A complete set of ladder operators for the hydrogen atom
|
|
- Lynette Tucker
- 5 years ago
- Views:
Transcription
1 A copete set of adder operators for the hydrogen ato C. E. Burkhardt St. Louis Counity Coege at Forissant Vaey 3400 Persha Road St. Louis, MO J. J. Leventha Departent of Physics University of Missouri - St. Louis St. Louis, MO 63 Abstract Ladder operators capabe of converting one hydrogen ato eigenfunction into another by raising the anguar oentu quantu nuber are derived using ony eeentary techniques. The derivation is perfored using forais no ore sophisticated than that used to derive the properties of the ordinary anguar oentu adder operators in undergraduate quantu echanics courses. The properties of these operators, which consist of coponents of the quantu echanica Len vector, deonstrate the accidenta degeneracy of the hydrogen ato. It is shown that, starting fro the 0 eigenfunction for a given principa quantu nuber n the copete set of eigenfunctions for that n can be obtained. PACS nubers: Ca ; Fd ; Ge
2 It is we-known that the Keper/Couob potentias endow panetary orbits and hydrogen atos with specia properties, properties not present in systes subject to other centra potentias. A pure Keperian orbit is fixed in space, that is, it does not precess. Moreover, the tota energy of the syste depends ony on the vaue of the sei-ajor axis a and not on that of the sei-inor axis b. There exists, therefore, an infinity of possibe orbits, a having the sae energy and sei-ajor axis, but having different vaues for the sei-inor axis. Whie a is independent of the anguar oentu, b is deterined by the anguar oentu so the energy is independent of the anguar oentu. This cassica degeneracy is the resut of the sae syetry of the /r potentia that causes the ceebrated "accidenta degeneracy" of the hydrogen ato, the independence of the energy eigenvaues on the anguar oentu quantu nuber. This syetry is different fro the spatia syetry extant for any centra potentia. It is often referred to as a "dynaica syetry"[]. The spatia syetry causes the energy to be independent of, the quantu nuber corresponding to the -coponent of the anguar oentu. Cassicay, the dynaica syetry anifests itsef as an additiona constant of the otion, the Len vector A, which points aong the ajor axis of the eipse[]. This resuts in an orbit that is fixed in space. Quantu echanicay, A corresponds to an additiona operator  that coutes with the Haitonian Ĥ as shown by Paui in his andark paper[3]. Of course the agnitude of the anguar oentu and each of its coponents aso coute with Ĥ for any centra potentia. Athough these consequences of the dynaica syetry of the /r potentia are rarey discussed at the undergraduate eve, they provide insight into a variety of cassica and quantu concepts. In this paper we wi concentrate on the quantu echanica consequences of the cassica constant A. We wi show that it is possibe to derive a set of adder operators invoving coponents of  that transfors certain spherica hydrogen eigenstates into other spherica eigenstates. Spherica eigenstates are those that resut fro separation of the Schrödinger
3 equation in spherica coordinates and is characteried by the quantu nubers n (energy), (anguar oentu) and (-coponent of anguar oentu). The dynaica syetry akes it possibe to separate the Schrödinger in paraboic coordinates as we[4]. The fact that the Lˆ change ony the vaue of and neither nor n is a anifestation of the degeneracy associated with the spatia syetry. Because of this the Lˆ are adder operators for any centra potentia. The  and  adder operators effect state-to-state changes in the quantu nuber on ony spherica eigenfunctions for the Couob potentia, and, as wi be shown, ony on states for which  Â. Because and change they refect the accidenta degeneracy of the hydrogen ato in which the energy eigenvaues are independent of. In this paper we wi derive the properties of and and obtain the exact expressions for the actions of these operators without having to resort to advanced concepts[5]. The derivation wi be carried out using forais no ore sophisticated than that used to derive the properties   of Lˆ in ost undergraduate quantu echanics courses. It wi be seen that, given the 0 eigenfuntion for a particuar n, a eigenfunctions for that n can be obtained by judicious appication a cobination of Lˆ,  and Â. Using atoic units for which h e e where e is the eectronic charge and e the eectronic ass, the Len vector in cassica echanics is defined in atoic units as A p L ˆr ( ) where p is the inear oentu, L the anguar oentu and rˆ the unit vector in the r direction. The direction of A, toward apocenter or pericenter, is a atter of choice. We eect the definition in Equation ()[6]. The quantu echanica operator corresponding to A cannot be constructed by erey repacing each quantity with its corresponding quantu echanica 3
4 operator because the resut is a non-heretian operator. Paui recognied that  ust be defined as[3] Note that the quantu echanica operators ( pˆ Lˆ ) ( Lˆ pˆ ) [( pˆ Lˆ ) ( Lˆ pˆ )] rˆ ( ). It is easiy shown that  as defined in Equation () is indeed Heretian. It can aso be shown, but with consideraby ore A ˆ Hˆ. abor, that [, ] 0 The anguar oentu adder operators are defined as Lˆ L ˆ Lˆ i ˆ x L y ( 3) and, when operating on a spherica eigenfunction for any centra potentia, which we designate as n C, cause the foowing state-to-state conversion. ( )( ) n ; ( ) ( 4) L ˆ n C C where we have inserted a sei-coon between quantu nubers in the ket on the right hand side for carity. Equation (4) is, of course, aso vaid for spherica hydrogen ato eigenfunctions since the Couob potentia is a centra potentia. The operators  are defined as A ˆ i ˆ x A y (5) For convenience, a nuber of reations between the various operators are copied in Tabe I. Soe require engthy agebraic anipuations, but a are straightforward. We investigate the action of  on the spherica hydrogen eigenfunction n. We specify that this is a hydrogen eigenfunction by oitting the subscript C that was used to designate an eigenfunction for an arbitrary centra potentia. Using the reations contained in Tabe I we find that 4
5 which shows that { A n Lˆ { n Lˆ ( x i y ) n ( Lˆ ) n ( 6) ( ){ n ˆ is an eigenfunction of with eigenvaue Lˆ ( ). Siiary { ( ˆ ˆ ˆ ˆ ˆ ˆ ˆ n A L A L A A L ) n { ( ) ( ) { n ( ) ( ) n; ( ) ( 7) ˆ L Note that the { A n n are not eigenfunctions of Â, but if ˆ is an eigenfunction of ˆL with eigenvaue  the ter with vanishes and Because { A n ( ) ( ) ( )( ) ( 8) ˆ is an eigenfunction of ˆL and with eigenvaues respectivey, we know that where D Lˆ ( )( ) and ( ) A ˆ D ; n ( ) ; ( ) ( 9) is a constant that depends on. But, to construct a copete set of eigenfunctions for a given n fro ony D n 00 it is necessary to evauate. We begin by foowing a siiar procedure to that used to evauate the constants when finding the action of Lˆ on the n. C We consider the atrix eeent n * ˆ ˆ A A ( D ) D ( 0) * ( D ) where the was obtained by operating to the eft with  and noting that the Heretian   conjugate of is. We specify for convenience, and by anaogy with the assuption epoyed in the Lˆ construction, that is rea. The operator  ay be expanded giving D  5
6 Lˆ Hˆ ( Lˆ Hˆ Hˆ ) Lˆ Hˆ ( ) Since the n are eigenfunctions of a operators on the right hand side except  we obtain n ( ) ( ) D n ˆ A ( ) where we have used Lˆ Lˆ Hˆ ( ) n ( 3) To copete the evauation of D we ust deterine the atrix eeent. In order to do this we first exaine the consequence of operating on  n with. The - coponent of the vector operator  ay be written A ˆ cosθ ( 4) where θ is the poar ange in spherica coordinates. Bearing in ind that the anguar parts of the n are erey the usua spherica haronics Y ( θ,φ ) n R n ( r) cosθy ( cosθ ) ( 5) where R n () r is the radia part of the hydrogen ato eigenfunction. Using a we-known recursion reation for the spherica haronics[4] we see that ( θ, φ ) ( )( ) ( )( 3) ( θ, φ ) ( )( ) ( )( ) cosθy Y Y φ ( θ, ) (6) 6
7 n ( )( ) ( )( 3) ( )( ) ( )( 3) ( ) ( )( ) ( )( ) Rn () r Y ( θ, φ ) Y ; ( )( ) ( )( ) ( θ, φ ) ( 7) ( ) ; Thus, the action of  on n is to raise and ower by unity whie eaving unchanged. In the present case, however, the resut is even siper because thus eiinating the second ter. We find then that ˆ A Y ( θ, φ ) ( 8) ( ) ; which shows that  operating on n is a raising operator for, but not for. Whie our interest at this tie is in the eigenvector n it shoud be noted that the second ter in Equation (7) aso vanishes for. In fact the coefficient of the first ter in Equation (7)  is the sae for. Thus, raises, but not for when operating on either n or n ;. Ared with Equation (8) we now exaine the atrix eeent A Using the coutator [, ] n ; n ; Lˆ ˆ we have ( ) ( ) ˆ ; A D ( 9) ( ) ;( ) n ( ) ( ) Lˆ Lˆ ; ; n ;( ) ; ( ) Lˆ ( 0) ( ) ( ) ; where L ˆ 0 and we have operated to the eft with Lˆ noting that the Heretian conjugate of Lˆ is Lˆ. We have then ( ) n ; ; D ( ) ( ) 7
8 This equation uniquey deterines ˆ in ters of the constant. Note that without A D Equation (8) additiona ters in Equation () woud be possibe as ong as they were orthogona to n ;( ) ;. We have then ˆ A D ( ) ( ) ; ( ) Inserting Equation () into Equation (), soving for ( ) D and taking the square root we have D n ( ) [ n ( ) ] ( 3) ( 3) It is iportant to note that, athough the operator  was present in our anaysis of the action of  on n, the ony property of that was used was that it is the heretian conjugate is Â.   The action of on an eigenfunction was not needed. Aso, because was obtained by D D ) taking the square root of ( there is no inforation on the sign of. In accordance with convention it is the inus sign that is retained[5]. D It is tepting to assue that the action of  on a spherica eigenfunction is as a owering operator. In fact, for a state-to-state conversion it does ower the -coponent of the anguar oentu, but it raises the tota anguar oentu as wi be shown. In a anner identica to that epoyed for { A ˆ n we find that { A n of Lˆ with eigenvaue (. Evauation of ) Lˆ { A ˆ n ˆ is an eigenfunction reveas it to be { n { ( ) ( ) { n ( )( ) ;( ) ( 4) Lˆ ˆ is an eigenfunction of with eigenvaue so that { A n ˆL ( )( ) if. Note that 8
9 this is the ony way to ake the ter containing  vanish because is prohibited. We concude therefore that ( ) D ( ) ; ( ) ( 5) A ˆ ; D can be evauated in a anner anaogous to that epoyed to evauate, but it can aso be D D ( θ, φ ) ( ) { ( θ, φ ) * Y evauated in ters of by aking use of the fact that Y. In the bra and ket notation used here this becoes n; ( ) ( ) n ( 6) which we use to evauate the copex conjugate of the atrix eeent A ˆ. We have * { ( ) Now, the eft hand side of Equation (7) can be written The right hand side of Equation (7) ay be written * { ; ˆ ˆ ( ) ( ) A A ( ) ( 7) { ; ( ) ; ( ) * * { D ( ) ;( ) D ( ) ;( ) A ˆ * { n; ( ) ; ( ) n; ( ) n; ( ) D n; ( ) ( ; )[ ( ) ] ( 9) D ( )( ) ; ( ) ( ) D ( ; )( ) where we have used ( ) Equations (8) and (9) we see that * { ( 8) in Equation (9). Coparing the right hand sides of D D n ( ) [ n ( ) ] ( 30) ( 3) 9
10 Notice that had we evauated using the sae technique epoyed to evauate we D woud not have obtained the inus sign in Equation (30). D The properties of the raising and owering operators are suaried in Tabe II. Whie the anguar oentu adder operators change ony the quantu nuber a of the raising operators consisting of coponents of the Len vector operator change the quantu nuber. This is a consequence of the accidenta degeneracy of the hydrogen ato in which the energy eigenvaues are independent of. Exaination of Tabe II shows that it is possibe to construct the entire set of hydrogen ato eigenfunctions for a given n if ony the 0 eigenfunction for that n is known. Figure iustrates two possibe sequences of appications of the adder operators required to generate the copete set of n 3 spherica eigenfunctions. ACKNOWLEDGEMENT The authors woud ike to thank their coeagues, Joseph F. Baugh and Ta-Pei Cheng for hepfu discussions. 0
11 REFERENCES. L. I. Schiff, Quantu Mechanics (McGraw-Hi; New York; third editio 968). P 34.. T. P. Hee, C. E. Burkhardt, M. Ciocca and J. J. Leventha, "Cassica View of the Stark Effect in Hydrogen Atos", A. J. Phys. 60, 34 (99). 3. W. Paui, "On the Hydrogen Spectru fro the Standpoint of the New Quantu Mechanics" The Engish transation is in: Sources of Quantu Mechanics edited by B. L.d Van Der Waerden (Dover; New York; 967) pp H. A. Bethe and E. E. Sapeter, Quantu Mechanics of One- and Two-Eectron Atos (Springer-Verag; Beri 957). pp L. C. Biedenharn, J. D. Louck and P. A. Carruthers, Anguar Moentu in Quantu Mechanics: Theory and Appication (Addison-Wesey; Reading, MA; 98). Pp H. Godstein, Cassica Mechanics (Addison-Wesey; Reading, MA; second editio 980).
12 FIGURE CAPTIONS Figure. Two possibe sequences of appications of the adder operators that generate the copete set of n 3 spherica eigenfunctions starting with 300.
13 Figure. Two possibe sequences of appications of the adder operators that generate the copete set of n 3 spherica eigenfunctions starting with
14 Tabe I. List of soe reations invoving the quantu echanica operators used in this work. ( ) ( ) [ Lˆ i, Lˆ j ] ilˆ k [ Lˆ, ] [ Lˆ i, j ] i k [ i, j ] ilˆ k Hˆ ˆ ( ˆ A L ) Hˆ 4
15 Tabe II. The resut of operations with the adder operators on the specified spherica eigenfunctions of the hydrogen ato. Operation L n Resut n; ˆ ( )( ) ( ) [ ] A n ( ) n 3 ( ) n ; ( ) ; n ; ; ( ) [ n ( ) ] n ; ( ) ; ( ) n ( 3) A ( ) n ( ) n ( 3) ; A ˆ ; ( ) ( ) n ( ) n ( 3) n ; ; [ ] ( ) ( ) [ ] ( ) ( ) 5
Part B: Many-Particle Angular Momentum Operators.
Part B: Man-Partice Anguar Moentu Operators. The coutation reations deterine the properties of the anguar oentu and spin operators. The are copete anaogous: L, L = i L, etc. L = L ± il ± L = L L L L =
More informationLecture 8 Symmetries, conserved quantities, and the labeling of states Angular Momentum
Lecture 8 Syetries, conserved quantities, and the labeling of states Angular Moentu Today s Progra: 1. Syetries and conserved quantities labeling of states. hrenfest Theore the greatest theore of all ties
More informationAngular Momentum Properties
Cheistry 460 Fall 017 Dr. Jean M. Standard October 30, 017 Angular Moentu Properties Classical Definition of Angular Moentu In classical echanics, the angular oentu vector L is defined as L = r p, (1)
More informationIdentites and properties for associated Legendre functions
Identites and properties for associated Legendre functions DBW This note is a persona note with a persona history; it arose out off y incapacity to find references on the internet that prove reations that
More information11 - KINETIC THEORY OF GASES Page 1. The constituent particles of the matter like atoms, molecules or ions are in continuous motion.
- KIETIC THEORY OF GASES Page Introduction The constituent partices of the atter ike atos, oecues or ions are in continuous otion. In soids, the partices are very cose and osciate about their ean positions.
More informationInvolutions and representations of the finite orthogonal groups
Invoutions and representations of the finite orthogona groups Student: Juio Brau Advisors: Dr. Ryan Vinroot Dr. Kaus Lux Spring 2007 Introduction A inear representation of a group is a way of giving the
More informationSession : Electrodynamic Tethers
Session : Eectrodynaic Tethers Eectrodynaic tethers are ong, thin conductive wires depoyed in space that can be used to generate power by reoving kinetic energy fro their orbita otion, or to produce thrust
More informationJoel Broida UCSD Fall 2009 Phys 130B QM II. Homework Set 2 DO ALL WORK BY HAND IN OTHER WORDS, DON T USE MATHEMAT- ICA OR ANY CALCULATORS.
Joe Broida UCSD Fa 009 Phys 30B QM II Homework Set DO ALL WORK BY HAND IN OTHER WORDS, DON T USE MATHEMAT- ICA OR ANY CALCULATORS. You may need to use one or more of these: Y 0 0 = 4π Y 0 = 3 4π cos Y
More informationAll you need to know about QM for this course
Introduction to Eleentary Particle Physics. Note 04 Page 1 of 9 All you need to know about QM for this course Ψ(q) State of particles is described by a coplex contiguous wave function Ψ(q) of soe coordinates
More informationWave Motion: revision. Professor Guy Wilkinson Trinity Term 2014
Wave Motion: revision Professor Gu Wiinson gu.wiinson@phsics.o.a.u Trinit Ter 4 Introduction Two ectures to reind ourseves of what we earned ast ter Wi restrict discussion to the topics on the sabus Wi
More informationi ij j ( ) sin cos x y z x x x interchangeably.)
Tensor Operators Michael Fowler,2/3/12 Introduction: Cartesian Vectors and Tensors Physics is full of vectors: x, L, S and so on Classically, a (three-diensional) vector is defined by its properties under
More informationVector Spherical Harmonics
Vector Spherica Haronics Lecture Introduction Previousy we have seen that the Lapacian operator is different when operating on a vector or a scaar function. We avoided this probe by etting the Lapacian
More informationTransforms, Convolutions, and Windows on the Discrete Domain
Chapter 3 Transfors, Convoutions, and Windows on the Discrete Doain 3. Introduction The previous two chapters introduced Fourier transfors of functions of the periodic and nonperiodic types on the continuous
More informationh mc What about matter? Louis de Broglie ( ) French 8-5 Two ideas leading to a new quantum mechanics
8-5 Two ideas eading to a new quantu echanics What about atter? Louis de Brogie (189 1989) French Louis de Brogie Werner Heisenberg Centra idea: Einstein s reativity E=c Matter & energy are reated Louis
More informationTranslation to Bundle Operators
Syetry, Integrabiity and Geoetry: Methods and Appications Transation to Bunde Operators SIGMA 3 7, 1, 14 pages Thoas P. BRANSON and Doojin HONG Deceased URL: http://www.ath.uiowa.edu/ branson/ Departent
More informationOSCILLATIONS. dt x = (1) Where = k m
OSCILLATIONS Periodic Motion. Any otion, which repeats itsef at reguar interva of tie, is caed a periodic otion. Eg: 1) Rotation of earth around sun. 2) Vibrations of a sipe penduu. 3) Rotation of eectron
More informationWork, Energy and Momentum
Work, Energy and Moentu Work: When a body oves a distance d along straight line, while acted on by a constant force of agnitude F in the sae direction as the otion, the work done by the force is tered
More informationPhysics 139B Solutions to Homework Set 3 Fall 2009
Physics 139B Solutions to Hoework Set 3 Fall 009 1. Consider a particle of ass attached to a rigid assless rod of fixed length R whose other end is fixed at the origin. The rod is free to rotate about
More informationElectron Spin. I = q T = e 2πr. (12.1)
ectron Spin I Introduction Our oution of the TIS in three dienion for one-eectron ato reuted in quantu tate that are uniquey pecified by the vaue of the three quantu nuber n,, Thi picture wa very uccefu
More informationMidterm 2 Review. Drew Rollins
Midterm 2 Review Drew Roins 1 Centra Potentias and Spherica Coordinates 1.1 separation of variabes Soving centra force probems in physics (physica systems described by two objects with a force between
More informationFactorizations of Invertible Symmetric Matrices over Polynomial Rings with Involution
Goba Journa of Pure and Appied Matheatics ISSN 0973-1768 Voue 13 Nuber 10 (017) pp 7073-7080 Research India Pubications http://wwwripubicationco Factorizations of Invertibe Syetric Matrices over Poynoia
More informationwhich is the moment of inertia mm -- the center of mass is given by: m11 r m2r 2
Chapter 6: The Rigid Rotator * Energy Levels of the Rigid Rotator - this is the odel for icrowave/rotational spectroscopy - a rotating diatoic is odeled as a rigid rotator -- we have two atos with asses
More informationThe state vectors j, m transform in rotations like D(R) j, m = m j, m j, m D(R) j, m. m m (R) = j, m exp. where. d (j) m m (β) j, m exp ij )
Anguar momentum agebra It is easy to see that the operat J J x J x + J y J y + J z J z commutes with the operats J x, J y and J z, [J, J i ] 0 We choose the component J z and denote the common eigenstate
More information13 Harmonic oscillator revisited: Dirac s approach and introduction to Second Quantization
3 Haronic oscillator revisited: Dirac s approach and introduction to Second Quantization. Dirac cae up with a ore elegant way to solve the haronic oscillator proble. We will now study this approach. The
More informationLECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October 7, 202 Prof. Aan Guth LECTURE NOTES 9 TRACELESS SYMMETRIC TENSOR APPROACH TO LEGENDRE POLYNOMIALS AND SPHERICAL
More informationChapter 6 1-D Continuous Groups
Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores:
More informationApplied Nuclear Physics (Fall 2006) Lecture 7 (10/2/06) Overview of Cross Section Calculation
22.101 Appied Nucear Physics (Fa 2006) Lecture 7 (10/2/06) Overview of Cross Section Cacuation References P. Roman, Advanced Quantum Theory (Addison-Wesey, Reading, 1965), Chap 3. A. Foderaro, The Eements
More informationConstruction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Atom
Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Ato Thoas S. Kuntzlean Mark Ellison John Tippin Departent of Cheistry Departent of Cheistry Departent
More informationFoundations of Complex Mechanics
Foundations of Compex Mechanics Chapter Outine ------------------------------------------------------------------------------------------------------------------------ 3. Quantum Hamiton Mechanics 56 3.
More informationOn the summations involving Wigner rotation matrix elements
Journal of Matheatical Cheistry 24 (1998 123 132 123 On the suations involving Wigner rotation atrix eleents Shan-Tao Lai a, Pancracio Palting b, Ying-Nan Chiu b and Harris J. Silverstone c a Vitreous
More informationAPPENDIX B. Some special functions in low-frequency seismology. 2l +1 x j l (B.2) j l = j l 1 l +1 x j l (B.3) j l = l x j l j l+1
APPENDIX B Soe specia functions in ow-frequency seisoogy 1. Spherica Besse functions The functions j and y are usefu in constructing ode soutions in hoogeneous spheres (fig B1). They satisfy d 2 j dr 2
More informationarxiv: v1 [quant-ph] 23 Dec 2018
THE CGLMP BELL INEQUALITIES AND QUANTUM THEORY B. J. Daton 1,2 arxiv:1812.09651v1 [quant-ph] 23 Dec 2018 1 Centre for Quantu and Optica Science, Swinburne University of Technoogy, Mebourne, Victoria 3122,
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationNuclear Size and Density
Nucear Size and Density How does the imited range of the nucear force affect the size and density of the nucei? Assume a Vecro ba mode, each having radius r, voume V = 4/3π r 3. Then the voume of the entire
More information14 - OSCILLATIONS Page 1
14 - OSCILLATIONS Page 1 14.1 Perioic an Osciator otion Motion of a sste at reguar interva of tie on a efinite path about a efinite point is known as a perioic otion, e.g., unifor circuar otion of a partice.
More informationSpherical harmonic representation of the geomagnetic fields in the Nigeria sector of the Niger delta basin
AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 010, Science Huβ, http://www.scihub.org/ajsir ISSN: 153-649X doi:10.551/ajsir.010.1..190.0 Spherica haronic representation of the geoagnetic fieds
More informationScattering and bound states
Chapter Scattering and bound states In this chapter we give a review of quantu-echanical scattering theory. We focus on the relation between the scattering aplitude of a potential and its bound states
More informationEigenvalues of the Angular Momentum Operators
Eigenvalues of the Angular Moentu Operators Toda, we are talking about the eigenvalues of the angular oentu operators. J is used to denote angular oentu in general, L is used specificall to denote orbital
More informationPhase Diagrams. Chapter 8. Conditions for the Coexistence of Multiple Phases. d S dt V
hase Diaras Chapter 8 hase - a for of atter that is unifor with respect to cheica coposition and the physica state of areation (soid, iquid, or aseous phases) icroscopicay and acroscopicay. Conditions
More informationa l b l m m w2 b l b G[ u/ R] C
Degree Probes for ence Graph Graars K. Skodinis y E. Wanke z Abstract The copexity of the bounded degree probe is anayzed for graph anguages generated by ence graph graars. In particuar, the bounded degree
More informationVIII. Addition of Angular Momenta
VIII Addition of Anguar Momenta a Couped and Uncouped Bae When deaing with two different ource of anguar momentum, Ĵ and Ĵ, there are two obviou bae that one might chooe to work in The firt i caed the
More informationPhysics 221B: Solution to HW # 6. 1) Born-Oppenheimer for Coupled Harmonic Oscillators
Physics B: Solution to HW # 6 ) Born-Oppenheier for Coupled Haronic Oscillators This proble is eant to convince you of the validity of the Born-Oppenheier BO) Approxiation through a toy odel of coupled
More informationPhysics 235 Chapter 8. Chapter 8 Central-Force Motion
Physics 35 Chapter 8 Chapter 8 Centra-Force Motion In this Chapter we wi use the theory we have discussed in Chapter 6 and 7 and appy it to very important probems in physics, in which we study the motion
More informationUniversity of Alabama Department of Physics and Astronomy. PH 105 LeClair Summer Problem Set 11
University of Aabaa Departent of Physics and Astronoy PH 05 LeCair Suer 0 Instructions: Probe Set. Answer a questions beow. A questions have equa weight.. Due Fri June 0 at the start of ecture, or eectronicay
More informationConvergence P H Y S I C S
+1 Test (Newton s Law of Motion) 1. Inertia is that property of a body by virtue of which the body is (a) Unabe to change by itsef the state of rest (b) Unabe to change by itsef the state of unifor otion
More informationV(R) = D e (1 e a(r R e) ) 2, (9.1)
Cheistry 6 Spectroscopy Ch 6 Week #3 Vibration-Rotation Spectra of Diatoic Molecules What happens to the rotation and vibration spectra of diatoic olecules if ore realistic potentials are used to describe
More informationA Simple Framework of Conservative Algorithms for the Coupled Nonlinear Schrödinger Equations with Multiply Components
Coun. Theor. Phys. 61 (2014) 703 709 Vo. 61, o. 6, June 1, 2014 A Sipe Fraework of Conservative Agoriths for the Couped oninear Schrödinger Equations with Mutipy Coponents QIA u ( ), 1,2, SOG Song-He (
More informationIntroduction to LMTO method
1 Introduction to MTO method 24 February 2011; V172 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method Ab initio Eectronic Structure Cacuations
More informationMore Scattering: the Partial Wave Expansion
More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction
More informationAN ANALYTICAL ESTIMATION OF THE CORIOLIS METER'S CHARACTERISTICS BASED ON MODAL SUPERPOSITION. J. Kutin *, I. Bajsić
Fow Measureent and Instruentation 1 (00) 345 351 doi:10.1016/s0955-5986(0)00006-7 00 Esevier Science Ltd. AN ANALYTICAL ESTIMATION OF THE CORIOLIS METER'S CHARACTERISTICS BASED ON MODAL SUPERPOSITION J.
More informationSEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l
Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationCancelling out systematic uncertainties
Mon. Not. R. Astron. Soc. 419, 1040 1050 (2012) doi:10.1111/.1365-2966.2011.19761.x Canceing out systeatic uncertainties Jorge Noreña, 1 Licia Verde, 1 Rau Jienez, 1 Caros Peña-Garay 2 and Cesar Goez 3
More informationMonomial MUBs. September 27, 2005
Monoia MUBs Seteber 7, 005 Abstract We rove that axia sets of couniting onoia unitaries are equivaent to a grou under utiication. We aso show that hadaard that underies this set of unitaries is equivaent
More informationAgenda Administrative Matters Atomic Physics Molecules
Fromm Institute for Lifeong Learning University of San Francisco Modern Physics for Frommies IV The Universe - Sma to Large Lecture 3 Agenda Administrative Matters Atomic Physics Moecues Administrative
More informationTRANSFORMATION OF REAL SPHERICAL HARMONICS UNDER ROTATIONS
Vo. 39 (008) ACTA PHYSICA POLONICA B No 8 TRANSFORMATION OF REAL SPHERICAL HARMONICS UNDER ROTATIONS Zbigniew Romanowski Interdiscipinary Centre for Materias Modeing Pawińskiego 5a, 0-106 Warsaw, Poand
More informationName: Partner(s): Date: Angular Momentum
Nae: Partner(s): Date: Angular Moentu 1. Purpose: In this lab, you will use the principle of conservation of angular oentu to easure the oent of inertia of various objects. Additionally, you develop a
More informationHubbard model with intersite kinetic correlations
PHYSICAL REVIEW B 79, 06444 2009 Hubbard ode with intersite kinetic correations Grzegorz Górski and Jerzy Mizia Institute of Physics, Rzeszów University, u. Rejtana 6A, 35-959 Rzeszów, Poand Received 29
More information12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015
18.354J Nonlinear Dynaics II: Continuu Systes Lecture 12 Spring 2015 12 Towards hydrodynaic equations The previous classes focussed on the continuu description of static (tie-independent) elastic systes.
More informationFRIEZE GROUPS IN R 2
FRIEZE GROUPS IN R 2 MAXWELL STOLARSKI Abstract. Focusing on the Eucidean pane under the Pythagorean Metric, our goa is to cassify the frieze groups, discrete subgroups of the set of isometries of the
More informationFormulas for Angular-Momentum Barrier Factors Version II
BNL PREPRINT BNL-QGS-06-101 brfactor1.tex Formuas for Anguar-Momentum Barrier Factors Version II S. U. Chung Physics Department, Brookhaven Nationa Laboratory, Upton, NY 11973 March 19, 2015 abstract A
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationLecture #8-3 Oscillations, Simple Harmonic Motion
Lecture #8-3 Oscillations Siple Haronic Motion So far we have considered two basic types of otion: translation and rotation. But these are not the only two types of otion we can observe in every day life.
More informationLegendre Polynomials - Lecture 8
Legendre Poynomias - Lecture 8 Introduction In spherica coordinates the separation of variabes for the function of the poar ange resuts in Legendre s equation when the soution is independent of the azimutha
More informationChapter 12. Quantum gases Microcanonical ensemble
Chapter 2 Quantu gases In classical statistical echanics, we evaluated therodynaic relations often for an ideal gas, which approxiates a real gas in the highly diluted liit. An iportant difference between
More informationFirst-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More informationField Mass Generation and Control. Chapter 6. The famous two slit experiment proved that a particle can exist as a wave and yet
111 Field Mass Generation and Control Chapter 6 The faous two slit experient proved that a particle can exist as a wave and yet still exhibit particle characteristics when the wavefunction is altered by
More informationOn Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1 x 2 ) and Their Byproducts
Commun. Theor. Phys. 66 (216) 369 373 Vo. 66, No. 4, October 1, 216 On Integras Invoving Universa Associated Legendre Poynomias and Powers of the Factor (1 x 2 ) and Their Byproducts Dong-Sheng Sun ( 孙东升
More informationAN INVESTIGATION ON SEISMIC ANALYSIS OF SHALLOW TUNEELS IN SOIL MEDIUM
The 4 th October -7, 8, Beijing, China AN INVESTIGATION ON SEISMIC ANALYSIS OF SHALLOW TUNEELS IN SOIL MEDIUM J. Boouri Bazaz and V. Besharat Assistant Professor, Dept. of Civi Engineering, Ferdowsi University,
More information4.3 Proving Lines are Parallel
Nae Cass Date 4.3 Proving Lines are Parae Essentia Question: How can you prove that two ines are parae? Expore Writing Converses of Parae Line Theores You for the converse of and if-then stateent "if p,
More informationChapter 1: Basics of Vibrations for Simple Mechanical Systems
Chapter 1: Basics of Vibrations for Siple Mechanical Systes Introduction: The fundaentals of Sound and Vibrations are part of the broader field of echanics, with strong connections to classical echanics,
More informationAN IMPROVED DOA ESTIMATION ALGORITHM FOR ASYNCHRONOUS MULTIPATH CDMA SYSTEM 1
Vo.23 No. JOURNA OF EERONIS (INA January 26 AN IMPROVED DOA ESIMAION AGORIM FOR ASYNRONOUS MUIPA DMA SYSEM Yang Wei hen Junshi an Zhenhui (Schoo of Eectronics and Info. Eng. Beijing Jiaotong University
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationQuantum Ground States as Equilibrium Particle Vacuum Interaction States
Quantu Ground States as Euilibriu article Vacuu Interaction States Harold E uthoff Abstract A rearkable feature of atoic ground states is that they are observed to be radiationless in nature despite (fro
More informationSome Perspective. Forces and Newton s Laws
Soe Perspective The language of Kineatics provides us with an efficient ethod for describing the otion of aterial objects, and we ll continue to ake refineents to it as we introduce additional types of
More informationApproximate dynamic programming using model-free Bellman Residual Elimination
Approxiate dynaic prograing using ode-free Bean Residua Eiination The MIT Facuty has ade this artice openy avaiabe. Pease share how this access benefits you. Your story atters. Citation As Pubished Pubisher
More informationLecture 16: Scattering States and the Step Potential. 1 The Step Potential 1. 4 Wavepackets in the step potential 6
Lecture 16: Scattering States and the Step Potential B. Zwiebach April 19, 2016 Contents 1 The Step Potential 1 2 Step Potential with E>V 0 2 3 Step Potential with E
More informationChaos in Dirac electron optics: Emergence of a relativistic quantum chimera
Suppeentary Inforation for Chaos in Dirac eectron optics: Eergence of a reativistic quantu chiera Hong-Ya Xu, Guang-Lei Wang, Liang Huang, and Ying-Cheng Lai Corresponding author: Y.-C. Lai Ying-Cheng.Lai@asu.edu
More information3D and 6D Fast Rotational Matching
3D and 6D Fast Rotationa Matching Juio Kovacs, Ph.D. Department of Moecuar Bioogy The Scripps Research Institute 10550 N. Torrey Pines Road, Mai TPC6 La Joa, Caifornia, 9037 Situs Modeing Workshop, San
More informationMassachusetts Institute of Technology Quantum Mechanics I (8.04) Spring 2005 Solutions to Problem Set 4
Massachusetts Institute of Technology Quantu Mechanics I (8.04) Spring 2005 Solutions to Proble Set 4 By Kit Matan 1. X-ray production. (5 points) Calculate the short-wavelength liit for X-rays produced
More informationVolume 13, MAIN ARTICLES
Voume 13, 2009 1 MAIN ARTICLES THE BASIC BVPs OF THE THEORY OF ELASTIC BINARY MIXTURES FOR A HALF-PLANE WITH CURVILINEAR CUTS Bitsadze L. I. Vekua Institute of Appied Mathematics of Iv. Javakhishvii Tbiisi
More informationPhysics 140 D100 Midterm Exam 2 Solutions 2017 Nov 10
There are 10 ultiple choice questions. Select the correct answer for each one and ark it on the bubble for on the cover sheet. Each question has only one correct answer. (2 arks each) 1. An inertial reference
More informationPerformance Evaluation of Space-Time Block Coding Using a Realistic Mobile Radio Channel Model *
Perforance Evauation of Space-Tie Bock Coding Using a Reaistic obie Radio Channe ode H. Carrasco Espinosa, J.. egado Penín Javier R. Fonoosa epartent of Signa Theory Counications Universitat Poitècnica
More informationSeparation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More informationc 2007 Society for Industrial and Applied Mathematics
SIAM REVIEW Vo. 49,No. 1,pp. 111 1 c 7 Society for Industria and Appied Mathematics Domino Waves C. J. Efthimiou M. D. Johnson Abstract. Motivated by a proposa of Daykin [Probem 71-19*, SIAM Rev., 13 (1971),
More information14 Separation of Variables Method
14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt
More informationORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION
J. Korean Math. Soc. 46 2009, No. 2, pp. 281 294 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION Hongying Xiao Abstract. Accuracy of the scaing function is very crucia in waveet theory, or correspondingy,
More informationHee = ~ dxdy\jj+ (x) 'IJ+ (y) u (x- y) \jj (y) \jj (x), V, = ~ dx 'IJ+ (x) \jj (x) V (x), Hii = Z 2 ~ dx dy cp+ (x) cp+ (y) u (x- y) cp (y) cp (x),
SOVIET PHYSICS JETP VOLUME 14, NUMBER 4 APRIL, 1962 SHIFT OF ATOMIC ENERGY LEVELS IN A PLASMA L. E. PARGAMANIK Khar'kov State University Subitted to JETP editor February 16, 1961; resubitted June 19, 1961
More informationAn Exactly Soluble Multiatom-Multiphoton Coupling Model
Brazilian Journal of Physics vol no 4 Deceber 87 An Exactly Soluble Multiato-Multiphoton Coupling Model A N F Aleixo Instituto de Física Universidade Federal do Rio de Janeiro Rio de Janeiro RJ Brazil
More informationPHYSICS 110A : CLASSICAL MECHANICS MIDTERM EXAM #2
PHYSICS 110A : CLASSICAL MECHANICS MIDTERM EXAM #2 [1] Two blocks connected by a spring of spring constant k are free to slide frictionlessly along a horizontal surface, as shown in Fig. 1. The unstretched
More informationLecture 6 Povh Krane Enge Williams Properties of 2-nucleon potential
Lecture 6 Povh Krane Enge Wiiams Properties of -nuceon potentia 16.1 4.4 3.6 9.9 Meson Theory of Nucear potentia 4.5 3.11 9.10 I recommend Eisberg and Resnik notes as distributed Probems, Lecture 6 1 Consider
More informationFFTs in Graphics and Vision. Spherical Convolution and Axial Symmetry Detection
FFTs in Graphics and Vision Spherica Convoution and Axia Symmetry Detection Outine Math Review Symmetry Genera Convoution Spherica Convoution Axia Symmetry Detection Math Review Symmetry: Given a unitary
More informationSimple Harmonic Motion
Chapter 3 Sipe Haronic Motion Practice Probe Soutions Student extboo pae 608. Conceptuaize the Probe - he period of a ass that is osciatin on the end of a sprin is reated to its ass and the force constant
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationSection 6: Magnetostatics
agnetic fieds in matter Section 6: agnetostatics In the previous sections we assumed that the current density J is a known function of coordinates. In the presence of matter this is not aways true. The
More informationPhysics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions
Physics 27c: Statistica Mechanics Fermi Liquid Theory: Coective Modes Botzmann Equation The quasipartice energy incuding interactions ε p,σ = ε p + f(p, p ; σ, σ )δn p,σ, () p,σ with ε p ε F + v F (p p
More informationUnimodality and Log-Concavity of Polynomials
Uniodaity and Log-Concavity of Poynoias Jenny Avarez University of Caifornia Santa Barbara Leobardo Rosaes University of Caifornia San Diego Agst 10, 2000 Mige Aadis Nyack Coege New York Abstract A poynoia
More information8.1 Force Laws Hooke s Law
8.1 Force Laws There are forces that don't change appreciably fro one instant to another, which we refer to as constant in tie, and forces that don't change appreciably fro one point to another, which
More informationLECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October, 202 Prof. Aan Guth LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
More informationLecture 6. Announcements. Conservation Laws: The Most Powerful Laws of Physics. Conservation Laws Why they are so powerful
Conseration Laws: The Most Powerful Laws of Physics Potential Energy gh Moentu p = + +. Energy E = PE + KE +. Kinetic Energy / Announceents Mon., Sept. : Second Law of Therodynaics Gie out Hoework 4 Wed.,
More information