Physics 228, Lecture 12 Thursday, March 3, 2005 Uncertainty
... and thus it is in principle impossible to know a particle’s position and velocity precisely at any instant of time. One way to understand this is to imagine that we have a free particle for which, somehow, we have measured its momentum precisely. Now if we can measure where it is without messing up ...
... and thus it is in principle impossible to know a particle’s position and velocity precisely at any instant of time. One way to understand this is to imagine that we have a free particle for which, somehow, we have measured its momentum precisely. Now if we can measure where it is without messing up ...
The buoyant force on an object totally submerged in a fluid depends
... "We believe in all truth, no matter to what subject it may refer. No sect or religious denomination [or, I may say, no searcher of truth] in the world possesses a single principle of truth that we do not accept or that we will reject. We are willing to receive all truth, from whatever source it may ...
... "We believe in all truth, no matter to what subject it may refer. No sect or religious denomination [or, I may say, no searcher of truth] in the world possesses a single principle of truth that we do not accept or that we will reject. We are willing to receive all truth, from whatever source it may ...
14-Research quantum mechanical methods of bioobjects
... 3. The wave function must be twice differentiable. This means that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.) 4. In order to normalize a wave function, it must approach zero as x approaches infinity. ...
... 3. The wave function must be twice differentiable. This means that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.) 4. In order to normalize a wave function, it must approach zero as x approaches infinity. ...
Subject Description Form
... (a) articulate the experimental basis for attributing particle properties to waves and wave properties to particles; elaborate on the de Broglie theory of matter waves; apply Heisenberg’s uncertainty principle to simple systems; (b) elaborate on the various forms of Schrödinger’s equation and identi ...
... (a) articulate the experimental basis for attributing particle properties to waves and wave properties to particles; elaborate on the de Broglie theory of matter waves; apply Heisenberg’s uncertainty principle to simple systems; (b) elaborate on the various forms of Schrödinger’s equation and identi ...
Quantum Mechanics
... hydrogen atom states can be described by the single quantum number n, the wave functions describing these states require three quantum numbers. Principal quantum number: n = 1, 2,3,... Orbital quantum number: l = 1, 2,3,..., n − 1 Orbital magnetic quantum number m = −l , −l + 1,...0,...l − 1, l The ...
... hydrogen atom states can be described by the single quantum number n, the wave functions describing these states require three quantum numbers. Principal quantum number: n = 1, 2,3,... Orbital quantum number: l = 1, 2,3,..., n − 1 Orbital magnetic quantum number m = −l , −l + 1,...0,...l − 1, l The ...
CR2
... predict what the system will do at any time after the initial conditions of the system. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebrai ...
... predict what the system will do at any time after the initial conditions of the system. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebrai ...
Physical Chemistry The hydrogen atom Center of mass
... Addition of spin The spatial part is incomplete One incorporates spin as a separate coordinate Wave function is a product n ,l ,m , I ,mI (r , , ) ...
... Addition of spin The spatial part is incomplete One incorporates spin as a separate coordinate Wave function is a product n ,l ,m , I ,mI (r , , ) ...
Physical Chemistry The hydrogen atom Center of mass
... 1 hartree 27.2114 electron volts 1 rydberg 13.606 electron volts ...
... 1 hartree 27.2114 electron volts 1 rydberg 13.606 electron volts ...
Chapter7Part3
... (for example: the path of a thrown ball) (for example: the motion of an electron in an atom) the path of the ball is given by - the electron is moving so fast and it has such a its position and its velocity at small mass, that its path cannot be predicted various times we think of the ball as moving ...
... (for example: the path of a thrown ball) (for example: the motion of an electron in an atom) the path of the ball is given by - the electron is moving so fast and it has such a its position and its velocity at small mass, that its path cannot be predicted various times we think of the ball as moving ...
Document
... terms of probabilities and not specific numbers. Therefore, instead of finding the average value of any term (for example position of particle x ), we find the expectation value of that.
...
... terms of probabilities and not specific numbers. Therefore, instead of finding the average value of any term (for example position of particle x ), we find the expectation value
Developing a new physics for atoms
... and they would spiral into the nucleus and collapse the atom. But, in the subatomic quantum world, atoms are subject to quantum mechanics rather than the Newtonian laws of motion. • Electrons have quantum energy and that level of energy can only be changed transiently. • Uncertainty says that if an ...
... and they would spiral into the nucleus and collapse the atom. But, in the subatomic quantum world, atoms are subject to quantum mechanics rather than the Newtonian laws of motion. • Electrons have quantum energy and that level of energy can only be changed transiently. • Uncertainty says that if an ...
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.