Document
... In mechanics and field theory (both classical and quantum), there are two main languages - Lagrangian and Hamiltonian. In the classical setting, the Lagrangian language is the language of vari-ational calculus (i.e. one studies extremals of the action functional), while the Hamiltonian language is t ...
... In mechanics and field theory (both classical and quantum), there are two main languages - Lagrangian and Hamiltonian. In the classical setting, the Lagrangian language is the language of vari-ational calculus (i.e. one studies extremals of the action functional), while the Hamiltonian language is t ...
Lecture 27: Quantum Mechanics (Continued)
... Scanning Tunneling Microscopy (STM) Since electronic wavefunctions have nodes and anti-nodes, is it possible to actually image the electronic wavefunction? The answer is definitely yes, but it took almost 60 years to experimentally demonstrate it. 1986 Nobel prize went to two scientists at IBM to p ...
... Scanning Tunneling Microscopy (STM) Since electronic wavefunctions have nodes and anti-nodes, is it possible to actually image the electronic wavefunction? The answer is definitely yes, but it took almost 60 years to experimentally demonstrate it. 1986 Nobel prize went to two scientists at IBM to p ...
Notes on total internal reflection and waveguides
... 2 For finite-dimensional Hermitian operators, you always have a basis of eigenvectors (diagonalizability). For infinite-dimensional operators (i.e. operators on functions), one can construct pathological counterexamples where this is not the case, but these are rarely relevant to physical problems—t ...
... 2 For finite-dimensional Hermitian operators, you always have a basis of eigenvectors (diagonalizability). For infinite-dimensional operators (i.e. operators on functions), one can construct pathological counterexamples where this is not the case, but these are rarely relevant to physical problems—t ...
English
... According to protective measurement, the charge of a charged quantum system such as an electron is distributed throughout space, and the charge density in each position is proportional to the modulus square of the wave function of the system there. Historically, the charge density interpretation for ...
... According to protective measurement, the charge of a charged quantum system such as an electron is distributed throughout space, and the charge density in each position is proportional to the modulus square of the wave function of the system there. Historically, the charge density interpretation for ...
Lecture Notes V: Spin, Pauli Exclusion Principle, Symmetric
... Electrons, because they satisfy the Pauli exclusion principle, don’t “like” each other and are actually rather good at being “noninteracting.” In a few minutes, we will see that there is a different take on this idea… If the particles are identical, it shouldn't make a difference to our measurements ...
... Electrons, because they satisfy the Pauli exclusion principle, don’t “like” each other and are actually rather good at being “noninteracting.” In a few minutes, we will see that there is a different take on this idea… If the particles are identical, it shouldn't make a difference to our measurements ...
Quantum Mechanics: Concepts and Applications, 2nd Edition
... The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electro ...
... The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electro ...
Chapter21 - Cobb Learning
... A mechanical wave is a physical disturbance in an elastic medium. Consider a stone dropped into a lake. Energy is transferred from stone to floating log, but only the disturbance travels. Actual motion of any individual water particle is small. Energy propagation via such a disturbance is known as m ...
... A mechanical wave is a physical disturbance in an elastic medium. Consider a stone dropped into a lake. Energy is transferred from stone to floating log, but only the disturbance travels. Actual motion of any individual water particle is small. Energy propagation via such a disturbance is known as m ...
Lecture 2
... photons are particles. If we now wait long enough such that very many photons leave marks on the screen eventually an intensity pattern will emerge and this pattern is an interference pattern!!! This is cleary very strange. If photons are particles, they should go thtrough only one slit. But if that ...
... photons are particles. If we now wait long enough such that very many photons leave marks on the screen eventually an intensity pattern will emerge and this pattern is an interference pattern!!! This is cleary very strange. If photons are particles, they should go thtrough only one slit. But if that ...
Discussion of Experimental Proof for the Paradox of Einstein, Rosen
... First of all, Bohr' has proposed that the observing apparatus plus what is observed form a single indivisible combined system not capable at the quantummechanical level of being analyzed correctly into separate and distinct parts. Each particular kind of apparatus then forms with an electron for exa ...
... First of all, Bohr' has proposed that the observing apparatus plus what is observed form a single indivisible combined system not capable at the quantummechanical level of being analyzed correctly into separate and distinct parts. Each particular kind of apparatus then forms with an electron for exa ...
the heisenberg uncertainty relation derived by multiplying matter
... (Received 14 July 2005; accepted 19 October 2005) ...
... (Received 14 July 2005; accepted 19 October 2005) ...
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.