A Study of Fractional Schrödinger Equation-Composed Via Jumarie Fractional Derivative
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Lecture 16 Matter Waves & Wave Functions
LECTURE 16 MATTER WAVES & WAVE FUNCTIONS Instructor: Kazumi Tolich Lecture 16 2 ¨ Reading chapter 34-5 to 34-8 & 34-10 ¤ Electrons and matter waves n The de Broglie hypothesis n Electron interference and diffraction ¤ Wave-particle duality ¤ Standing waves and energy quantization ¤ Wave function ¤ Uncertainty principle ¤ A particle in a box n Standing wave functions de Broglie hypothesis 3 ¨ In 1924 Louis de Broglie hypothesized: ¤ Since light exhibits particle-like properties and act as a photon, particles could exhibit wave-like properties and have a definite wavelength. ¨ The wavelength and frequency of matter: # & � = , and � = $ # ¤ For macroscopic objects, de Broglie wavelength is too small to be observed. Example 1 4 ¨ One of the smallest composite microscopic particles we could imagine using in an experiment would be a particle of smoke or soot. These are about 1 µm in diameter, barely at the resolution limit of most microscopes. A particle of this size with the density of carbon has a mass of about 10-18 kg. What is the de Broglie wavelength for such a particle, if it is moving slowly at 1 mm/s? Diffraction of matter 5 ¨ In 1927, C. J. Davisson and L. H. Germer first observed the diffraction of electron waves using electrons scattered from a particular nickel crystal. ¨ G. P. Thomson (son of J. J. Thomson who discovered electrons) showed electron diffraction when the electrons pass through a thin metal foils. ¨ Diffraction has been seen for neutrons, hydrogen atoms, alpha particles, and complicated molecules. ¨ In all cases, the measured λ matched de Broglie’s prediction. X-ray diffraction electron diffraction neutron diffraction Interference of matter 6 ¨ If the wavelengths are made long enough (by using very slow moving particles), interference patters of particles can be observed. -
Dirac Particle in a Square Well and in a Box
Dirac particle in a square well and in a box A. D. Alhaidari Saudi Center for Theoretical Physics, Dhahran, Saudi Arabia We obtain an exact solution of the 1D Dirac equation for a square well potential of depth greater then twice the particle’s mass. The energy spectrum formula in the Klein zone is surprisingly very simple and independent of the depth of the well. This implies that the same solution is also valid for the potential box (infinitely deep well). In the nonrelativistic limit, the well-known energy spectrum of a particle in a box is obtained. We also provide in tabular form the elements of the complete solution space of the problem for all energies. PACS numbers: 03.65.Pm, 03.65.Ge, 03.65.Nk Keywords: Dirac equation, square well, potential box, energy spectrum, Klein paradox Aside from the mathematically “trivial” free case, particle in a box is usually the first problem that an undergraduate student of nonrelativistic quantum mechanics is asked to solve. Normally, he goes further into obtaining the bound states solution of a particle in a finite square well. It is much later that he works out more involved exercises like the 3D Coulomb problem [1]. Unfortunately, in relativistic quantum mechanics the story is reversed. One can hardly find a textbook on relativistic quantum mechanics where the 1D problem of a particle in a potential box is solved before the relativistic Hydrogen atom is [2]. The difficulty is that if this is to be done from the start, then one is forced to get into subtle issues like the Klein paradox, electron-positron pair production, stability of the vacuum, appropriate boundary conditions, …etc. -
L the Probability Current Or Flux
Atomic and Molecular Quantum Theory Course Number: C561 L The Probability Current or flux 1. Consider the time-dependent Schr¨odinger Equation and its complex conjugate: 2 ∂ h¯ ∂2 ıh¯ ψ(x, t)= Hψ(x, t)= − 2 + V ψ(x, t) (L.26) ∂t " 2m ∂x # 2 2 ∂ ∗ ∗ h¯ ∂ ∗ −ıh¯ ψ (x, t)= Hψ (x, t)= − 2 + V ψ (x, t) (L.27) ∂t " 2m ∂x # 2. Multiply Eq. (L.26) by ψ∗(x, t), andEq. (L.27) by ψ(x, t): ∂ ψ∗(x, t)ıh¯ ψ(x, t) = ψ∗(x, t)Hψ(x, t) ∂t 2 2 ∗ h¯ ∂ = ψ (x, t) − 2 + V ψ(x, t) (L.28) " 2m ∂x # ∂ −ψ(x, t)ıh¯ ψ∗(x, t) = ψ(x, t)Hψ∗(x, t) ∂t 2 2 h¯ ∂ ∗ = ψ(x, t) − 2 + V ψ (x, t) (L.29) " 2m ∂x # 3. Subtract the two equations to obtain (see that the terms involving V cancels out) 2 2 ∂ ∗ h¯ ∗ ∂ ıh¯ [ψ(x, t)ψ (x, t)] = − ψ (x, t) 2 ψ(x, t) ∂t 2m " ∂x 2 ∂ ∗ − ψ(x, t) 2 ψ (x, t) (L.30) ∂x # or ∂ h¯ ∂ ∂ ∂ ρ(x, t)= − ψ∗(x, t) ψ(x, t) − ψ(x, t) ψ∗(x, t) (L.31) ∂t 2mı ∂x " ∂x ∂x # 4. If we make the variable substitution: h¯ ∂ ∂ J = ψ∗(x, t) ψ(x, t) − ψ(x, t) ψ∗(x, t) 2mı " ∂x ∂x # h¯ ∂ = I ψ∗(x, t) ψ(x, t) (L.32) m " ∂x # (where I [···] represents the imaginary part of the quantity in brackets) we obtain ∂ ∂ ρ(x, t)= − J (L.33) ∂t ∂x or in three-dimensions ∂ ρ(x, t)+ ∇·J =0 (L.34) ∂t Chemistry, Indiana University L-37 c 2003, Srinivasan S. -
Relativistic Quantum Mechanics 1
Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4). -
Analytic Solution of the Schrцdinger Equation: Particle in A
Analytic solution of the Schrödinger equation: Particle in a box Notes on Quantum Mechanics http://quantum.bu.edu/notes/QuantumMechanics/ParticleInABox.pdf Last updated Tuesday, September 26, 2006 16:04:15-05:00 Copyright © 2004 Dan Dill ([email protected]) Department of Chemistry, Boston University, Boston MA 02215 One way to work with the curvature form of the Schrödinger equation, curvature of y at x ∂-kinetic energy at x μyat x, is try to guess what wavefunctions, y, satisfy it. For systems in which the kinetic energy changes with position in complicated ways, systematic implementation of this so-called analytic approach typically requires sophisticated mathematical machinery and so it not as easy to understand as the visual approach we will use. However, there is one example of the analytic approach that is very easy to understand, namely the case where a particle of mass m is confined in a one-dimensional region of width L; in this region it moves freely but it is not able to move outside this region. Such a system is called a particle in a box. It is one of the most important example quantum systems in chemistry, because it helps us develop intuition about the behavior on electrons confined in molecules. To use the analytical method, we begin by writing the Schrödinger equation more precisely as 2 m curvature of y at x =-ÅÅÅÅÅÅÅÅÅÅÅÅ μ kinetic energy at x μyat x. Ñ2 The reason the particle is able to move freely in the specified region, 0 § x § L (inside the "box"), is that its potential energy there is zero. -
Photon Wave Mechanics: a De Broglie - Bohm Approach
PHOTON WAVE MECHANICS: A DE BROGLIE - BOHM APPROACH S. ESPOSITO Dipartimento di Scienze Fisiche, Universit`adi Napoli “Federico II” and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli Mostra d’Oltremare Pad. 20, I-80125 Napoli Italy e-mail: [email protected] Abstract. We compare the de Broglie - Bohm theory for non-relativistic, scalar matter particles with the Majorana-R¨omer theory of electrodynamics, pointing out the impressive common pecu- liarities and the role of the spin in both theories. A novel insight into photon wave mechanics is envisaged. 1. Introduction Modern Quantum Mechanics was born with the observation of Heisenberg [1] that in atomic (and subatomic) systems there are directly observable quantities, such as emission frequencies, intensities and so on, as well as non directly observable quantities such as, for example, the position coordinates of an electron in an atom at a given time instant. The later fruitful developments of the quantum formalism was then devoted to connect observable quantities between them without the use of a model, differently to what happened in the framework of old quantum mechanics where specific geometrical and mechanical models were investigated to deduce the values of the observable quantities from a substantially non observable underlying structure. We now know that quantum phenomena are completely described by a complex- valued state function ψ satisfying the Schr¨odinger equation. The probabilistic in- terpretation of it was first suggested by Born [2] and, in the light of Heisenberg uncertainty principle, is a pillar of quantum mechanics itself. All the known experiments show that the probabilistic interpretation of the wave function is indeed the correct one (see any textbook on quantum mechanics, for 2 S. -
Schrödinger Equation - Particle in a 1-D Box
Particle in a Box Outline - Review: Schrödinger Equation - Particle in a 1-D Box . Eigenenergies . Eigenstates . Probability densities 1 TRUE / FALSE 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. 2 Schrodinger: A Wave Equation for Electrons Schrodinger guessed that there was some wave-like quantity that could be related to energy and momentum … wavefunction 3 Schrodinger: A Wave Equation for Electrons (free-particle) (free-particle) ..The Free-Particle Schrodinger Wave Equation ! Erwin Schrödinger (1887–1961) 4 Image in the Public Domain Schrodinger Equation and Energy Conservation ... The Schrodinger Wave Equation ! Total E term K.E. term P.E . t e rm ... In physics notation and in 3-D this is how it looks: Electron Maximum height Potential and zero speed Energy Zero speed start Incoming Electron Fastest Battery 5 Time-Dependent Schrodinger Wave Equation To t a l E K.E. term P.E . t e rm PHYSICS term NOTATION Time-Independent Schrodinger Wave Equation 6 Particle in a Box e- 0.1 nm The particle the box is bound within certain regions of space. If bound, can the particle still be described as a wave ? YES … as a standing wave (wave that does not change its with time) 7 A point mass m constrained to move on an infinitely-thin, frictionless wire which is strung tightly between two impenetrable walls a distance L apart m 0 L WE WILL HAVE MULTIPLE SOLUTIONS FOR , SO WE INTRODUCE LABEL IS CONTINUOUS 8 WE WILL HAVE e- MULTIPLE SOLUTIONS FOR , SO WE INTRODUCE LABEL n L REWRITE AS: WHERE GENERAL SOLUTION: OR 9 USE BOUNDARY CONDITIONS TO DETERMINE COEFFICIENTS A and B e- since NORMALIZE THE INTEGRAL OF PROBABILITY TO 1 L 10 EIGENENERGIES for EIGENSTATES for PROBABILITY 1-D BOX 1-D BOX DENSITIES 11 Today’s Culture Moment Max Planck • Planck was a gifted musician. -
“Particle-In-A-Box” Problems in Quantum Mechanics
2-DIMENSIONAL “PARTICLE-IN-A-BOX” PROBLEMS IN QUANTUM MECHANICS Part I: Propagator & eigenfunctions by the method of images Nicholas Wheeler, Reed College Physics Department July 1997 Introduction. All strings make the same music, and support the same physics, because all strings have the same shape. The vibrational physics of a drumhead is, on the other hand, shape-dependent—whence Mark Kac’famous question: “Can one hear the shape of a drum?”1 Relatedly, all instances of the quantum mechanical “particle-in-a-box” problem are, in the one-dimensional case, scale- equivalent to one another, but higher-dimensional instances of the same problem are (in general) inequivalent. Though the one-dimensional box-problem yields quite readily to exact closed-form analysis—the simplest (but only the simplest!) aspects of which can be found in every quantum text—the higher-dimensional problem (N ≥ 2) is in most of its aspects analytically intractable except in certain special cases. In previous work2 I have identified a class of cases that yield to analysis by what might be called the “quantum mechanical method of images.” I undertake the present review of the essentials of that work partly to make it more readily available to students of quantum mechanics, but mainly to be responsive to an expressed need of my colleague Oz Bonfim, who speculates that some of my results may be relevent to his own work in connection with the so-called “Bohm interpretation of quantum mechanics.”3 I have also an ulterior motivation, 1 Amer. Math. Monthly 73, 1 (1966). This classic paper is reprinted in Mark Kac: Selected Papers on Probability, Number Theory & Statistical Physics () 2 feynman formalism for polygonal domains (–); circular boxes & rigid wavepackets (–). -
Electron Spin and Probability Current Density in Quantum Mechanics W
Electron spin and probability current density in quantum mechanics W. B. Hodgea) and S. V. Migirditch Department of Physics, Davidson College, Davidson, North Carolina 28035 W. C. Kerrb) Olin Physical Laboratory, Wake Forest University, Winston-Salem, North Carolina 27109 (Received 13 August 2013; accepted 26 February 2014) This paper analyzes how the existence of electron spin changes the equation for the probability current density in the quantum-mechanical continuity equation. A spinful electron moving in a potential energy field experiences the spin-orbit interaction, and that additional term in the time- dependent Schrodinger€ equation places an additional spin-dependent term in the probability current density. Further, making an analogy with classical magnetostatics hints that there may be an additional magnetization current contribution. This contribution seems not to be derivable from a non-relativistic time-dependent Schrodinger€ equation, but there is a procedure described in the quantum mechanics textbook by Landau and Lifschitz to obtain it. We utilize and extend their procedure to obtain this magnetization term, which also gives a second derivation of the spin-orbit term. We conclude with an evaluation of these terms for the ground state of the hydrogen atom with spin-orbit interaction. The magnetization contribution is generally the larger one, except very near an atomic nucleus. VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4868094] I. INTRODUCTION occurs early in the course (and the textbook) when only the simplest form of the time-dependent Schrodinger€ equation has The probability interpretation of the wave function w of a been introduced. -
Quantum Mechanics (Surprising !!!) the Analysis of Electronic Flow)
2/6/2019 INE: Revision INTRODUCTION TO NANOELECTRONICS Week-6, Lecture-2 Sneh Saurabh 6th February, 2019 Introduction to Nanoelectronics: S. Saurabh Overview 2 A free particle: Significance in Solid State Physics Seminal concept in solid state physics Impact on Analysis . In crystalline solids electrons behave as if . Reduces the complicated problem of they are in vacuum, but with an effective electron waves in a crystalline solid to that mass different from their natural mass of waves in vacuum (Dramatically eases Quantum Mechanics (Surprising !!!) the analysis of electronic flow) . Energy-momentum relation 푝 . Valid for several materials such as Copper (metal) and Silicon (semiconductor) in wide 퐸 푝 = 퐸 + ∗ 2푚 range of energies . Energy-wavenumber (퐸 − 푘) relation (using . Parabolic 퐸 − 푘 푟푒푙푎푡푖표푛푠ℎ푖푝 푝 = ℏ푘) A Free particle ℏퟐ풌ퟐ (퐷푖푠푝푒푟푠푖표푛 푟푒푙푎푡푖표푛푠ℎ푖푝) is not valid for 퐸 푘 = 퐸 + some materials such as graphene 2푚∗ Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 3 Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 4 1 2/6/2019 A free particle + particle in a box ! Wave function of an electron in a confined nanoelectronic device can be treated as combined effect of (a free particle + a particle in a box) Quantum Mechanics Bulk Quantum Quantum Quantum (3D) Well (2D) Wire (1D) Dots (0D) 3-D Schrödinger Equation Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 5 Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 6 Schrödinger Equation: 3-D Schrödinger Equation: 3-D Separable solution (1) Three-dimensional time-independent Schrödinger equation: . Applying quantum mechanics to nanostructures (which are naturally 3-D objects) requires 퐻휓(푥, 푦, 푧)= 퐸휓(푥, 푦, 푧) extending Schrödinger Equation to 3-D where: ℏ 휕 휕 휕 . -
Optical Physics of Quantum Wells
Optical Physics of Quantum Wells David A. B. Miller Rm. 4B-401, AT&T Bell Laboratories Holmdel, NJ07733-3030 USA 1 Introduction Quantum wells are thin layered semiconductor structures in which we can observe and control many quantum mechanical effects. They derive most of their special properties from the quantum confinement of charge carriers (electrons and "holes") in thin layers (e.g 40 atomic layers thick) of one semiconductor "well" material sandwiched between other semiconductor "barrier" layers. They can be made to a high degree of precision by modern epitaxial crystal growth techniques. Many of the physical effects in quantum well structures can be seen at room temperature and can be exploited in real devices. From a scientific point of view, they are also an interesting "laboratory" in which we can explore various quantum mechanical effects, many of which cannot easily be investigated in the usual laboratory setting. For example, we can work with "excitons" as a close quantum mechanical analog for atoms, confining them in distances smaller than their natural size, and applying effectively gigantic electric fields to them, both classes of experiments that are difficult to perform on atoms themselves. We can also carefully tailor "coupled" quantum wells to show quantum mechanical beating phenomena that we can measure and control to a degree that is difficult with molecules. In this article, we will introduce quantum wells, and will concentrate on some of the physical effects that are seen in optical experiments. Quantum wells also have many interesting properties for electrical transport, though we will not discuss those here. -
Quantum Mechanics Part 2
Quantum Mechanics Part 2 Ch 40 The Schoedinger Eqn. Making sense of The Schoedinger Equation There are physics laws which are encoded in equations e.g. F=ma; Einitial=Efinal etc Is there a rule for ψ ????? Schoedinger ~ 1924 dx2ψ () 2 m =−[()]()EUx − ψ x dx22= Why do we believe it Because it works! What is the program? Justifying it This isn’t the real reason we believe it real reason is because of experiment Making models of real life and solving Periodic Table of the elements Molecular binding (chemistry) tunneling (decay) transistors Zero point energy of the vacuum! … A “ justification” of the Schroedinger eqn this represents matter waves hmm Note: I cheated a bit, since K is really K(x) potential energy dx2ψ () 2 m =−[()]()E −Uxψ x dx22= This is the 1-D eqn. We should really use a 3-D version (in this class we will stick to 1-D) what we need to do is to find appropriate U(x) which describe real situations and then solve for ψ(x) Boundary Conditions We have to require some things about ψ(x) since P(x)= ψ2 ψ(x) must be continuous ψ(x) →0 as x→∞ and x→-∞ ψ(x)=0 in areas it should not be ψ(x) must be normalized (i.e. its total probability = 1 or 100%) PROBLEM-SOLVING STRATEGY Quantum-mechanics problems MODEL Determine a potential-energy function that describes the particle’s interactions. Make simplifying assumptions. VISUALIZE The potential-energy curve is the pictorial representation. Draw the potential-energy curve. Identify known information.