Wavepacket evolution in an unbound system
|
|
- Katherine Cain
- 5 years ago
- Views:
Transcription
1 J. Phys. B: At. Mol. Opt. Phys. 31 (1998) L891 L897. Printed in the UK PII: S (98)9711- LETTER TO THE EDITOR Wavepacket evolution in an unbound system Xiangyang Wang and Dan Dill Department of Chemistry, Boston University, 59 Commonwealth Avenue, Boston, MA 2215, USA Received 26 August 1998 Abstract. We report a method to efficiently describe wavepacket evolution in an unbound system for arbitrary large times. The problem is difficult using current methods due to the spatial expansion of wavepacket with time. Our approach circumvents this difficulty by solving the time development operator in a small interaction region outside of which the wavepacket evolution is described analytically. The formulation is illustrated by comparing numerical results of wavepacket propagation in a Morse potential with results using the pseudo-spectral Chebyshev expansion method. The evolution of wavepacket states has been studied extensively both theoretically and experimentally (e.g. Mallalieu and Stroud 1995, Cao and Wilson 1997, Bardeen et al 1995, Stapelfeldt et al 1995). It not only provides a uniform model for time-dependent processes such as photon atom interaction (e.g. Buchleitner and Delande 1995, Burke and Burke 1997), but also probes fundamental principles of the quantum world through wavepacket interference and entanglement (e.g. Monroe et al 1996, Noel and Stroud 1995). For a system involving unbound states, such as in electron molecule scattering and molecular photoionization, the experiment occurs in a rather large volume relative to the size of the molecules involved and so the particle can propagate and dissipate to virtually infinite distance during a long time period. To predict the time-dependent result of such experiments, some methods use a bound-system approximation by building a box which is large enough for all physics of interest to be confined within the box, during the interaction time (Zhang and Lambropoulos 1996), while other methods introduce an imaginary absorbing potential (e.g. Grozdanov and McCarrol 1996). Here we instead partition the molecular space into two regions: an interior interaction region (i) and a decomposition region (d), as in the time-independent R-matrix theory (e.g. Aymar et al 1996) and multiple-scattering theory (Dill and Dehmer 1974). The two regions are separated by a spherical surface of radius R. The potential is classified into two parts, a short-range potential V i and a long-range potential V. The radius R is chosen to make V i zero in the decomposition region. The long-range potential V is usually zero for short-range scattering problems and is a Coulomb potential for photoionization problems. The time-development operator U(t,) describes the evolution of the wavepacket ψ(t) = U(t,)ψ() (1) and it satisfies the time-dependent Schrödinger equation U(t,) = H U(t, ) (2) t /98/ $19.5 c 1998 IOP Publishing Ltd L891
2 L892 where H is the Hamiltonian of the system H = H + V i (3) H = T + V. (4) First we find the solution U (t, ) for a special case V i =, t U (t, ) = H U (t, ). (5) For some analytical potentials V, this solution is available in path-integral form. Next we write U(t,) = U (t, )U I (t) (6) and substitute this expression into equation (2) to obtain t U I(t) = U 1 (t, )V i U(t,). (7) Integration of this equation yields U I (t) = U 1 (s, )V i U(s,)ds. (8) Using this result in equation (6) we obtain an integral equation for U(t,), U(t,) = U (t, ) + 1 U (t,s)v i U(s,)ds (9) where we have used the relation U (t, )U 1 (s, ) = U (t, s). (1) Our goal is to exploit the fact that V i vanishes outside the interaction region. First we rewrite equation (9) in matrix form by partitioning the basis set spanning the whole space into two subsets x x = x i x i + x d x d =1 (11) x i x d x where x i denotes a point in the interaction region and x d denotes a point in the decomposition region. With this partitioning we get { } U U(t,) = ii (t, ) U id (t, ) U di (t, ) U dd (12) (t, ) { U ii id } (t, s) U (t, s) U (t, s) = U di (13) dd (t, s) U (t, s) { } V ii V i = i. (14) Substituting these expressions in equation (9), we obtain four integral equations. Because we have defined the interaction potential so that it vanishes outside the interaction region, the first of these equations, U ii (t, ) = U ii 1 (t, ) + U ii (t, s)v ii i U ii (s, ) ds (15)
3 L893 has the same form as the general equation (9), but depends only on interaction-region quantities. This is the key feature of our approach, for it allows us to work in a limited region of space. The second equation, U di (t, ) = U di 1 (t, ) + U di (t,s)vii i U ii (s, ) ds (16) determines the escape from the interaction region into the decomposition region. We can solve this equation once we know the solution to equation (15). In the example described below, the wavepacket is initially localized in the interaction region and so these two equations are sufficient to describe its evolution. The remaining two equations, U id (t, ) = U id 1 (t, ) + U dd (t, ) = U dd 1 (t, ) + U ii (t,s)vii i U id (s, ) ds (17) U di (t,s)vii i U id (s, ) ds (18) describe propagation from the decomposition region into the interaction region. These equations thus pertain to scattering processes, and we will discuss their application elsewhere. In the following discussion we will first discuss the solution of equation (15). We will drop all the superscripts and assume it is understood that all the operators below are actually their respective projected operators in the interaction region. Generally speaking it is difficult to solve integral equations such as equation (15) either directly or using recursive methods. In the case that both the long-range potential V and short-range potential V i are time independent, we have, U (t, s) = U (t s, ) (19) U(t,) = U (t, ) + 1 U (t s, )V i U(s,)ds (2) this equation can be simplified by applying the Laplace transform technique and the Faltung theorem f(s)= exp( st) f (t) dt (21) { } L f(t s)g(s)ds = f g (22) where f denotes the Laplace transform of function f. Applying the Laplace transform to both sides of equation (15), we have Ũ(s) = U (s) + 1 U (s) V i Ũ(s) (23) { Ũ(s) = U 1 (s) V i } 1. (24) The Laplace transform of U(t,) can be found easily from this equation. Consequently, U(t,) can be calculated from Ũ(s) using an inverse Laplace transform U(t,) = L 1{ Ũ(s) }. (25) As an example, we propagate a wavepacket state in a Morse potential (all parameters are listed in table 1). The initial state is chosen to be the ground state (E = au) with a group velocity k = 6. au, ψ(x, t = ) = (x) exp(ikx), (26)
4 L894 Table 1. Parameters for Morse potential, calculation using our theory and Chebyshev expansion method (atomic unit). Morse potential This work Chebyshev expansion method α = 1. R = 15. R = 15. D = 3. basis = 75 points = 248 r e = 2. truncation error = 1 12 truncation error = 1 12 Figure 1. Morse potential and initial wavepacket. The energy of the wavepacket is au. The arrow indicates the initial momentum direction. the minus sign indicates that the wavepacket initially moves toward the negative x-direction (figure 1). Such a state has the same density distribution as the ground state and its energy is { p 2 } E = (x) exp( ikx) 2m + V(x) (x) exp(ikx) dx = E + k 2 /2 p k = au. (27)
5 L895 We choose the radius R of the interaction region to be 15 au, the long-range potential to be { x> V (x) = (28) x and V i (x) to be the Morse potential. The probability for a wavepacket with an energy of au to penetrate the half-space x < in this Morse potential is negligible. The time-development operator for V can be found in path-integral form. In the coordinate representation it corresponds to a free-particle propagator with a topological constraint (e.g. Kleinert 199), x 2 U (t, ) x 1 =g (x 2,t;x 1,) = 1 sin(kx 1 ) sin(kx 2 ) exp( ik 2 t/2)dk. (29) 2π Using a function basis is more stable numerically than using the coordinate representation and so we choose the basis functions for the interaction region to be φ k (x) = 2/R sin {( k 1 2) πx/r }. (3) The matrix elements in this basis set are U mn (t, ) = Ũ mn (s) = R For the basis defined in equation (3), we find for n m R φ m (x 2 ) dx 2 φ n (x 1 )g (x 2,t;x 1,)dx 1 (31) U mn (t, ) exp( st)dt. (32) k m k n φ m (R )φ n (R ) Ũ mn (s) = (i + 1) s(km 2 + sr e2 (i 1) } (33) 2is)(k2 n 2is){1 and for n = m where km 2 Ũ mm (s) = φ2 m (R ) (i + 1) {1 + sr s(km 2 e2 (i 1) }+ R2 φ4 m (R ) 2is)2 2i(km 2 2is) (34) k m = ( m 1 2) π/r. (35) The inverse Laplace transform is carried out using Durbin s formula and the ɛ-algorithm is used to accelerate the convergence (Piessens and Huysmans 1984). As a comparison, we also solved the same problem using the pseudo-spectral Chebyshev expansion method (Tal-Ezer and Kosloff 1984, Kosloff 1992). We choose the box length R to be 15 au to cover all the space the particle can reach in a time period of 25 au, and 248 discrete points to acquire a reasonable precision. We analyse the results in terms of the probability of the particle either remaining in the ground state or escaping into the decomposition region. We compute the probability of remaining in the ground state as R 2 P (t) = (x) ψ(x, t) dx. (36)
6 L896 Figure 2. Absolute value of the wavefunction at t = 25 au for the method of this paper (full curve) and the Chebyshev expansion method (dotted curve). Since the ground state is an eigenstate of H, this probability does not change with time, and we use this property as a test of the numerical precision of our calculation. The probability of escape into the decomposition region is P d (t) = 1 R ψ(x, t) 2 dx. (37) The probability increases with time and describes the time evolution of the escape process. The numerical results clearly indicate that the two independent methods match each other with good precision. It should be noted that for the current problem, our method takes only one third of the CPU time of the Chebyshev expansion method. In theory, the computational effort required for our method is roughly proportional to the evolution time while that of the Chebyshev method is proportional to the evolution time squared. In figure 2 we present the graph of the wavepacket at time 25 au. The results from these two methods are almost identical. The small difference around R = 15. auisdue to the boundary condition of our basis set. In summary, we have shown that the time-development operator for an unbound system can be solved in a small interaction region. We express the operator in terms of the shortrange potential and the propagator of the long-range potential. The formulation is valid
7 L897 Table 2. Probability of particle in decomposition region (P d ) and particle in the ground state (P ). This work Chebyshev Time (au) P P d P P d for both bound and unbound systems. When the potential is time independent, our theory provides an efficient numerical method to describe wavepacket evolution in an unbound potential. We are grateful to the Center of Scientific Computing and Visualization at Boston University where the calculations of this work were carried out. References Aymar M, Greene C H and Luc-Koenig E 1996 Rev. Mod. Phys Bardeen C J, Wang Q and Shank C V 1995 Phys. Rev. Lett Buchleitner A and Delande D 1995 Phys. Rev. Lett Burke P G and Burke V M 1997 J. Phys. B: At. Mol. Opt. Phys. 3 L383 Cao J and Wilson K 1997 J. Chem. Phys Dill D and Dehmer J L 1974 J. Chem. Phys Grozdanov T P and McCarrol R 1996 J. Phys. B: At. Mol. Opt. Phys Kleinert H 199 Path Integral in Quantum Mechanics, Statistics and Polymer Physics (Singapore: World Scientific) p 212 Kosloff R 1992 Time-Dependent Quantum Molecular Dynamics (New York: Plenum) p 97 Mallalieu M and Stroud CRJr1995 Phys. Rev. A Monroe C, Meekhof D M, King B E and Wineland D J 1996 Science Noel M W and Stroud CRJr1995 Phys. Rev. Lett Piessens R and Huysmans R 1984 ACM Trans. Math. Software Stapelfeldt H, Constant E and Corkum P B 1995 Phys. Rev. Lett Tal-Ezer H and Kosloff R 1984 J. Chem. Phys Zhang J and Lambropoulos P 1996 Phys. Rev. Lett
Dominance of short-range correlations in photoejection-induced excitation processes
J. Phys. B: At. Mol. Opt. Phys. 30 (1997) L641 L647. Printed in the UK PII: S0953-4075(97)86594-7 LETTER TO THE EDITOR Dominance of short-range correlations in photoejection-induced excitation processes
More information4. Supplementary Notes on Time and Space Evolution of a Neutrino Beam
Lecture Notes for Quantum Physics II & III 8.05 & 8.059 Academic Year 1996/1997 4. Supplementary Notes on Time and Space Evolution of a Neutrino Beam c D. Stelitano 1996 As an example of a two-state system
More informationComplex WKB analysis of energy-level degeneracies of non-hermitian Hamiltonians
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 4 (001 L1 L6 www.iop.org/journals/ja PII: S005-4470(01077-7 LETTER TO THE EDITOR Complex WKB analysis
More informationAbsolute differential cross sections for electron elastic scattering and vibrational excitation in nitrogen in the angular range from 120 to 180
J. Phys. B: At. Mol. Opt. Phys. 33 (2000) L527 L532. Printed in the UK PII: S0953-4075(00)50902-X LETTER TO THE EDITOR Absolute differential cross sections for electron elastic scattering and vibrational
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More informationThe Particle in a Box
Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:
More informationThe infinite square well in a reformulation of quantum mechanics without potential function
The infinite square well in a reformulation of quantum mechanics without potential function A.D. Alhaidari (a), T.J. Taiwo (b) (a) Saudi Center for Theoretical Physics, P.O Box 32741, Jeddah 21438, Saudi
More informationUNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS. BSc and MPhys Undergraduate Programmes in Physics LEVEL HE2
Phys/Level /1/9/Semester, 009-10 (1 handout) UNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS BSc and MPhys Undergraduate Programmes in Physics LEVEL HE PAPER 1 MATHEMATICAL,
More informationNotes on wavefunctions IV: the Schrödinger equation in a potential and energy eigenstates.
Notes on wavefunctions IV: the Schrödinger equation in a potential and energy eigenstates. We have now seen that the wavefunction for a free electron changes with time according to the Schrödinger Equation
More informationTwo-pulse alignment of molecules
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 37 (2004) L43 L48 PII: S0953-4075(04)70990-6 LETTER TO THE EDITOR Two-pulse alignment
More informationStrongly correlated systems in atomic and condensed matter physics. Lecture notes for Physics 284 by Eugene Demler Harvard University
Strongly correlated systems in atomic and condensed matter physics Lecture notes for Physics 284 by Eugene Demler Harvard University September 18, 2014 2 Chapter 5 Atoms in optical lattices Optical lattices
More informationJ10M.1 - Rod on a Rail (M93M.2)
Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight
More informationAppendix B: The Transfer Matrix Method
Y D Chong (218) PH441: Quantum Mechanics III Appendix B: The Transfer Matrix Method The transfer matrix method is a numerical method for solving the 1D Schrödinger equation, and other similar equations
More informationMathematical Tripos Part IB Michaelmas Term Example Sheet 1. Values of some physical constants are given on the supplementary sheet
Mathematical Tripos Part IB Michaelmas Term 2015 Quantum Mechanics Dr. J.M. Evans Example Sheet 1 Values of some physical constants are given on the supplementary sheet 1. Whenasampleofpotassiumisilluminatedwithlightofwavelength3
More informationModels for Time-Dependent Phenomena
Models for Time-Dependent Phenomena I. Phenomena in laser-matter interaction: atoms II. Phenomena in laser-matter interaction: molecules III. Model systems and TDDFT Manfred Lein p. Outline Phenomena in
More informationHighly accurate evaluation of the few-body auxiliary functions and four-body integrals
INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICSB: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 1857 1867 PII: S0953-4075(03)58918-0 Highly accurate evaluation of the
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent
More informationPhysics 606, Quantum Mechanics, Final Exam NAME ( ) ( ) + V ( x). ( ) and p( t) be the corresponding operators in ( ) and x( t) : ( ) / dt =...
Physics 606, Quantum Mechanics, Final Exam NAME Please show all your work. (You are graded on your work, with partial credit where it is deserved.) All problems are, of course, nonrelativistic. 1. Consider
More informationDark pulses for resonant two-photon transitions
PHYSICAL REVIEW A 74, 023408 2006 Dark pulses for resonant two-photon transitions P. Panek and A. Becker Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187 Dresden,
More informationModels for Time-Dependent Phenomena. I. Laser-matter interaction: atoms II. Laser-matter interaction: molecules III. Model systems and TDDFT
Models for Time-Dependent Phenomena I. Laser-matter interaction: atoms II. Laser-matter interaction: molecules III. Model systems and TDDFT Manfred Lein, TDDFT school Benasque 22 p. Outline Laser-matter
More information1 Infinite-Dimensional Vector Spaces
Theoretical Physics Notes 4: Linear Operators In this installment of the notes, we move from linear operators in a finitedimensional vector space (which can be represented as matrices) to linear operators
More informationEmpirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 38 (2005) 2593 2600 doi:10.1088/0953-4075/38/15/001 Empirical formula for static
More information2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.
Section 5.1 Simple One-Dimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and
More informationTransmission across potential wells and barriers
3 Transmission across potential wells and barriers The physics of transmission and tunneling of waves and particles across different media has wide applications. In geometrical optics, certain phenomenon
More informationSeparation of Variables in Linear PDE: One-Dimensional Problems
Separation of Variables in Linear PDE: One-Dimensional Problems Now we apply the theory of Hilbert spaces to linear differential equations with partial derivatives (PDE). We start with a particular example,
More informationCALCULATING TRANSITION AMPLITUDES FROM FEYNMAN DIAGRAMS
CALCULATING TRANSITION AMPLITUDES FROM FEYNMAN DIAGRAMS LOGAN T. MEREDITH 1. Introduction When one thinks of quantum field theory, one s mind is undoubtedly drawn to Feynman diagrams. The naïve view these
More informationQuantum Final Project by Anustup Poddar and Cody Tripp 12/10/2013
Quantum Final Project by Anustup Poddar and Cody Tripp 12102013 Introduction The Hamiltonian in the Schrӧdinger equation is the sum of a kinetic and potential energy operator. The Fourier grid Hamiltonian
More informationDavid J. Starling Penn State Hazleton PHYS 214
Not all chemicals are bad. Without chemicals such as hydrogen and oxygen, for example, there would be no way to make water, a vital ingredient in beer. -Dave Barry David J. Starling Penn State Hazleton
More informationPage 684. Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02
Page 684 Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02 Time Transformations Section 12.5 Symmetries: Time Transformations Page 685 Time Translation
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationGeneralized PT symmetry and real spectra
INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICSA: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 35 (2002) L467 L471 PII: S0305-4470(02)36702-7 LETTER TO THE EDITOR Generalized PT symmetry and real spectra
More informationIntroduction to particle physics Lecture 2
Introduction to particle physics Lecture 2 Frank Krauss IPPP Durham U Durham, Epiphany term 2009 Outline 1 Quantum field theory Relativistic quantum mechanics Merging special relativity and quantum mechanics
More informationGeneration of Glauber Coherent State Superpositions via Unitary Transformations
Proceedings of Institute of Mathematics of NAS of Ukraine 004, Vol. 50, Part, 881 885 Generation of Glauber Coherent State Superpositions via Unitary Transformations Antonino MESSINA, Benedetto MILITELLO
More informationEnergy spectrum for a short-range 1/r singular potential with a nonorbital barrier using the asymptotic iteration method
Energy spectrum for a short-range 1/r singular potential with a nonorbital barrier using the asymptotic iteration method A. J. Sous 1 and A. D. Alhaidari 1 Al-Quds Open University, Tulkarm, Palestine Saudi
More informationNon-relativistic scattering
Non-relativistic scattering Contents Scattering theory 2. Scattering amplitudes......................... 3.2 The Born approximation........................ 5 2 Virtual Particles 5 3 The Yukawa Potential
More informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More informationChapter 1 Recollections from Elementary Quantum Physics
Chapter 1 Recollections from Elementary Quantum Physics Abstract We recall the prerequisites that we assume the reader to be familiar with, namely the Schrödinger equation in its time dependent and time
More informationQuantum Mechanics. p " The Uncertainty Principle places fundamental limits on our measurements :
Student Selected Module 2005/2006 (SSM-0032) 17 th November 2005 Quantum Mechanics Outline : Review of Previous Lecture. Single Particle Wavefunctions. Time-Independent Schrödinger equation. Particle in
More informationPHYS-454 The position and momentum representations
PHYS-454 The position and momentum representations 1 Τhe continuous spectrum-a n So far we have seen problems where the involved operators have a discrete spectrum of eigenfunctions and eigenvalues.! n
More informationXI. INTRODUCTION TO QUANTUM MECHANICS. C. Cohen-Tannoudji et al., Quantum Mechanics I, Wiley. Outline: Electromagnetic waves and photons
XI. INTRODUCTION TO QUANTUM MECHANICS C. Cohen-Tannoudji et al., Quantum Mechanics I, Wiley. Outline: Electromagnetic waves and photons Material particles and matter waves Quantum description of a particle:
More informationAnalogous comments can be made for the regions where E < V, wherein the solution to the Schrödinger equation for constant V is
8. WKB Approximation The WKB approximation, named after Wentzel, Kramers, and Brillouin, is a method for obtaining an approximate solution to a time-independent one-dimensional differential equation, in
More informationNotes on Quantum Mechanics
Notes on Quantum Mechanics Kevin S. Huang Contents 1 The Wave Function 1 1.1 The Schrodinger Equation............................ 1 1. Probability.................................... 1.3 Normalization...................................
More informationThe rotating Morse potential energy eigenvalues solved by using the analytical transfer matrix method
Chin. Phys. B Vol. 21, No. 1 212 133 The rotating Morse potential energy eigenvalues solved by using the analytical transfer matrix method He Ying 何英, Tao Qiu-Gong 陶求功, and Yang Yan-Fang 杨艳芳 Department
More informationarxiv: v1 [math.cv] 18 Aug 2015
arxiv:508.04376v [math.cv] 8 Aug 205 Saddle-point integration of C bump functions Steven G. Johnson, MIT Applied Mathematics Created November 23, 2006; updated August 9, 205 Abstract This technical note
More informationProblems and Multiple Choice Questions
Problems and Multiple Choice Questions 1. A momentum operator in one dimension is 2. A position operator in 3 dimensions is 3. A kinetic energy operator in 1 dimension is 4. If two operator commute, a)
More informationLecture 4. 1 de Broglie wavelength and Galilean transformations 1. 2 Phase and Group Velocities 4. 3 Choosing the wavefunction for a free particle 6
Lecture 4 B. Zwiebach February 18, 2016 Contents 1 de Broglie wavelength and Galilean transformations 1 2 Phase and Group Velocities 4 3 Choosing the wavefunction for a free particle 6 1 de Broglie wavelength
More informationA few principles of classical and quantum mechanics
A few principles of classical and quantum mechanics The classical approach: In classical mechanics, we usually (but not exclusively) solve Newton s nd law of motion relating the acceleration a of the system
More informationarxiv:quant-ph/ v2 20 Nov 1999
A General Type of a Coherent State with Thermal Effects Wen-Fa Lu Department of Applied Physics, Shanghai Jiao Tong University, Shanghai 200030, China (August 3, 208) arxiv:quant-ph/9903084v2 20 Nov 999
More information(e, 2e) spectroscopy of atomic clusters
J. Phys. B: At. Mol. Opt. Phys. 30 (1997) L703 L708. Printed in the UK PII: S0953-4075(97)86235-9 LETTER TO THE EDITOR (e, 2e) spectroscopy of atomic clusters S Keller, E Engel, H Ast and R M Dreizler
More informationNew Shape Invariant Potentials in Supersymmetric. Quantum Mechanics. Avinash Khare and Uday P. Sukhatme. Institute of Physics, Sachivalaya Marg,
New Shape Invariant Potentials in Supersymmetric Quantum Mechanics Avinash Khare and Uday P. Sukhatme Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India Abstract: Quantum mechanical potentials
More informationTransient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation
Symmetry, Integrability and Geometry: Methods and Applications Vol. (5), Paper 3, 9 pages Transient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation Marcos MOSHINSKY and Emerson SADURNÍ
More informationNon-sequential and sequential double ionization of NO in an intense femtosecond Ti:sapphire laser pulse
J. Phys. B: At. Mol. Opt. Phys. 30 (1997) L245 L250. Printed in the UK PII: S0953-4075(97)80013-2 LETTER TO THE EDITOR Non-sequential and sequential double ionization of NO in an intense femtosecond Ti:sapphire
More informationAbnormal pulse duration dependence of the ionization probability of Na atoms in intense laser fields
INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICSB: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 1121 1127 PII: S0953-4075(03)53141-8 Abnormal pulse duration dependence
More informationA. F. J. Levi 1 EE539: Engineering Quantum Mechanics. Fall 2017.
A. F. J. Levi 1 Engineering Quantum Mechanics. Fall 2017. TTh 9.00 a.m. 10.50 a.m., VHE 210. Web site: http://alevi.usc.edu Web site: http://classes.usc.edu/term-20173/classes/ee EE539: Abstract and Prerequisites
More information221A Lecture Notes Steepest Descent Method
Gamma Function A Lecture Notes Steepest Descent Method The best way to introduce the steepest descent method is to see an example. The Stirling s formula for the behavior of the factorial n! for large
More informationPhysics 221A Fall 2017 Notes 27 The Variational Method
Copyright c 2018 by Robert G. Littlejohn Physics 221A Fall 2017 Notes 27 The Variational Method 1. Introduction Very few realistic problems in quantum mechanics are exactly solvable, so approximation methods
More informationQuantum Mechanics for Scientists and Engineers
Quantum Mechanics for Scientists and Engineers Syllabus and Textbook references All the main lessons (e.g., 1.1) and units (e.g., 1.1.1) for this class are listed below. Mostly, there are three lessons
More informationPhysics 217 Problem Set 1 Due: Friday, Aug 29th, 2008
Problem Set 1 Due: Friday, Aug 29th, 2008 Course page: http://www.physics.wustl.edu/~alford/p217/ Review of complex numbers. See appendix K of the textbook. 1. Consider complex numbers z = 1.5 + 0.5i and
More informationNon-relativistic Quantum Electrodynamics
Rigorous Aspects of Relaxation to the Ground State Institut für Analysis, Dynamik und Modellierung October 25, 2010 Overview 1 Definition of the model Second quantization Non-relativistic QED 2 Existence
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m.
PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m.- 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms
More informationTheoretical Photochemistry WiSe 2016/17
Theoretical Photochemistry WiSe 2016/17 Lecture 8 Irene Burghardt burghardt@chemie.uni-frankfurt.de) http://www.theochem.uni-frankfurt.de/teaching/ Theoretical Photochemistry 1 Topics 1. Photophysical
More informationNear horizon geometry, Brick wall model and the Entropy of a scalar field in the Reissner-Nordstrom black hole backgrounds
Near horizon geometry, Brick wall model and the Entropy of a scalar field in the Reissner-Nordstrom black hole backgrounds Kaushik Ghosh 1 Department of Physics, St. Xavier s College, 30, Mother Teresa
More informationLecture 7. 1 Wavepackets and Uncertainty 1. 2 Wavepacket Shape Changes 4. 3 Time evolution of a free wave packet 6. 1 Φ(k)e ikx dk. (1.
Lecture 7 B. Zwiebach February 8, 06 Contents Wavepackets and Uncertainty Wavepacket Shape Changes 4 3 Time evolution of a free wave packet 6 Wavepackets and Uncertainty A wavepacket is a superposition
More informationNonstandard Finite Difference Time Domain (NSFDTD) Method for Solving the Schrödinger Equation
Pramana J. Phys. (2018)?: #### DOI??/?? c Indian Academy of Sciences Nonstandard Finite Difference Time Domain (NSFDTD) Method for Solving the Schrödinger Equation I WAYAN SUDIARTA mitru et al. [11] have
More informationSolving the Schrödinger equation for the Sherrington Kirkpatrick model in a transverse field
J. Phys. A: Math. Gen. 30 (1997) L41 L47. Printed in the UK PII: S0305-4470(97)79383-1 LETTER TO THE EDITOR Solving the Schrödinger equation for the Sherrington Kirkpatrick model in a transverse field
More informationarxiv: v1 [cond-mat.mes-hall] 24 May 2013
arxiv:35.56v [cond-mat.mes-hall] 4 May 3 Effects of excitation frequency on high-order terahertz sideband generation in semiconductors Xiao-Tao Xie Department of Physics, The Chinese University of Hong
More informationChapter 18: Scattering in one dimension. 1 Scattering in One Dimension Time Delay An Example... 5
Chapter 18: Scattering in one dimension B. Zwiebach April 26, 2016 Contents 1 Scattering in One Dimension 1 1.1 Time Delay.......................................... 4 1.2 An Example..........................................
More informationPHYS. LETT. A 285, , 2001 THE INTERFERENCE TERM IN THE WIGNER DISTRIBUTION FUNCTION AND THE AHARONOV-BOHM EFFECT
1 PHYS. LETT. A 85, 19-114, 1 THE INTERFERENCE TERM IN THE WIGNER DISTRIBUTION FUNCTION AND THE AHARONOV-BOHM EFFECT D. Dragoman * Univ. Bucharest, Physics Dept., P.O. Box MG-11, 769 Bucharest, Romania
More informationElectronic correlation studied by neutron scattering
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 35 (2002) L31 L36 PII: S0953-4075(02)29617-0 LETTER TO THE EDITOR Electronic
More informationk m Figure 1: Long problem L2 2 + L2 3 I 1
LONG PROBLEMS 1: Consider the system shown in Figure 1: Two objects, of mass m 1 and m, can be treated as point-like. Each of them is suspended from the ceiling by a wire of negligible mass, and of length
More information221A Lecture Notes Convergence of Perturbation Theory
A Lecture Notes Convergence of Perturbation Theory Asymptotic Series An asymptotic series in a parameter ɛ of a function is given in a power series f(ɛ) = f n ɛ n () n=0 where the series actually does
More informationGRAPH QUANTUM MECHANICS
GRAPH QUANTUM MECHANICS PAVEL MNEV Abstract. We discuss the problem of counting paths going along the edges of a graph as a toy model for Feynman s path integral in quantum mechanics. Let Γ be a graph.
More informationPhysics 215 Quantum Mechanics 1 Assignment 5
Physics 15 Quantum Mechanics 1 Assignment 5 Logan A. Morrison February 10, 016 Problem 1 A particle of mass m is confined to a one-dimensional region 0 x a. At t 0 its normalized wave function is 8 πx
More informationYou may not start to read the questions printed on the subsequent pages of this question paper until instructed that you may do so by the Invigilator.
NTURL SCIENCES TRIPOS Part I Saturday 9 June 2007 1.30 pm to 4.30 pm PHYSICS nswer the whole of Section and four questions from Sections B, C and D, with at least one question from each of these Sections.
More informationQuantum Mechanics C (130C) Winter 2014 Final exam
University of California at San Diego Department of Physics Prof. John McGreevy Quantum Mechanics C (130C Winter 014 Final exam Please remember to put your name on your exam booklet. This is a closed-book
More informationNonlinear instability of half-solitons on star graphs
Nonlinear instability of half-solitons on star graphs Adilbek Kairzhan and Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada Workshop Nonlinear Partial Differential Equations on
More informationElectronic structure of solids
Electronic structure of solids Eigenvalue equation: Áf(x) = af(x) KNOWN: Á is an operator. UNKNOWNS: f(x) is a function (and a vector), an eigenfunction of Á; a is a number (scalar), the eigenvalue. Ackowledgement:
More informationAdiabatic Approximation
Adiabatic Approximation The reaction of a system to a time-dependent perturbation depends in detail on the time scale of the perturbation. Consider, for example, an ideal pendulum, with no friction or
More informationarxiv:physics/ v1 [physics.comp-ph] 26 Oct 2001
The wave packet propagation using wavelets Andrei G. BORISOV a and Sergei V. SHABANOV b arxiv:physics/0110077v1 [physics.comp-ph] 26 Oct 2001 a Laboratoire des Collisions Atomiques et Moléculaires, Unité
More informationDepartment of Physics PRELIMINARY EXAMINATION 2014 Part II. Long Questions
Department of Physics PRELIMINARY EXAMINATION 2014 Part II. Long Questions Friday May 16th, 2014, 14-17h Examiners: Prof. A. Clerk, Prof. M. Dobbs, Prof. G. Gervais (Chair), Prof. T. Webb, Prof. P. Wiseman
More informationG : Quantum Mechanics II
G5.666: Quantum Mechanics II Notes for Lecture 5 I. REPRESENTING STATES IN THE FULL HILBERT SPACE Given a representation of the states that span the spin Hilbert space, we now need to consider the problem
More informationInterference, vector spaces
Interference, vector spaces Sourendu Gupta TIFR Graduate School Quantum Mechanics 1 August 6, 2008 Sourendu Gupta (TIFR Graduate School) Interference, vector spaces QM I 1 / 16 Outline 1 The double-slit
More informationAn Algebraic Approach to Reflectionless Potentials in One Dimension. Abstract
An Algebraic Approach to Reflectionless Potentials in One Dimension R.L. Jaffe Center for Theoretical Physics, 77 Massachusetts Ave., Cambridge, MA 02139-4307 (Dated: January 31, 2009) Abstract We develop
More informationLecture-05 Perturbation Theory and Feynman Diagrams
Lecture-5 Perturbation Theory and Feynman Diagrams U. Robkob, Physics-MUSC SCPY639/428 September 3, 218 From the previous lecture We end up at an expression of the 2-to-2 particle scattering S-matrix S
More informationFully differential cross sections for transfer ionization a sensitive probe of high level correlation effects in atoms
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 37 (2004) L201 L208 PII: S0953-4075(04)75475-9 LETTER TO THE EDITOR Fully differential
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More information( ) in the interaction picture arises only
Physics 606, Quantum Mechanics, Final Exam NAME 1 Atomic transitions due to time-dependent electric field Consider a hydrogen atom which is in its ground state for t < 0 For t > 0 it is subjected to a
More informationChapter 38 Quantum Mechanics
Chapter 38 Quantum Mechanics Units of Chapter 38 38-1 Quantum Mechanics A New Theory 37-2 The Wave Function and Its Interpretation; the Double-Slit Experiment 38-3 The Heisenberg Uncertainty Principle
More informationMinimum Uncertainty for Entangled States
Minimum Uncertainty for Entangled States Tabish Qureshi Centre for Theoretical Physics Jamia Millia Islamia New Delhi - 110025. www.ctp-jamia.res.in Collaborators: N.D. Hari Dass, Aditi Sheel Tabish Qureshi
More informationCalculations of the binding energies of weakly bound He He H, He He H and He H H molecules
J. Phys. B: At. Mol. Opt. Phys. 32 (1999) 4877 4883. Printed in the UK PII: S0953-4075(99)05621-7 Calculations of the binding energies of weakly bound He He H, He He H and He H H molecules Yong Li and
More informationLaser Control of Atom-Molecule Reaction: Application to Li +CH 4 Reaction
Adv. Studies Theor. Phys., Vol. 3, 2009, no. 11, 439-450 Laser Control of Atom-Molecule Reaction: Application to Li +CH 4 Reaction Hassan Talaat Physics Dept., Faculty of Science, Ain Shams University,
More informationAnalysis of recombination in high-order harmonic generation in molecules
Analysis of recombination in high-order harmonic generation in molecules B. Zimmermann, M. Lein,* and J. M. Rost Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden,
More informationTHE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 2912 PHYSICS 2B (ADVANCED) ALL QUESTIONS HAVE THE VALUE SHOWN
CC0936 THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 91 PHYSICS B (ADVANCED) SEMESTER, 014 TIME ALLOWED: 3 HOURS ALL QUESTIONS HAVE THE VALUE SHOWN INSTRUCTIONS: This paper consists
More informationProblem Set 6: Solutions Math 201A: Fall a n x n,
Problem Set 6: Solutions Math 201A: Fall 2016 Problem 1. Is (x n ) n=0 a Schauder basis of C([0, 1])? No. If f(x) = a n x n, n=0 where the series converges uniformly on [0, 1], then f has a power series
More informationWave Mechanics in One Dimension
Wave Mechanics in One Dimension Wave-Particle Duality The wave-like nature of light had been experimentally demonstrated by Thomas Young in 1820, by observing interference through both thin slit diffraction
More informationDepartment of Physics and Astronomy University of Georgia
Department of Physics and Astronomy University of Georgia August 2007 Written Comprehensive Exam Day 1 This is a closed-book, closed-note exam. You may use a calculator, but only for arithmetic functions
More informationModern physics. 4. Barriers and wells. Lectures in Physics, summer
Modern physics 4. Barriers and wells Lectures in Physics, summer 016 1 Outline 4.1. Particle motion in the presence of a potential barrier 4.. Wave functions in the presence of a potential barrier 4.3.
More informationCovariance of the Schrödinger equation under low velocity boosts.
Apeiron, Vol. 13, No. 2, April 2006 449 Covariance of the Schrödinger equation under low velocity boosts. A. B. van Oosten, Theor. Chem.& Mat. Sci. Centre, University of Groningen, Nijenborgh 4, Groningen
More informationCandidacy Exam Department of Physics February 6, 2010 Part I
Candidacy Exam Department of Physics February 6, 2010 Part I Instructions: ˆ The following problems are intended to probe your understanding of basic physical principles. When answering each question,
More informationQuantum Annealing and the Schrödinger-Langevin-Kostin equation
Quantum Annealing and the Schrödinger-Langevin-Kostin equation Diego de Falco Dario Tamascelli Dipartimento di Scienze dell Informazione Università degli Studi di Milano IQIS Camerino, October 28th 2008
More information