 (t−t1 )/n + a91 w91 (x)e−iE91 ...
Wave Function as Geometric Entity
... set of Dirac spinors preserves the structure of the linear vector space, but does not preserve the ring structure since defining the composition operation involves some complications. Thus allowed states are depleted inasmuch as wave function behavior under the parallel translation cannot be calcula ...
... set of Dirac spinors preserves the structure of the linear vector space, but does not preserve the ring structure since defining the composition operation involves some complications. Thus allowed states are depleted inasmuch as wave function behavior under the parallel translation cannot be calcula ...
Course Outline Template Word Document - Physics for All
... This course is intended to be a first introduction to quantum phenomena in nature. Quatum Mechanics forms the basis of our description of nature at small scales and a clear understanding of it is required to understand phenomena ranging from atoms and chemical bonding to semiconductors and nuclear p ...
... This course is intended to be a first introduction to quantum phenomena in nature. Quatum Mechanics forms the basis of our description of nature at small scales and a clear understanding of it is required to understand phenomena ranging from atoms and chemical bonding to semiconductors and nuclear p ...
PPT
... • Observation is very complicated and depends on processes which theory is supposed to explain. ...
... • Observation is very complicated and depends on processes which theory is supposed to explain. ...
10.5.1. Density Operator
... When dealing with a large quantum system, we need to take 2 averages, one over the inherent quantum uncertainties and one over the uninteresting microscopic details. Consider then an isolated system described, in the Schrodinger picture, by a complete set of orthonormal eigenstates n t ...
... When dealing with a large quantum system, we need to take 2 averages, one over the inherent quantum uncertainties and one over the uninteresting microscopic details. Consider then an isolated system described, in the Schrodinger picture, by a complete set of orthonormal eigenstates n t ...
Document
... The above could be done for 2 by 2 matrices if there is no mass. Massless fermion contains only half the degrees of freedom. ...
... The above could be done for 2 by 2 matrices if there is no mass. Massless fermion contains only half the degrees of freedom. ...
Particle in a box (PPT - 6.9MB)
... The Schrodinger equation is given above. 1. The wavefunction Ψ can be complex, so we should remember to take the Real part of Ψ. 2. Time-harmonic solutions to Schrodinger equation are of the form: 3. Ψ(x,t) is a measurable quantity and represents the probability distribution of finding the particle. ...
... The Schrodinger equation is given above. 1. The wavefunction Ψ can be complex, so we should remember to take the Real part of Ψ. 2. Time-harmonic solutions to Schrodinger equation are of the form: 3. Ψ(x,t) is a measurable quantity and represents the probability distribution of finding the particle. ...
Particle in a box - MIT OpenCourseWare
... The Schrodinger equation is given above. 1. The wavefunction Ψ can be complex, so we should remember to take the Real part of Ψ. 2. Time-harmonic solutions to Schrodinger equation are of the form: 3. Ψ(x,t) is a measurable quantity and represents the probability distribution of finding the particle. ...
... The Schrodinger equation is given above. 1. The wavefunction Ψ can be complex, so we should remember to take the Real part of Ψ. 2. Time-harmonic solutions to Schrodinger equation are of the form: 3. Ψ(x,t) is a measurable quantity and represents the probability distribution of finding the particle. ...
7 - Physics at Oregon State University
... 1. Normalized ket vector contains all the information about the state of a quantum mechanical system. 2. Operator A describes a physical observable and acts on kets. 3. One of the eigenvalues an of A is the only possible result of a measurement. 4. The probability of obtaining the eigenvalue an : ...
... 1. Normalized ket vector contains all the information about the state of a quantum mechanical system. 2. Operator A describes a physical observable and acts on kets. 3. One of the eigenvalues an of A is the only possible result of a measurement. 4. The probability of obtaining the eigenvalue an : ...
BWilliamsPaper - FSU High Energy Physics
... think of light as being a stream of particles that together, act like a wave, with some photons canceling each other out, and some photons building on each other. Unfortunately, this doesn’t work. Let’s say we slow down the source to emit only one photon at a time, say one every ten seconds. With o ...
... think of light as being a stream of particles that together, act like a wave, with some photons canceling each other out, and some photons building on each other. Unfortunately, this doesn’t work. Let’s say we slow down the source to emit only one photon at a time, say one every ten seconds. With o ...
PHYSICAL MEANING OF IMAGINARY UNIT i
... Fig. 1. Distribution of domains of maxima of the wave function modulus ̂ at l =1 and m=0 in a spherical space-field of the hydrogen atom; a) Yˆl ,m (, ) is the surface of the modulus of the polar-azimuth factor of the wave function, describes a surface shaped like a dumbbell; s1 and s2 (b) are do ...
... Fig. 1. Distribution of domains of maxima of the wave function modulus ̂ at l =1 and m=0 in a spherical space-field of the hydrogen atom; a) Yˆl ,m (, ) is the surface of the modulus of the polar-azimuth factor of the wave function, describes a surface shaped like a dumbbell; s1 and s2 (b) are do ...
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.