Time Evolution of Matter States
|
|
- Walter Shaw
- 5 years ago
- Views:
Transcription
1 Tie Evolution of Matter States W. M. Hetherington February 15, 1 The Tie-Evolution Operat The tie-evolution of a wavefunction is deterined by the effect of a tie evolution operat through the relation Ψ r, t) = Ut)Ψ r, t), with U) = 1. 1) Using this in the Schrödinger equation i t Ψ r, t) = HΨ r, t) = i t Ut) ) Ψ r, ) = HUt)Ψ r, ), ) yields the differential equation f U i Ut) = HUt). 3) t If H is independent of tie, the solution is the siple tie dependent phase fact of a stationary state Ut) = e i ~ Ht. 4) Since the stationary state Ψ) is an eigenfunction of H with eigenvalue E, Ut)Ψ) = e i ~ Ht Ψ) = e i ~ Et Ψ). 5) When the Hailtonian is a general function of tie Ut) = U) i H dt )Ut )dt. 6) The solution to this integral equation can be obtained by interation. The first der approxiation can be obtained by substituting the zero der approxiation U) = 1 into the integral Ut) = 1 i Ht ) dt. 7) The second der approxiation is obtained by replacing Ut) in the integral with the first der expression Ut) = 1 i Ht )dt + i ) Ht )Ht )dt dt, 8) 1
2 and so fth Ut) = 1 i Ht 1 ) dt 1 + i ) + i ) 3 Ht 1 )Ht ) dt 1 dt Ht 1 )Ht )Ht 3 ) dt 1 dt dt 3 +, 9) When H = H o + V t), Ut) can be written in the f Ut) = U o t)u v t), where U o t) = e i ~ Hot. 1) Using this definition in the Schrodinger equation yields ) ) i t U o U v + U o t U v = HU o U v = H o U o U v + V t)u o U v. 11) Then,??) leads to siply Operating on the left with U o yields the relation H o U o U v + iu o t U v = H o U o U v + V t)u o U v, 1) iu o t U v = V t)u o U v. 13) i t U v = U V t)u ou v = V H t)u v, 14) where V H t) is the Heisenberg representation of the operat Vt). Solving the resulting integral equation by iteration yields U v t) = 1 i V H t 1 ) dt 1 + i ) + i ) 3 V H t 1 )V H t ) dt 1 dt V H t 1 )V H t )V H t 3 ) dt 1 dt dt ) When there is no applied tie-dependent electroagnetic field other perturbation, then the Hailtonian ay consist of two ters: one large operat H o f which the eigenstates can be found, and one sall operat V = constant operat considered to be a perturbation. Then the expression f U is again?? but the tie dependence of the integrands is particulary siple.
3 Expansion of Ψ in a basis of pseudo-stationary states Coonly, we know only the set of eigenstates φ l : l = 1,,... of an approxiate tie-independent Hailtonian H a, with H a φ l = ε l φ l. Even when the total Hailtonian has no tie dependence, Ψ will be a tie-dependent linear cobination of the basis functions. Suppose that H o = H a + H p, where H p represents a sall tie-independent perturbation, such as spin-bit coupling. Then Ψt) = k c k t)φ k = k c k t) k a, 16) where the set of coefficients c k is deterined by perturbation they a variational approach, and the subscript a on the index k indicates that the function k a is an eigenstate of H a. Using the concept of a vect space, the set k a is presued to be a coplete basis f the description of any state vect Ψ. An iptant general expression f the coefficients is c k t) = k a Ψt), 17) which is siply the projection of Ψ on to the k a axis. Now suppose that we know that at tie t = the syste is in a definite eigenstate of H a, that is Ψ) = l a. We also know that So, Ψt) = Ut)Ψ) = UΨ o. 18) c k t) = k Ψt) = k UΨ o = k a U l a = U kl. 19) The zero-der expression f c k t) arises fro the zero-der approxiation f U, that is Ut) = U o U v = U o. Then c k t) = k a U o l a = k a e i ~ H at l a = e i ~ ε lt k a l a = e i ~ ε lt δ kl. ) So, the state of the syste does not change. The first der ter in U v contributes c k t) = k a U o i c k t) = i e i ~ ε lt k a H p l a c k t) = i e i ~ ε lt The probability of the state k a is c k t) = e i ~ H at H p e i ~ H at dt ) l a, 1) k a e i ~ Ha t H p e i ~ Ha t l a dt, ) e~ i ε k ε l )t dt = i ) ~ e i i εlt k a H p l a e~ i ε k ε l )t 1) ɛ k ɛ l ɛ k ɛ l ) k a H p l a )) ɛk ɛ l ) 1 cos t. 3), 4) which is just the first-der tie-independent perturbation result. This looks pathological as ɛ k ɛ l ), but it really is not since cos x = 1 x / +. The ost useful application of this expression is f the case of ɛ k ɛ l. Then c k t) = k a H p l a t. 5) 3
4 Ut) f the EM Field-Matter Interation It is custoary to consider the interaction between the electroagnetic field and atter to be weak relative to the interactions aong the atter particles theselves, and this is certainly the case f typical nonlinear optical phenoena. Therefe, we write H = H + H I t) + H F, 6) where H describes the atter, H F describes the field and H I t) describes the interaction. If H I t) =, then H has two independent ters, so the total wavefunction is a siple product of atter and field states, Ψt) = ψt)ξt). 7) We will assue that the Schrödinger equation f the atter has been solved and that the wavefunction f the atter is Ψt) = c l t)φ l = c l t) l, 8) l l which describes a wave packet coherent superposition of the stationary state solutions k. The field wavefunction is ξ = n k k, 9) k k where k ranges over odes of the radiation field that are of interest and k refers to all other odes. A coherent state function would be prefered, but the nuber state function akes the algebra sipler. The coherence of the radiation field can be handled at a later stage. The total Hailtonian is H = n e +n n 1 n e +n n p l q la rl, t) + l i j odes q i q j + 4πɛ r ij k ω k a k a k + 1 ). 3) Considering the interaction of the radiation field only with the electrons, the interaction energy operat is n e e H I = p l A r l, t) + A r ) n e e l, t) p l + A r l, t) A r l, t), 31) and the total Hailtonian is H = n e +n n p n e +n n l + l i j A tie-evolution operat is now odes q i q j + H I t) + 4πɛ r ij k ω k a k a k + 1 ), 3) H = H + H I t) + H F. 33) Ut) = U t)u v t), where U t) = e i ~ H +H F )t 34) and U v is defined by eq. 16. The operat V H t) = U t)h I t)u t) 35) ust now be carefully defined. First, specify H I precisely. The ter involving A will be igned as not being iptant f this discussion. Since we are wking in the Coulob gauge A =, and p A = i A = i A) + A = ia = A p 36) 4
5 as an operat relation. Therefe, n e H I = A r l, t) p l. 37) This is refered to as the oentu f of the interaction Hailtonian. The vect potential A is a siple ultiplicative operat as far as the atter states are concerned but contains annihilation and creation operats f the field states. So, consider only the atter operats f the oent. Using p l = i l as an operat on a atter state l chills the spine. Ftunately, the siple coutat relations x, p x = y, p y = z, p z = i, 38) x, p x = ip x, y, p y = ip y, z, p z = ip z 39) provide a way to replace p with r. We will need to evaluate integrals of the f k p l k. Using we find that r j, H = n e 1 r j, p l = i p j, 4) k p l k = i k r l, H k = i k r l H H r l k, 41) k p l k = i E k E k ) k r l k = iω k ω k) k r l k = iω k k k r l k. 4) Me thoughtful analyses yield the sae result. This replaceent of p with r will be used later. Notice that H I is a su of one-electron operats. Electrons interact with the field individually, and the results are additive. Without any loss in generality, the interaction of only a single particle will be considered.h I. Thus H I = A r, t) p. 43) Now, turn to the quantized field operat A. Fro an earlier discussion, A r, t) = odes j A j a j ˆɛ j e i k j r ω j t) + a j ˆɛ j e i k j r ω j t). 44) Both creation a and annihilation a operats appear f all the polarizations and wave vects k. In a two-photon process, governed by the second ter in equation 16, any products of these operats would appear. We can restrict the contributions to A to only those operats which yield changes in the field which are easurable expected. Thus, f a two-photon absption of photons of odes ω 1, ˆɛ 1 ) and ω, ˆɛ ), we can use A r, t) = A 1 a 1 ˆɛ 1 e i k 1 r ω 1 t) + A a ˆɛ e i k r ω t). 45) Siilarly, f su-frequency generation second haronic generation), photons of odes ω 1, ˆɛ 1 ) and ω, ˆɛ ) are annihilated, and one photon of ode ω 3, ˆɛ 3 ) is created. So, we can use A r, t) = A 1 a 1 ˆɛ 1 e i k 1 r ω 1 t) + A a ˆɛ e i k r ω t) + A 3 a 3 ˆɛ 3 e i k 3 r ω 3 t). 46) All this can now be used to construct V H, V H t) = U t)h I t)u t) = e i ~ H +H F )t H I t)e i ~ H +H F )t. 47) 5
6 Continuing, V H t) = e ~ i H +H F )t A r, t) p e~ i H +H F )t. 48) We are now ready to ake use of the second der ter f U v t), U v t) = i ) V H t 1 )V H t ) dt 1 dt. 49) First, consider how this operat will be used. Fro??, Ψt) = Ut)ψ)ξ), 5) we see that we need to specify the initial state of the syste. Ordinarily, this would be given as a ξ a, where a is a stationary state of the atter Hailtonian and ξ a is a product of nuber states of the field. We will be looking f evidence of a particular final state f ξ f. Thus, we want to project Ψt) onto the f ξ f axis. Generally, Ψt) = l c l t) k a = Ut)ψ)ξ), 51) so, Using the product f f U yields Operating to the left with U, c f t) = fξ f Ψt) = fξ f Ut) aξ a = U fa. 5) fξ f U t)u v t) aξ a = fξ f e i ~ H +H F )t U v t) aξ a. 53) Using only the second der ter f U v, Continuing we arrive at the glious equation i ) e i ~ E f t fξ f U v t) aξ a. 54) e i ~ E f t fξ f U v t) aξ a. 55) i ) fξ f V H t 1 )V H t ) aξ a dt 1 dt, 56) fξ f e ~ i H +H F )t 1 A r, t 1 ) p A r, t ) p e~ i H +H F )t aξ a dt 1 dt. 57) If we use the third-der ter f U v, we arrive at the even e glious equation i ) 3 fξ f e i ~ H +H F )t 1 A r, t 1 ) p A r, t 3 ) p A r, t ) p e i ~ H +H F )t 3 aξ a dt 1 dt dt 3. 58) 6
7 And gliouser sill is the fourth-der equation i ) 4 3 Two-Photon Absption fξ f e ~ i H +H F )t 1 A r, t 3 ) p A r, t 4 ) p A r, t 1 ) p A r, t ) p e i ~ H +H F )t 4 aξ a dt 1 dt dt 3 dt 4. 59) To find the probability of finding the syste in state fξ f at tie t we need to take the absolute square of the glious equation. To evaluate i ) 1 fξ f A r, t 1 ) p A r, t ) p aξ a e~ i E at dt 1 dt, 6) we will insert a coplete set of states between the two operats fξ f A r, t 1 ) p jξ j jξ j A r, t ) p aξ a. 61) j Focusing on just the t integral, jξ j A 1 a 1 ˆɛ 1 e i k 1 r ω 1 t ) + A a ˆɛ e i k r ω t ) p aξ a e~ i Ea t dt = jξ j A 1 a 1 ˆɛ 1 e i k 1 r ω 1 t ) p aξ a e i ~ E a t dt jξ j A a ˆɛ e i k r ω t ) p aξ a e i ~ E a t dt. 6) Thus, we have two different tie-derings: ω 1 followed by ω and vice versa. 7
Physics 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 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 informationIII. Quantization of electromagnetic field
III. Quantization of electroagnetic field Using the fraework presented in the previous chapter, this chapter describes lightwave in ters of quantu echanics. First, how to write a physical quantity operator
More informationFirst of all, because the base kets evolve according to the "wrong sign" Schrödinger equation (see pp ),
HW7.nb HW #7. Free particle path integral a) Propagator To siplify the notation, we write t t t, x x x and work in D. Since x i, p j i i j, we can just construct the 3D solution. First of all, because
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 informationSupporting Information for Supression of Auger Processes in Confined Structures
Supporting Inforation for Supression of Auger Processes in Confined Structures George E. Cragg and Alexander. Efros Naval Research aboratory, Washington, DC 20375, USA 1 Solution of the Coupled, Two-band
More information(a) As a reminder, the classical definition of angular momentum is: l = r p
PHYSICS T8: Standard Model Midter Exa Solution Key (216) 1. [2 points] Short Answer ( points each) (a) As a reinder, the classical definition of angular oentu is: l r p Based on this, what are the units
More informationPhysics 215 Winter The Density Matrix
Physics 215 Winter 2018 The Density Matrix The quantu space of states is a Hilbert space H. Any state vector ψ H is a pure state. Since any linear cobination of eleents of H are also an eleent of H, it
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 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 informationSOLUTIONS for Homework #3
SOLUTIONS for Hoework #3 1. In the potential of given for there is no unboun states. Boun states have positive energies E n labele by an integer n. For each energy level E, two syetrically locate classical
More informationDispersion. February 12, 2014
Dispersion February 1, 014 In aterials, the dielectric constant and pereability are actually frequency dependent. This does not affect our results for single frequency odes, but when we have a superposition
More informationMechanics Physics 151
Mechanics Physics 5 Lecture Oscillations (Chapter 6) What We Did Last Tie Analyzed the otion of a heavy top Reduced into -diensional proble of θ Qualitative behavior Precession + nutation Initial condition
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 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 information2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all
Lecture 6 Introduction to kinetic theory of plasa waves Introduction to kinetic theory So far we have been odeling plasa dynaics using fluid equations. The assuption has been that the pressure can be either
More information5.2. Example: Landau levels and quantum Hall effect
68 Phs460.nb i ħ (-i ħ -q A') -q φ' ψ' = + V(r) ψ' (5.49) t i.e., using the new gauge, the Schrodinger equation takes eactl the sae for (i.e. the phsics law reains the sae). 5.. Eaple: Lau levels quantu
More informationCausality and the Kramers Kronig relations
Causality and the Kraers Kronig relations Causality describes the teporal relationship between cause and effect. A bell rings after you strike it, not before you strike it. This eans that the function
More informationStern-Gerlach Experiment
Stern-Gerlach Experient HOE: The Physics of Bruce Harvey This is the experient that is said to prove that the electron has an intrinsic agnetic oent. Hydrogen like atos are projected in a bea through a
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 informationElectromagnetic Waves
Electroagnetic Waves Physics 4 Maxwell s Equations Maxwell s equations suarize the relationships between electric and agnetic fields. A ajor consequence of these equations is that an accelerating charge
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 information1 Graded problems. PHY 5246: Theoretical Dynamics, Fall November 23 rd, 2015 Assignment # 12, Solutions. Problem 1
PHY 546: Theoretical Dynaics, Fall 05 Noveber 3 rd, 05 Assignent #, Solutions Graded probles Proble.a) Given the -diensional syste we want to show that is a constant of the otion. Indeed,.b) dd dt Now
More information2 Quantization of the Electromagnetic Field
2 Quantization of the Electromagnetic Field 2.1 Basics Starting point of the quantization of the electromagnetic field are Maxwell s equations in the vacuum (source free): where B = µ 0 H, D = ε 0 E, µ
More informationQuantum Chemistry Exam 2 Take-home Solutions
Cheistry 60 Fall 07 Dr Jean M Standard Nae KEY Quantu Cheistry Exa Take-hoe Solutions 5) (0 points) In this proble, the nonlinear variation ethod will be used to deterine an approxiate solution for the
More informationA Simple Regression Problem
A Siple Regression Proble R. M. Castro March 23, 2 In this brief note a siple regression proble will be introduced, illustrating clearly the bias-variance tradeoff. Let Y i f(x i ) + W i, i,..., n, where
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 informationPhysics 202H - Introductory Quantum Physics I Homework #12 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/12/13
Physics 0H - Introctory Quantu Physics I Hoework # - Solutions Fall 004 Due 5:0 PM, Monday 004//3 [70 points total] Journal questions. Briefly share your thoughts on the following questions: What aspects
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search
Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths
More information0.1 Schrödinger Equation in 2-dimensional system
0.1 Schrödinger Equation in -dimensional system In HW problem set 5, we introduced a simpleminded system describing the ammonia (NH 3 ) molecule, consisting of a plane spanned by the 3 hydrogen atoms and
More informationCHAPTER 15: Vibratory Motion
CHAPTER 15: Vibratory Motion courtesy of Richard White courtesy of Richard White 2.) 1.) Two glaring observations can be ade fro the graphic on the previous slide: 1.) The PROJECTION of a point on a circle
More informationBALLISTIC PENDULUM. EXPERIMENT: Measuring the Projectile Speed Consider a steel ball of mass
BALLISTIC PENDULUM INTRODUCTION: In this experient you will use the principles of conservation of oentu and energy to deterine the speed of a horizontally projected ball and use this speed to predict the
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 information13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices
CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay
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 informationDepartment of Physics Preliminary Exam January 3 6, 2006
Departent of Physics Preliinary Exa January 3 6, 2006 Day 1: Classical Mechanics Tuesday, January 3, 2006 9:00 a.. 12:00 p.. Instructions: 1. Write the answer to each question on a separate sheet of paper.
More informationMulti-Scale/Multi-Resolution: Wavelet Transform
Multi-Scale/Multi-Resolution: Wavelet Transfor Proble with Fourier Fourier analysis -- breaks down a signal into constituent sinusoids of different frequencies. A serious drawback in transforing to the
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 informationPeriodic Motion is everywhere
Lecture 19 Goals: Chapter 14 Interrelate the physics and atheatics of oscillations. Draw and interpret oscillatory graphs. Learn the concepts of phase and phase constant. Understand and use energy conservation
More informationIII.H Zeroth Order Hydrodynamics
III.H Zeroth Order Hydrodynaics As a first approxiation, we shall assue that in local equilibriu, the density f 1 at each point in space can be represented as in eq.iii.56, i.e. f 0 1 p, q, t = n q, t
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 information72. (30.2) Interaction between two parallel current carrying wires.
7. (3.) Interaction between two parallel current carrying wires. Two parallel wires carrying currents exert forces on each other. Each current produces a agnetic field in which the other current is placed.
More informationA toy model of quantum electrodynamics in (1 + 1) dimensions
IOP PUBLISHING Eur. J. Phys. 29 (2008) 815 830 EUROPEAN JOURNAL OF PHYSICS doi:10.1088/0143-0807/29/4/014 A toy odel of quantu electrodynaics in (1 + 1) diensions ADBoozer Departent of Physics, California
More informationCSE525: Randomized Algorithms and Probabilistic Analysis May 16, Lecture 13
CSE55: Randoied Algoriths and obabilistic Analysis May 6, Lecture Lecturer: Anna Karlin Scribe: Noah Siegel, Jonathan Shi Rando walks and Markov chains This lecture discusses Markov chains, which capture
More informationGeneralized eigenfunctions and a Borel Theorem on the Sierpinski Gasket.
Generalized eigenfunctions and a Borel Theore on the Sierpinski Gasket. Kasso A. Okoudjou, Luke G. Rogers, and Robert S. Strichartz May 26, 2006 1 Introduction There is a well developed theory (see [5,
More information(a) Why cannot the Carnot cycle be applied in the real world? Because it would have to run infinitely slowly, which is not useful.
PHSX 446 FINAL EXAM Spring 25 First, soe basic knowledge questions You need not show work here; just give the answer More than one answer ight apply Don t waste tie transcribing answers; just write on
More informationForce and dynamics with a spring, analytic approach
Force and dynaics with a spring, analytic approach It ay strie you as strange that the first force we will discuss will be that of a spring. It is not one of the four Universal forces and we don t use
More informationma x = -bv x + F rod.
Notes on Dynaical Systes Dynaics is the study of change. The priary ingredients of a dynaical syste are its state and its rule of change (also soeties called the dynaic). Dynaical systes can be continuous
More informationThe Chebyshev Matching Transformer
/9/ The Chebyshev Matching Transforer /5 The Chebyshev Matching Transforer An alternative to Binoial (Maxially Flat) functions (and there are any such alternatives!) are Chebyshev polynoials. Pafnuty Chebyshev
More informationThe Weierstrass Approximation Theorem
36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined
More informationU V. r In Uniform Field the Potential Difference is V Ed
SPHI/W nit 7.8 Electric Potential Page of 5 Notes Physics Tool box Electric Potential Energy the electric potential energy stored in a syste k of two charges and is E r k Coulobs Constant is N C 9 9. E
More informationThe accelerated expansion of the universe is explained by quantum field theory.
The accelerated expansion of the universe is explained by quantu field theory. Abstract. Forulas describing interactions, in fact, use the liiting speed of inforation transfer, and not the speed of light.
More informationlecture 36: Linear Multistep Mehods: Zero Stability
95 lecture 36: Linear Multistep Mehods: Zero Stability 5.6 Linear ultistep ethods: zero stability Does consistency iply convergence for linear ultistep ethods? This is always the case for one-step ethods,
More informationPhysically Based Modeling CS Notes Spring 1997 Particle Collision and Contact
Physically Based Modeling CS 15-863 Notes Spring 1997 Particle Collision and Contact 1 Collisions with Springs Suppose we wanted to ipleent a particle siulator with a floor : a solid horizontal plane which
More informationPhys463.nb. Many electrons in 1D at T = 0. For a large system (L ), ΕF =? (6.7) The solutions of this equation are plane waves (6.
â â x Ψn Hx Ε Ψn Hx 35 (6.7) he solutions of this equation are plane waves Ψn Hx A exphä n x (6.8) he eigen-energy Εn is n (6.9) Εn For a D syste with length and periodic boundary conditions, Ψn Hx Ψn
More informationIn the session you will be divided into groups and perform four separate experiments:
Mechanics Lab (Civil Engineers) Nae (please print): Tutor (please print): Lab group: Date of lab: Experients In the session you will be divided into groups and perfor four separate experients: (1) air-track
More informationPHY307F/407F - Computational Physics Background Material for Expt. 3 - Heat Equation David Harrison
INTRODUCTION PHY37F/47F - Coputational Physics Background Material for Expt 3 - Heat Equation David Harrison In the Pendulu Experient, we studied the Runge-Kutta algorith for solving ordinary differential
More informationPrinciples of Optimal Control Spring 2008
MIT OpenCourseWare http://ocw.it.edu 16.323 Principles of Optial Control Spring 2008 For inforation about citing these aterials or our Ters of Use, visit: http://ocw.it.edu/ters. 16.323 Lecture 10 Singular
More informationUMPC mercredi 19 avril 2017
UMPC ercrei 19 avril 017 M Mathéatiques & Applications UE ANEDP, COCV: Analyse et contrôle e systèes quantiques Contrôle es connaissances, urée heures. Sujet onné par M. Mirrahii et P. Rouchon Les ocuents
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 informationAnisotropic reference media and the possible linearized approximations for phase velocities of qs waves in weakly anisotropic media
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS J. Phys. D: Appl. Phys. 5 00 007 04 PII: S00-770867-6 Anisotropic reference edia and the possible linearized approxiations for phase
More informationSOLUTIONS. PROBLEM 1. The Hamiltonian of the particle in the gravitational field can be written as, x 0, + U(x), U(x) =
SOLUTIONS PROBLEM 1. The Hailtonian of the particle in the gravitational field can be written as { Ĥ = ˆp2, x 0, + U(x), U(x) = (1) 2 gx, x > 0. The siplest estiate coes fro the uncertainty relation. If
More informationNuclear Physics (10 th lecture)
~Theta Nuclear Physics ( th lecture) Content Nuclear Collective Model: Rainwater approx. (reinder) Consequences of nuclear deforation o Rotational states High spin states and back bending o Vibrational
More informationPhysics 2210 Fall smartphysics 20 Conservation of Angular Momentum 21 Simple Harmonic Motion 11/23/2015
Physics 2210 Fall 2015 sartphysics 20 Conservation of Angular Moentu 21 Siple Haronic Motion 11/23/2015 Exa 4: sartphysics units 14-20 Midter Exa 2: Day: Fri Dec. 04, 2015 Tie: regular class tie Section
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 informationPh 20.3 Numerical Solution of Ordinary Differential Equations
Ph 20.3 Nuerical Solution of Ordinary Differential Equations Due: Week 5 -v20170314- This Assignent So far, your assignents have tried to failiarize you with the hardware and software in the Physics Coputing
More informationFeshbach Resonances in Ultracold Gases
Feshbach Resonances in Ultracold Gases Sara L. Capbell MIT Departent of Physics Dated: May 5, 9) First described by Heran Feshbach in a 958 paper, Feshbach resonances describe resonant scattering between
More informationEffects of an Inhomogeneous Magnetic Field (E =0)
Effects of an Inhoogeneous Magnetic Field (E =0 For soe purposes the otion of the guiding centers can be taken as a good approxiation of that of the particles. ut it ust be recognized that during the particle
More informationA new type of lower bound for the largest eigenvalue of a symmetric matrix
Linear Algebra and its Applications 47 7 9 9 www.elsevier.co/locate/laa A new type of lower bound for the largest eigenvalue of a syetric atrix Piet Van Mieghe Delft University of Technology, P.O. Box
More information1. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants.
Sample final questions.. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants. 2. A one-dimensional harmonic oscillator, originally in the ground state,
More information16. GAUGE THEORY AND THE CREATION OF PHOTONS
6. GAUGE THEORY AD THE CREATIO OF PHOTOS In the previous chapter the existence of a gauge theory allowed the electromagnetic field to be described in an invariant manner. Although the existence of this
More informationA Note on Scheduling Tall/Small Multiprocessor Tasks with Unit Processing Time to Minimize Maximum Tardiness
A Note on Scheduling Tall/Sall Multiprocessor Tasks with Unit Processing Tie to Miniize Maxiu Tardiness Philippe Baptiste and Baruch Schieber IBM T.J. Watson Research Center P.O. Box 218, Yorktown Heights,
More informationIn this chapter we will start the discussion on wave phenomena. We will study the following topics:
Chapter 16 Waves I In this chapter we will start the discussion on wave phenoena. We will study the following topics: Types of waves Aplitude, phase, frequency, period, propagation speed of a wave Mechanical
More informationPH 221-2A Fall Waves - I. Lectures Chapter 16 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition)
PH 1-A Fall 014 Waves - I Lectures 4-5 Chapter 16 (Halliday/Resnick/Walker, Fundaentals of Physics 9 th edition) 1 Chapter 16 Waves I In this chapter we will start the discussion on wave phenoena. We will
More informationNon-Parametric Non-Line-of-Sight Identification 1
Non-Paraetric Non-Line-of-Sight Identification Sinan Gezici, Hisashi Kobayashi and H. Vincent Poor Departent of Electrical Engineering School of Engineering and Applied Science Princeton University, Princeton,
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 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 informationAn Approximate Model for the Theoretical Prediction of the Velocity Increase in the Intermediate Ballistics Period
An Approxiate Model for the Theoretical Prediction of the Velocity... 77 Central European Journal of Energetic Materials, 205, 2(), 77-88 ISSN 2353-843 An Approxiate Model for the Theoretical Prediction
More informationP (t) = P (t = 0) + F t Conclusion: If we wait long enough, the velocity of an electron will diverge, which is obviously impossible and wrong.
4 Phys520.nb 2 Drude theory ~ Chapter in textbook 2.. The relaxation tie approxiation Here we treat electrons as a free ideal gas (classical) 2... Totally ignore interactions/scatterings Under a static
More informationTopic 5a Introduction to Curve Fitting & Linear Regression
/7/08 Course Instructor Dr. Rayond C. Rup Oice: A 337 Phone: (95) 747 6958 E ail: rcrup@utep.edu opic 5a Introduction to Curve Fitting & Linear Regression EE 4386/530 Coputational ethods in EE Outline
More informationFour-vector, Dirac spinor representation and Lorentz Transformations
Available online at www.pelagiaresearchlibrary.co Advances in Applied Science Research, 2012, 3 (2):749-756 Four-vector, Dirac spinor representation and Lorentz Transforations S. B. Khasare 1, J. N. Rateke
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationGeneralized r-modes of the Maclaurin spheroids
PHYSICAL REVIEW D, VOLUME 59, 044009 Generalized r-odes of the Maclaurin spheroids Lee Lindblo Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, California 9115 Jaes R. Ipser
More informationA method to determine relative stroke detection efficiencies from multiplicity distributions
A ethod to deterine relative stroke detection eiciencies ro ultiplicity distributions Schulz W. and Cuins K. 2. Austrian Lightning Detection and Inoration Syste (ALDIS), Kahlenberger Str.2A, 90 Vienna,
More informationdt dt THE AIR TRACK (II)
THE AIR TRACK (II) References: [] The Air Track (I) - First Year Physics Laoratory Manual (PHY38Y and PHYY) [] Berkeley Physics Laoratory, nd edition, McGraw-Hill Book Copany [3] E. Hecht: Physics: Calculus,
More informationTEST OF HOMOGENEITY OF PARALLEL SAMPLES FROM LOGNORMAL POPULATIONS WITH UNEQUAL VARIANCES
TEST OF HOMOGENEITY OF PARALLEL SAMPLES FROM LOGNORMAL POPULATIONS WITH UNEQUAL VARIANCES S. E. Ahed, R. J. Tokins and A. I. Volodin Departent of Matheatics and Statistics University of Regina Regina,
More informationIn this chapter, we consider several graph-theoretic and probabilistic models
THREE ONE GRAPH-THEORETIC AND STATISTICAL MODELS 3.1 INTRODUCTION In this chapter, we consider several graph-theoretic and probabilistic odels for a social network, which we do under different assuptions
More informationWhat is the instantaneous acceleration (2nd derivative of time) of the field? Sol. The Euler-Lagrange equations quickly yield:
PHYSICS 75: The Standard Model Midter Exa Solution Key. [3 points] Short Answer (6 points each (a In words, explain how to deterine the nuber of ediator particles are generated by a particular local gauge
More informationPolygonal Designs: Existence and Construction
Polygonal Designs: Existence and Construction John Hegean Departent of Matheatics, Stanford University, Stanford, CA 9405 Jeff Langford Departent of Matheatics, Drake University, Des Moines, IA 5011 G
More informationPhysicsAndMathsTutor.com
. A raindrop falls vertically under gravity through a cloud. In a odel of the otion the raindrop is assued to be spherical at all ties and the cloud is assued to consist of stationary water particles.
More informationlecture 37: Linear Multistep Methods: Absolute Stability, Part I lecture 38: Linear Multistep Methods: Absolute Stability, Part II
lecture 37: Linear Multistep Methods: Absolute Stability, Part I lecture 3: Linear Multistep Methods: Absolute Stability, Part II 5.7 Linear ultistep ethods: absolute stability At this point, it ay well
More informationEE5900 Spring Lecture 4 IC interconnect modeling methods Zhuo Feng
EE59 Spring Parallel LSI AD Algoriths Lecture I interconnect odeling ethods Zhuo Feng. Z. Feng MTU EE59 So far we ve considered only tie doain analyses We ll soon see that it is soeties preferable to odel
More informationTHE ROCKET EXPERIMENT 1. «Homogenous» gravitational field
THE OCKET EXPEIENT. «Hoogenous» gravitational field Let s assue, fig., that we have a body of ass Μ and radius. fig. As it is known, the gravitational field of ass Μ (both in ters of geoetry and dynaics)
More informationRadiating Dipoles in Quantum Mechanics
Radiating Dipoles in Quantum Mechanics Chapter 14 P. J. Grandinetti Chem. 4300 Oct 27, 2017 P. J. Grandinetti (Chem. 4300) Radiating Dipoles in Quantum Mechanics Oct 27, 2017 1 / 26 P. J. Grandinetti (Chem.
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 informationOcean 420 Physical Processes in the Ocean Project 1: Hydrostatic Balance, Advection and Diffusion Answers
Ocean 40 Physical Processes in the Ocean Project 1: Hydrostatic Balance, Advection and Diffusion Answers 1. Hydrostatic Balance a) Set all of the levels on one of the coluns to the lowest possible density.
More informationBiostatistics Department Technical Report
Biostatistics Departent Technical Report BST006-00 Estiation of Prevalence by Pool Screening With Equal Sized Pools and a egative Binoial Sapling Model Charles R. Katholi, Ph.D. Eeritus Professor Departent
More informationMULTIPLAYER ROCK-PAPER-SCISSORS
MULTIPLAYER ROCK-PAPER-SCISSORS CHARLOTTE ATEN Contents 1. Introduction 1 2. RPS Magas 3 3. Ites as a Function of Players and Vice Versa 5 4. Algebraic Properties of RPS Magas 6 References 6 1. Introduction
More informationδ 12. We find a highly accurate analytic description of the functions δ 11 ( δ 0, n)
Coplete-return spectru for a generalied Rosen-Zener two-state ter-crossing odel T.A. Shahverdyan, D.S. Mogilevtsev, V.M. Red kov, and A.M Ishkhanyan 3 Moscow Institute of Physics and Technology, 47 Dolgoprudni,
More information5.7 Chebyshev Multi-section Matching Transformer
3/8/6 5_7 Chebyshev Multisection Matching Transforers / 5.7 Chebyshev Multi-section Matching Transforer Reading Assignent: pp. 5-55 We can also build a ultisection atching network such that Γ f is a Chebyshev
More information