Table: Various One Dimensional Potentials System Physical Potential Total Energies and Probability Density Significant Feature Correspondence Free Particle I.E
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Particle-On-A-Ring” Suppose a Diatomic Molecule Rotates in Such a Way That the Vibration of the Bond Is Unaffected by the Rotation
University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015 Lecture 17 2/25/15 Recommended Reading Atkins & DePaula 9.5 A. Background in Rotational Mechanics Two masses m1 and m2 connected by a weightless rigid “bond” of length r rotates with angular velocity rv where v is the linear velocity. As with vibrations we can render this rotation motion in simpler form by fixing one end of the bond to the origin terminating the other end with reduced mass , and allowing the reduced mass to rotate freely For two masses m1 and m2 separated by r the center of mass (CoM) is defined by the condition mr11 mr 2 2 (17.1) where m2,1 rr1,2 mm12 (17.2) Figure 17.1: Location of center of mass (CoM0 for two masses separated by a distance r. The moment of inertia I plays the role in the rotational energy that is played by mass in translational energy. The moment of inertia is 22 I mr11 mr 2 2 (17.3) Putting 17.3 into 17.4 we obtain I r 2 (17.4) mm where 12 . mm12 We can express the rotational kinetic energy K in terms of the angular momentum. The angular momentum is defined in terms of the cross product between the position vector r and the linear momentum vector p: Lrp (17.5) prprvrsin where the angle between p and r is ninety degrees for circular motion. As a cross product, the angular momentum vector L is perpendicular to the plane defined by the r and p vectors as shown in Figure 17.2 We can use equation 17.6 to obtain an expression for the kinetic energy in terms of the angular momentum L: Figure 17.2: The angular momentum L is a cross product of the position r vector and the linear momentum p=mv vector. -
Quantum Mechanics
Quantum Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Intendedaudience................................ 5 1.2 MajorSources .................................. 5 1.3 AimofCourse .................................. 6 1.4 OutlineofCourse ................................ 6 2 Probability Theory 7 2.1 Introduction ................................... 7 2.2 WhatisProbability?.............................. 7 2.3 CombiningProbabilities. ... 7 2.4 Mean,Variance,andStandardDeviation . ..... 9 2.5 ContinuousProbabilityDistributions. ........ 11 3 Wave-Particle Duality 13 3.1 Introduction ................................... 13 3.2 Wavefunctions.................................. 13 3.3 PlaneWaves ................................... 14 3.4 RepresentationofWavesviaComplexFunctions . ....... 15 3.5 ClassicalLightWaves ............................. 18 3.6 PhotoelectricEffect ............................. 19 3.7 QuantumTheoryofLight. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 3.8 ClassicalInterferenceofLightWaves . ...... 21 3.9 QuantumInterferenceofLight . 22 3.10 ClassicalParticles . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 3.11 QuantumParticles............................... 25 3.12 WavePackets .................................. 26 2 QUANTUM MECHANICS 3.13 EvolutionofWavePackets . 29 3.14 Heisenberg’sUncertaintyPrinciple . ........ 32 3.15 Schr¨odinger’sEquation . 35 3.16 CollapseoftheWaveFunction . 36 4 Fundamentals of Quantum Mechanics 39 4.1 Introduction .................................. -
Electric Polarizability in the Three Dimensional Problem and the Solution of an Inhomogeneous Differential Equation
Electric polarizability in the three dimensional problem and the solution of an inhomogeneous differential equation M. A. Maizea) and J. J. Smetankab) Department of Physics, Saint Vincent College, Latrobe, PA 15650 USA In previous publications, we illustrated the effectiveness of the method of the inhomogeneous differential equation in calculating the electric polarizability in the one-dimensional problem. In this paper, we extend our effort to apply the method to the three-dimensional problem. We calculate the energy shift of a quantum level using second-order perturbation theory. The energy shift is then used to calculate the electric polarizability due to the interaction between a static electric field and a charged particle moving under the influence of a spherical delta potential. No explicit use of the continuum states is necessary to derive our results. I. INTRODUCTION In previous work1-3, we employed the simple and elegant method of the inhomogeneous differential equation4 to calculate the energy shift of a quantum level in second-order perturbation theory. The energy shift was a result of an interaction between an applied static electric field and a charged particle moving under the influence of a one-dimensional bound 1 potential. The electric polarizability, , was obtained using the basic relationship ∆퐸 = 휖2 0 2 with ∆퐸0 being the energy shift and is the magnitude of the applied electric field. The method of the inhomogeneous differential equation devised by Dalgarno and Lewis4 and discussed by Schwartz5 can be used in a large variety of problems as a clever replacement for conventional perturbation methods. As we learn in our introductory courses in quantum mechanics, calculating the energy shift in second-order perturbation using conventional methods involves a sum (which can be infinite) or an integral that contains all possible states allowed by the transition. -
Revisiting Double Dirac Delta Potential
Revisiting double Dirac delta potential Zafar Ahmed1, Sachin Kumar2, Mayank Sharma3, Vibhu Sharma3;4 1Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India 2Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India 3;4Amity Institute of Applied Sciences, Amity University, Noida, UP, 201313, India∗ (Dated: June 14, 2016) Abstract We study a general double Dirac delta potential to show that this is the simplest yet versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission (T = 1). Perfect transmission energies turn out to be the critical property of symmetric and anti- symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission [or zero reflectivity, R(E)] at energy E = 0 is non-intuitive. However, earlier this has been found and called \threshold anomaly". Here we show that it is a critical phenomenon and we can have 0 ≤ R(0) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke zero-energy and zero curvature eigenstate ( (x) = Ax + B) of delta well between two symmetric rigid walls for R(0) = 0. We resolve that the resonant energies and the perfect transmission energies are different and they arise differently. arXiv:1603.07726v4 [quant-ph] 13 Jun 2016 ∗Electronic address: 1:[email protected], 2: [email protected], 3: [email protected], 4:[email protected] 1 I. INTRODUCTION The general one-dimensional Double Dirac Delta Potential (DDDP) is written as [see Fig. -
Aromaticity As a Guiding Concept for Spectroscopic Features and Nonlinear Optical Properties of Porphyrinoids
molecules Article Aromaticity as a Guiding Concept for Spectroscopic Features and Nonlinear Optical Properties of Porphyrinoids Tatiana Woller 1, Paul Geerlings 1, Frank De Proft 1, Benoît Champagne 2 ID and Mercedes Alonso 1,* ID 1 Eenheid Algemene Chemie (ALGC), Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium; [email protected] (T.W.); [email protected] (P.G.); [email protected] (F.D.P.) 2 Laboratoire de Chimie Théorique, Unité de Chimie Physique Théorique et Structurale, University of Namur, Rue de Bruxelles 61, B-5000 Namur, Belgium; [email protected] * Correspondence: [email protected] Academic Editors: Luis R. Domingo and Miquel Solà Received: 5 March 2018; Accepted: 24 May 2018; Published: 1 June 2018 Abstract: With their versatile molecular topology and aromaticity, porphyrinoid systems combine remarkable chemistry with interesting photophysical properties and nonlinear optical properties. Hence, the field of application of porphyrinoids is very broad ranging from near-infrared dyes to opto-electronic materials. From previous experimental studies, aromaticity emerges as an important concept in determining the photophysical properties and two-photon absorption cross sections of porphyrinoids. Despite a considerable number of studies on porphyrinoids, few investigate the relationship between aromaticity, UV/vis absorption spectra and nonlinear properties. To assess such structure-property relationships, we performed a computational study focusing on a series of Hückel porphyrinoids to: (i) assess their (anti)aromatic character; (ii) determine the fingerprints of aromaticity on the UV/vis spectra; (iii) evaluate the role of aromaticity on the NLO properties. Using an extensive set of aromaticity descriptors based on energetic, magnetic, structural, reactivity and electronic criteria, the aromaticity of [4n+2] π-electron porphyrinoids was evidenced as was the antiaromaticity for [4n] π-electron systems. -
Course of Study
Introduction to Quantum Chemistry and Spectroscopy CY41009; Autumn Semester 2016-2017 Course of Study 1. Birth of Quantum Mechanics Failures of Classical Mechanics in Black Body Radition, Photoelectric Effects, Heat capacity of Solids, and Atomic Spectra; Quantum Mechanics - the Saviour; Birth of Quantum Chemistry. 2. Principles and Postulates of Quantum Mechanics State Functions; Operators; Eigenfunctions; Expectation Values; Time Evolution of Expectation Values; Ehrenfest Theorem; Hermitian Property; Schmidt Orthogonalization; Dirac Notation; Dirac Delta Function; Commutation of Operators; Heisenberg's Uncertainty Principle; Parity Operator. 3. Exactly Solvable Models • Translational Motions Particle in a 1D box; Bohr's Correspondence Principle; Free Particle; Particle in a 3D box; Degeneracies; Particle in a Rectangular Well; Tunneling Through Barrier; Scanning Tunneling Microscopy. • Vibrational Motions Harmonic Oscillators; Creation-Annihilation Operators; Hermite Polynomials; Vibrational Spectroscopy • Angular Motions Angular Momentum Operators; Ladder Operators; Spherical Harmonics. • Rotational Motions Particle in a ring; Particle on a sphere; Rigid Rotor; • Hydrogen Atom Solution of H-atom; Bound-state H-atom Wave Functions; Radial Distribution Functions; H-like Orbitals; Zeeman Effect; H-atom with Electron Spin; Spin-Orbit Interaction; Atomic Spectra with Spin-Orbit Interac- tion in Magnetic Field. 4. Approximate Methods Variational Theorem; He atom with Variational Method; Perturbation Theory; 1st and 2nd Order Perturbation Correction -
LESSON PLAN B.Sc
DEPARTMENT OF PHYSICS AND NANOTECHNOLOGY LESSON PLAN B.Sc. – THIRD YEAR (2015-2016 REGULATION) FIFTH SEMESTER . SRM UNIVERSITY FACULTY OF SCIENCE AND HUMANITIES SRM NAGAR, KATTANKULATHUR – 603 203 SRM UNIVERSITY FACULTY OF SCIENCE AND HUMANITIES DEPARTMENT OF PHYSICS AND NANOTECHNOLOGY Third Year B.Sc Physics (2015-2016 Regulation) Course Code: UPY15501 Course Title: Quantum Mechanics Semester: V Course Time: JUL 2016 – DEC 2017 Location: S.R.M. UNIVERSITY OBJECTIVES 1. To understand the dual nature of matter wave. 2. To apply the Schrodinger equation to different potential. 3. To understand the Heisenberg Uncertainty Relation and its application. 4. To emphasize the significance of Harmonic Oscillator Potential and Hydrogen atom. Assessment Details: Cycle Test – I : 10 Marks Cycle Test – II : 10 Marks Model Exam : 20 Marks Assignments : 5 Marks Attendance : 5 Marks SRM UNIVERSITY FACULTY OF SCIENCE AND HUMANITIES DEPARTMENT OF PHYSICS AND NANOTECHNOLOGY Third Year B.Sc Physics (2015-2016 Regulation) Total Course Semester Course Title L T P of C Code LTP II UPY15501 QUANTUM MECHANICS 4 1 - 5 4 UNIT I - WAVE NATURE OF MATTER Inadequacy of classical mechanics - Black body radiation - Quantum theory – Photo electric effect -Compton effect -Wave nature of matter-Expressions for de-Broglie wavelength - Davisson and Germer's experiment - G.P. Thomson experiment - Phase and group velocity and relation between them - Wave packet - Heisenberg's uncertainity principle – Its consequences (free electron cannot reside inside the nucleus and gamma ray microscope). UNIT II - QUANTUM POSTULATES Basic postulates of quantum mechanics - Schrodinger's equation - Time Independent -Time Dependant - Properties of wave function - Operator formalism – Energy - Momentum and Hamiltonian Operators - Interpretation of Wave Function - Probability Density and Probability - Conditions for Physical Acceptability of Wave Function -. -
Piecewise Potentials II
Physics 342 Lecture 13 Piecewise Potentials II Lecture 13 Physics 342 Quantum Mechanics I Monday, February 25th, 2008 We have seen a few different types of behavior for the stationary states of piecewise potentials { we can have oscillatory solutions on one or both sides of a potential discontinuity, we can also have growing and decaying exponentials. In general, we will select between the oscillatory and decaying exponential by choosing an energy scale for the stationary state. Our final pass at this subject will begin with a finite step, where we set up some matrix machinery to avoid the tedious algebra that accompanies boundary matching. Aside from the result, the important observation is that the matrix form we develop is independent of the potential, so that we can apply this type of approach to any potential, provided it does not extend to infinity. Remember the process we are going through over and over (and over): Given a potential, find the stationary states, use those to form the general solution to Schr¨odinger'sequation by appending the appropriate temporal factor −i E t (e ~ ), and exploit completeness to decompose some initial ¯(x) waveform. In practice, as should be evident from our studies so far, this program is sensible but difficult. The initial distribution of choice, a Gaussian, has unwieldy decomposition in bases other than ei k x (the natural, Fourier set) { is there another/easier way to get a basic idea of what an initial Gaussian distribution does when evolved in time under the influence of a potential? Yes, and we discuss the simplest possible numerical solution at the end. -
Quantum Mechanics Recap
Chapter 2 Quantum Mechanics in a Nutshell 2.1 Introduction 2.2 Photons Time: end of the 19th century. Maxwell’s equations have established Faraday’s hunch that light is an electromagnetic wave. However, by early 20th century, experimental evidence mounted pointing towards the fact that light is carried by ‘particles’ that pack a definite momentum and energy. Here is the crux of the problem: consider the double-slit experiment. Monochromatic light of wavelength λ passing through two slits separated by a distance d λ forms a di↵raction pattern on a photographic plate. If one ⇠ tunes down the intensity of light in a double-slit experiment, one does not get a ‘dimmer’ interference pattern, but discrete strikes on the photographic plate and illumination at specific points. That means light is composed of ‘particles’ whose energy and momentum are concentrated in one point which leads to discrete hits. But their wavelength extends over space, which leads to di↵raction patterns. Planck postulated that light is composed of discrete lumps of momemtum p = ~k and energy E = ~!. Here k =(2⇡/λ)ˆn , ˆn the direction of propagation, ~ is Planck’s constant, and ! = c k with c the speed of light. Planck’s hypothesis explained spectral | | features of the blackbody radiation. It was used by Einstein to explain the photoelectric e↵ect. Einstein was developing the theory of relativity around the same time. In this theory, the momentum of a particle of mass m and velocity v is p = mv/ 1 (v/c)2, − where c is the speed of light. Thus if a particle has m = 0, the only way itp can pack a momentum is if its velocity is v = c. -
Quantum Mechanics for Several Dimensions • Solutions for Set #10 Are Posted
Quantum Mechanics for Several Dimensions • Solutions for set #10 are posted. • Still plan a Tuesday 1-3pm session. • Some Material Covered today is not in the book- particle in ring – 2D Square Well in Chapter 8 – Coulomb Potential • Homework Set #11 is available per our vote last Friday – it is due Dec. 2. • Simple Harmonic Oscillator • Particle in a Ring -- idea of energy degeneracy • Two Dimensional Square Well – again energy degeneracy • Multiparticles in 3D http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 1 The wave functions of the first three states are Where ω = (k/m)–½ is the classical angular frequency, and n is the quantum number http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 2 http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 3 Clicker question 1 Set frequency to AD What happens to the node spacing away from x=0 for higher energy values in the harmonic oscillator potential? Think about the deBroglie wavelength. A. The node spacing doesn’t change. B. The node spacing decreases. C. The node spacing increases. http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 4 Clicker question 1 Set frequency to AD What happens to the node spacing away from x=0 for higher energy values in the harmonic oscillator potential? Think about the deBroglie wavelength. A. The node spacing doesn’t change. B. The node spacing decreases. C. The node spacing increases. http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 5 Clicker question 2 Set frequency to AD What happens to the probability density away from the equilibrium position at x=0 in the harmonic oscillator potential? A. -
Schrödinger Equation in Spherical Coordinate System and Angular
Chapter 4 (Lecture 9-10) Schrödinger Equation in Spherical coordinate system and Angular Momentum Operator In this section we will construct 3D Schrödinger equation and we give some simple examples. In this course we will consider problems where the partial differential equations are separable. Cartesian coordinate In previous chapters we have solved one dimensional problems, by using the time dependent Schrödinger equation ( ) Time part of the equation can be separated by substituting and we obtain time independent equation ( ) The equation can be extended in three dimensions (3D) by introducing 3D kinetic energy operator and potential: ( ) ( ) Where the operator ( ) Is square of gradient operator. It is obvious that 3D momentum operator can be written as ( ) Angular Momentum In quantum mechanics, the angular momentum operator is an operator analogous to classical angular momentum. ⃗ Where is position vector and is the momentum vector. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. In quantum mechanics angular momentum is quantized. This is because at the scale of quantum mechanics, the matter analyzed is best described by a wave equation or probability amplitude, rather than as a collection of fixed points or as a rigid body. Quantum angular momentum (cartesian) As it is known, observables in quantum physics are represented by operators. In quantum mechanics we get linear Hermitian angular momentum operators from the classical expressions using the postulates ⃗ ⃗⃗ When using Cartesian coordinates, it is customary to refer to the three spatial components of the angular momentum operator as: ( ) ( ) ( ) Square of the total angular momentum is defined as the square of the components: Commutation relation Different components of the angular momentum do not commute with another while all of the components commute with square of the total angular momentum. -
Arxiv:2008.08022V2 [Quant-Ph] 24 Feb 2021 Time) flow “To the Left”; This Is Clearly Impossible in the Line Classical World
Quantum backflow in a ring Arseni Goussev School of Mathematics and Physics, University of Portsmouth, Portsmouth PO1 3HF, United Kingdom (Dated: February 25, 2021) Free motion of a quantum particle with the wave function entirely comprised of plane waves with non-negative momenta may be accompanied by negative probability current, an effect called quantum backflow. The effect is weak and fragile, and has not yet been observed experimentally. Here we show that quantum backflow becomes significantly more pronounced and more amenable to experimental observation if, instead of letting the particle move along a straight line, one forces it to move in a circular ring. I. INTRODUCTION is the so-called Bracken-Melloy bound. Finding the exact value of cline remains an open challenge. It is interesting The probability density of a quantum particle may flow to note that cline is independent of the time window T , in the direction opposite to that of the particle's momen- the particle's mass µ, or Planck's constant ~. tum [1, 2], an effect called quantum backflow (QB) [3] [4]. Many questions related to QB have been addressed The effect is inconceivable from the viewpoint of classi- in the literature. These include QB against a constant cal physics, and in this respect can be paralleled with force [7], the pertinence of QB to the arrival-time prob- other genuinely quantum phenomena such as tunneling lem [8{11], position dependence of the backflow cur- or Schr¨odinger'scat states. Unlike the latter, however, rent [5, 12, 13], probability backflow in relativistic quan- QB is relatively unexplored and yet to be observed ex- tum systems [14{16], QB in escape problems [17, 18], and perimentally.