Method of Rotated Reference Frames
|
|
- Dale Riley
- 6 years ago
- Views:
Transcription
1 Method of Rotated Reference Frames Vadim Markel in collaboration with George Panasyuk Manabu Machida John Schotland Radiology/Bioengineering UPenn, Philadelphia
2 Outline
3 MOTIVATION (the inverse problems perspective)
4 ρ s S zs α(r) zd z Given a data function φ( ρ, ρ ) s d which is measured for multiple pairs ( ρ, ρ ), find the absorption s d D ρ d coefficient α( r) inside the slab 4
5 Linearized Integral Equation φ( ρ, ρ ) = Γ( ρ, ρ ; r) ( r)d 3 s d s d δα r φ ( ρ, ρ ) s d = I ( ρs, zs; ρd, zd) I0 ( ρs, zs; ρd, zd) I ( ρ, z ; ρ, z ) (measurable data-function) 0 α ( r) = α + δα( r) 0 s s d d Γ ( ρ, ρ ) = G ( ρ, z ; r) G ( r; ρ, z ) s d 0 s s 0 d d (first Born approximation)
6 How To Invert? Numerical SVD approach: φ = Γ δα i ij j j j φ i φ =Γδα Γ δα φ = Γ Γδα * * + * 1 * ( ) = ΓΓ Γ reg φ
7 Analytical SVD approach: Making use of the translational invariance φ ( q, q ) = φ ( ρ, ρ ) e d ρ d ρ i ( q s ρ s+ q d ρ d ) 2 2 s d s d s d qs = q /2 + p, qd = q /2 p; Data function: ψ ( q, p) = φ ( q / 2 + p, q / 2 p) L ψ ( q, p) = g ( q / 2 + p; z) g ( q / 2 p; z) δα ( q; z)dz 0 iρ q 2 δα ( q; ) = δα ( ρ, ) d ρ z dz 2 d q G ( ρ, z ; ρ, z) = g s z e ( 2π ) 0 s s 2 s ( 2π ) d ( q; z) e 2 d q G0 ( ρ, z; ρd, zd) = g ( ; ) 2 d q z e iq ( ρ ρ ) s iq ( ρ ρ) d 7
8 6cm S.D.Konecky, G.Y.Panasyuk, K.Lee, V.Markel, A.G.Yodh and J.C.Schotland Imaging complex structures with diffuse light Optics Express 16(7), (2008)
9 In the case of RTE, we need the plane-wave decomposition of the Greens function, of the form 2 d q G (,; rsr ˆ, sˆ ) = g(;,; qz sˆ z, sˆ ) e ( 2π ) 0 2 iq ( ρ ρ ) and the integral kernel 2 Γ ( q, p; z) = g( q/2 + p; z, zˆ ; z, sˆ ) g( q/2 p; z,; sˆ z, zˆ )ds We then get the integral equations of the form z d ψ( qp, ) = Γ( qp, ; z) δα ( q; z)dz z s s d
10 Motivation Continued (the spectral methods)
11 Solve system of equations : ( z+ W) x = b Find σ ( ) = Im = Im ( + ) 1 z bx b z W b for M different values of z, where z W = x+ iδ is a complex number is an N N matrix x, b are vectors of length N
12 The "naive" method: 1) Choose z 2) Make a matrix A= z+ W 3) Solve Ax = b 4) Goto step (1) Computational complexity: M N 3
13 The spectral method: 1) Diagonalize W (find eigenvectors n and eigenvalues w ) 2) For every z, x n 2 n n b cnδ = ; σ = 2 2 z+ w ( x+ w ) + δ n n n n Computational complexity: N + M N 3 2 However, if the whole vector x is not needed, the complexity may be as low as 3 N M N +
14 Spectral Method for the RTE? 2 sˆ μ ˆ ˆ ˆ ˆ ˆ ˆ t I r s μs A s s I r s s ε r s RTE: ( + )(,) = (, )(, )d + (,) Where is the "spectral variable"? How can we write this equation in the form ( z+ W) I = ε? μ and μ do not qualify... t s We can try to expand I( rs, ˆ) into a a 3D Fourier integral with respect to r and into the basis of ordinary spherical harmonics Y ( θϕ, ) with respect to sˆ... lm...and see if the equation can be cast into the desired form.
15 The Conventional Method of Spherical Harmonics This results in the following system of equations with respect to the vector of the expansion coefficients I( k) ( k - the Fourier variable) : ia k I( k) + ia k I( k) + ia k I( k) + D I( k) = ε ( k) ( x) ( y) ( z) x y z A, A, ( x) ( y) A ( z) are different matrices. ( x) * Alm, l m = sinθcos ϕylm( θϕ, ) Yl m ( θϕ, )sin θdθdϕ, etc.... This rather awe-inspiring set of equations has perhaps only academic interest. K.M.Case, P.F.Zweifel, Linear Transport Theory
16 Rotated Reference Frames The usual sperical harmonics are defined in the laboratory reference frame. Then θ and ϕ are the polar angles of the unit vector sˆ in that frame. THE MAIN IDEA: For each value of the Fourier variable k, use spherical harmonics defined in a reference frame whose z-axis is aligned with the direction of k. We call such frames "rotated". Spherical harmonics defined in the rotaded frame are denoted by Y ( sk ˆ; ˆ ).
17 Rotation of the Laboratory Frame (x,y,z). z y' z' l ˆ ˆ l Y (; ) (,,0) () ˆ lm sk = Dm m ϕk θk Ylm s m = l θ k y Wigner D-functions Euler angles x ϕ θ Spherical functions in the laboratory frame x'
18 ( sˆ + μ ) I = μ AI + ε t (,) ˆ (,) ˆ d ikr 3 I rs = I kse k ikr ε(,) rsˆ = ε( ks,) ˆ e d ( ik sˆ + μ ) I = μ AI + ε t I ( ks, ˆ) = F ( k) Y ( sk ˆ; ˆ ) lm, ε ( ks, ˆ) = E ( k) Y ( sk ˆ; ˆ ) lm, lm lm A(, ss) AY (; skˆ) Y (; skˆ) * ˆˆ = ˆ ˆ l lm lm lm, s s lm lm 3 k
19 lm ik R F ( k) + σ F ( k) = E ( k) lm lm l m l lm lm σ = μ + μ (1 A ) l a s l R = sˆ kˆ Y (; sk ˆ ˆ) Y (; sk ˆ ˆ)ds= l lm * 2 lm lm l m b( m) = [ b( m) b ( m) ] = δ δ + δ mm l l = l 1 l+ 1 l = l+ 1 l m l
20 RTE in the Angular Basis of Rotated Spherical Functions ikr I( k) + D I( k) = E( k) Scalar spectral parameter Block-tridiagonal real symmetric matrix Source term Diagonal matrix Slm, l m = δ ll δmm [ μa + μs(1 Al )] Parameters of the phase function. (For the HG l model, A = g ) l
21 Let D W = SS = S RS 1 1 ikr I( k) + D I( k) = E( k) ikw S I k S E 1 ( + 1) ( ) = ( ) I k S ikw S E ( ) = (1 + ) ( ) k k W ψ = λ ψ μ μ μ I ( k) = μ 1 1 S ψ ψ μ μ S E 1+ ikλ μ ( k)
22 The Spectral Solution ˆ ˆ (,) d σ 1+ ikλ 1 1 lm ψμ ψμ S E( k) Ylm( s; k) ikr 3 I rsˆ = e k lm ε(,) rsˆ = δ( r r) δ( sˆ sˆ ) ikr 0 * ˆ E ˆ lm( k) = e Y ( 3 lm s0; k) (2 π ) l μ μ The integral is not easy but doable. Details in J.Phys.A 39, 115 (2006)
23 ˆ m * ˆ ˆ ˆ ˆ = lm χll l m 0 m= l, l = m G (,; rsr, s) Y (; sr) ( R) Y ( s; Rˆ) m m ( 1) χll ( R) = ( 1) 2π σ σ l l l M n 3 M= l n, λ > 0 λn l φ ( M) φ ( M) l ( M ) l l l + 2 j,0 l l + 2 j,0 R ClMl,,, MClml,,, m kl l + 2 j j= 0 λn( M ) l = min( l, l ) μ = ( Mn, ) lm ψ ( ) Mn = δmm l φn M n n r ŝ ŝ 0 R r 0
24 Some Properties of the Eigenvalues
25 If λ < 1/ μ - continuous spectrum t If λ > 1/ μ - discrete spectrum t Bounds on the diffuse eigenvalue λ =max[ λ (0)] d D= cμλ 2 2 β1 + β2 λd β1 + β2 bl (0) 1 βl = ; β1 = ; σσ 3 μ [ μ + (1 A ) μ ] l l 1 a a 1 s β 2 = 2 15[ μ + (1 A) μ ][ μ + (1 A ) μ ] a 2 d a 1 s a 2 s n Bound on the gap: Δλ= λ λ d next largest Δλ max[ β + β β β,0]
26 Density, Current, and the Fick s Law 1 1 ( ) (, ˆ)d ; ( ) (, ˆ) ˆd c c 1 0 u( r) = 2l+ 1 χ ˆ ˆ 0l( r) Pl( s0 r) c 2 2 u r = I r s s J r = I r s s s l= 0 rˆ 0 Jr ( ) = 2l+ 1 χ ˆ ˆ 1l( r) Pl( s0 r) 3 l= 0 r λd If κ = 1, then Δ λλ J next largest = D u, where D= cμλ a 2 d
27 Infinite Space, Point Uni-Directional (Sharply-Peaked) Source (a) forward and backward propagation Angular dependence of the specific intensity for forward (a) and backward (b) μ μ -5 propagation obtained at lmax = 21, g = 0.98 and a/ s = The distance to the source z is assumed to be positive for forward propagation and negative for backward propagation.
28 Infinite Space, Point Uni-Directional (Sharply-Peaked) Source (b) off-axis propagation ŝ 0 z α ŝ Two cases: a) sˆ in the xy plane b) sˆ in the yz plane y ŝ β x
29 Angular distribution of specific intensity for off-axis propagation (relatively small absorption) -5 Parameters: g = 0.98 and μa/ μs = 6 10 (a), (b), = 0.03 (c), (d).
30 Angular distribution of specific intensity for off-axis propagation (relatively large absorption) Parameters: g = 0.98 and μ / μ = 0.2. a s
31 Plane-Wave Decomposition ˆ ˆ (,) d σ 1+ ikλ 1 1 lm ψμ ψμ S E( k) Ylm( s; k) ikr 3 I rsˆ = e k 2 dq ( 0 ) * ˆ ˆ iq ρ ρ lm ˆ ˆ ˆ lm lm 0 l m 0 lm, l m G (,; rsr, s) = e Y (;) szg (;, qz z ) Y ( s;) zˆ 2π ( ) dk g q z z ˆ ( ) ˆ 0 = e D q+ zk K q + k D q+ zk 2π z ikz ( z z0 ) 2 2 ( ;, ) ( ) ( ) Kw ( ) = lm μ S 1 l ψ μ ψ 1+ iwλ S μ z z z 1 μ μ iϕ J iθ J μ ; D ( kˆ ) = e cos θ = k / q + k k z k z e k y 2 2 z
32 i( m m ) ϕq l l lm e g ( q; z, z ) = sgn( z z ) d [ iτ ( qλ )] lm Q ( q) = q + 1/ λ μ 0 0 σσ l l 2 2 μ 2 cos[ τ( )] 1, sin[ τ( )] l+ l + m+ m l [ ] mm1 m = l m = l μλ, > Qμ ( q) z z0 e l 1ψ μ ψ 2 μ 2 mm τ λ 2 μ λμqμ( q) lm l m d [ i ( q )] i x = + x i x = ix α D ( αβγ,, ) = e d ( β) e l im l im γ mm mm μ μ
33 Plane waves: Evanescent Waves kr I = e F sˆ k k = k = knˆ k 2 k( ), 1/ λμ, nˆ - real unit vector F ( ˆ) = lm Y ( ˆ; ˆ k s ψ μ lm s k) lm Evanescent waves: k = q± zˆ + q zˆ = k k = i q 1/ λμ, 0, 1/ λμ iq k =± q + 1/ λ z z 2 2 n
34 kˆ r exp( imϕ k ) l Iˆ = exp Y ( ˆ; ˆ lm ) dmm ( θ ) l φn ( M) Mn sz k k λmn lm σ l (plane wave modes) exp( imϕ ) I = exp i Q ( q) z Y ( sz ˆ; ˆ) ( 1) d ( θ ) l φ ( M) [ q ρ ] ( ± ) q l+ m l qmn Mn lm m, M k n lm σ l (evanescent modes) G V 0 2 dq ( ) ( ) (,; ˆ 0, ˆ ˆ ˆ 0) 2 I ± rsr s = qmn(,) rs V Mn I q qmn( r0, s0) (2 π ) qmn = Q Mn 1 ( q) λ Mn 2 Mn
35 Half-space problem Z>0 Scattering medium Propagationd escribed by the RTE z<0 Nonscattering medium d 2 q I = A ( q) I (2 ) 2 Mn π Mn ( + ) qmn
36
37
38
39 CONCLUSIONS The method of rotated reference frames takes advantage of all symmetries of the RTE (symmetry with respect to rotations and inversions of the reference frame). The angular and spatial dependence of the obtained solutions is expressed in terms of analytical functions. The analytical part of the solution is of considerable mathematical complexity. This is traded for relative simplicity of the numerical part. We believe that we have reduced the numerical part of the computations to the absolute minimum allowed by the mathematical structure of the RTE.
40 Publications: 1. V.A.Markel, Modified spherical harmonics method for solving the radiative transport equation, Letter to the Editor, Waves in Random Media 14(1), L13-L19 (2004). 2. G.Y.Panasyuk, J.C.Schotland, and V.A.Markel, Radiative transport equation in rotated reference frames, Journal of Physics A, 39(1), (2006). 3. J.C.Schotland and V.A.Markel, Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation, Inverse Problems and Imaging 1(1), (2007). Available on the web at
Tomography of Highly Scattering Media with the Method of Rotated Reference Frames. Vadim Markel Manabu Machida George Panasyuk John Schotland
Tomography of Highly Scattering Media with the Method of Rotated Reference Frames Vadim Markel Manabu Machida George Panasyuk John Schotland Radiology/Bioengineering UPenn, Philadelphia MOTIVATION (The
More informationRadiative transport equation in rotated reference frames
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 39 (26) 115 137 doi:1.188/35-447/39/1/9 Radiative transport equation in rotated reference frames George
More informationGreen's functions for the two-dimensional radiative transfer equation in bounded media
Home Search Collections Journals About Contact us My IOPscience Green's functions for the two-dimensional radiative transfer equation in bounded media This article has been downloaded from IOPscience Please
More informationRepresentation theory and quantum mechanics tutorial Spin and the hydrogen atom
Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition
More informationSolutions to exam : 1FA352 Quantum Mechanics 10 hp 1
Solutions to exam 6--6: FA35 Quantum Mechanics hp Problem (4 p): (a) Define the concept of unitary operator and show that the operator e ipa/ is unitary (p is the momentum operator in one dimension) (b)
More informationSolution via Laplace transform and matrix exponential
EE263 Autumn 2015 S. Boyd and S. Lall Solution via Laplace transform and matrix exponential Laplace transform solving ẋ = Ax via Laplace transform state transition matrix matrix exponential qualitative
More informationQuantum Mechanics II Lecture 11 (www.sp.phy.cam.ac.uk/~dar11/pdf) David Ritchie
Quantum Mechanics II Lecture (www.sp.phy.cam.ac.u/~dar/pdf) David Ritchie Michaelmas. So far we have found solutions to Section 4:Transitions Ĥ ψ Eψ Solutions stationary states time dependence with time
More informationPreliminary Examination - Day 1 Thursday, August 10, 2017
UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, August, 7 This test covers the topics of Quantum Mechanics (Topic ) and Electrodynamics (Topic ). Each topic has 4 A questions
More information6. LIGHT SCATTERING 6.1 The first Born approximation
6. LIGHT SCATTERING 6.1 The first Born approximation In many situations, light interacts with inhomogeneous systems, in which case the generic light-matter interaction process is referred to as scattering
More informationFFTs in Graphics and Vision. Fast Alignment of Spherical Functions
FFTs in Graphics and Vision Fast Alignment of Spherical Functions Outline Math Review Fast Rotational Alignment Review Recall 1: We can represent any rotation R in terms of the triplet of Euler angles
More informationComputation of the scattering amplitude in the spheroidal coordinates
Computation of the scattering amplitude in the spheroidal coordinates Takuya MINE Kyoto Institute of Technology 12 October 2015 Lab Seminar at Kochi University of Technology Takuya MINE (KIT) Spheroidal
More informationFINITE SELF MASS OF THE ELECTRON II
FINITE SELF MASS OF THE ELECTRON II ABSTRACT The charge of the electron is postulated to be distributed or extended in space. The differential of the electron charge is set equal to a function of electron
More informationdoes not change the dynamics of the system, i.e. that it leaves the Schrödinger equation invariant,
FYST5 Quantum Mechanics II 9..212 1. intermediate eam (1. välikoe): 4 problems, 4 hours 1. As you remember, the Hamilton operator for a charged particle interacting with an electromagentic field can be
More informationAkira Ishimaru, Sermsak Jaruwatanadilok and Yasuo Kuga
INSTITUTE OF PHYSICS PUBLISHING Waves Random Media 4 (4) 499 5 WAVES IN RANDOMMEDIA PII: S959-774(4)789- Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including
More informationAppendix A Spin-Weighted Spherical Harmonic Function
Appendix A Spin-Weighted Spherical Harmonic Function Here, we review the properties of the spin-weighted spherical harmonic function. In the past, this was mainly applied to the analysis of the gravitational
More informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler April, 20 Lecture 2. Scattering Theory Reviewed Remember what we have covered so far for scattering theory. We separated the Hamiltonian similar to the way in
More informationNotes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n.
Chapter. Electrostatic II Notes: Most of the material presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartolo, Chap... Mathematical Considerations.. The Fourier series and the Fourier
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 Radiative Transfer Equation
The Radiative Transfer Equation R. Wordsworth April 11, 215 1 Objectives Derive the general RTE equation Derive the atmospheric 1D horizontally homogenous RTE equation Look at heating/cooling rates in
More informationI. Rayleigh Scattering. EE Lecture 4. II. Dipole interpretation
I. Rayleigh Scattering 1. Rayleigh scattering 2. Dipole interpretation 3. Cross sections 4. Other approximations EE 816 - Lecture 4 Rayleigh scattering is an approximation used to predict scattering from
More informationSurface Sensitive X-ray Scattering
Surface Sensitive X-ray Scattering Introduction Concepts of surfaces Scattering (Born approximation) Crystal Truncation Rods The basic idea How to calculate Examples Reflectivity In Born approximation
More information3. Calculating Electrostatic Potential
3. Calculating Electrostatic Potential 3. Laplace s Equation 3. The Method of Images 3.3 Separation of Variables 3.4 Multipole Expansion 3.. Introduction The primary task of electrostatics is to study
More informationLecture 4. Diffusing photons and superradiance in cold gases
Lecture 4 Diffusing photons and superradiance in cold gases Model of disorder-elastic mean free path and group velocity. Dicke states- Super- and sub-radiance. Scattering properties of Dicke states. Multiple
More information( ) ( ) QM A1. The operator ˆR is defined by R ˆ ψ( x) = Re[ ψ( x)] ). Is ˆR a linear operator? Explain. (it returns the real part of ψ ( x) SOLUTION
QM A The operator ˆR is defined by R ˆ ψ( x) = Re[ ψ( x)] (it returns the real part of ψ ( x) ). Is ˆR a linear operator? Explain. SOLUTION ˆR is not linear. It s easy to find a counterexample against
More informationFORMULA SHEET FOR QUIZ 2 Exam Date: November 8, 2017
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II November 5, 207 Prof. Alan Guth FORMULA SHEET FOR QUIZ 2 Exam Date: November 8, 207 A few items below are marked
More informationSpherical Harmonics on S 2
7 August 00 1 Spherical Harmonics on 1 The Laplace-Beltrami Operator In what follows, we describe points on using the parametrization x = cos ϕ sin θ, y = sin ϕ sin θ, z = cos θ, where θ is the colatitude
More informationNotes 19 Gradient and Laplacian
ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can
More informationPHYS 705: Classical Mechanics. Rigid Body Motion Introduction + Math Review
1 PHYS 705: Classical Mechanics Rigid Body Motion Introduction + Math Review 2 How to describe a rigid body? Rigid Body - a system of point particles fixed in space i r ij j subject to a holonomic constraint:
More informationEffects of Entanglement during Inflation on Cosmological Observables
Effects of Entanglement during Inflation on Cosmological Observables Nadia Bolis 1 Andreas Albrecht 1 Rich Holman 2 1 University of California Davis 2 Carnegie Mellon University September 5, 2015 Inflation
More informationTight-Binding Model of Electronic Structures
Tight-Binding Model of Electronic Structures Consider a collection of N atoms. The electronic structure of this system refers to its electronic wave function and the description of how it is related to
More informationOptimized diffusion approximation
356 Vol. 35, No. 2 / February 208 / Journal of the Optical Society of America A Research Article Optimized diffusion approximation UGO TRICOLI, CALLUM M. MACDONALD, ANABELA DA SILVA, AND VADIM A. MARKEL*
More informationLocal currents in a two-dimensional topological insulator
Local currents in a two-dimensional topological insulator Xiaoqian Dang, J. D. Burton and Evgeny Y. Tsymbal Department of Physics and Astronomy Nebraska Center for Materials and Nanoscience University
More informationChimica Inorganica 3
A symmetry operation carries the system into an equivalent configuration, which is, by definition physically indistinguishable from the original configuration. Clearly then, the energy of the system must
More informationReconstructing a Function from its V-line Radon Transform in a D
Reconstructing a Function from its V-line Radon Transform in a Disc University of Texas at Arlington Inverse Problems and Application Linköping, Sweden April 5, 2013 Acknowledgements The talk is based
More informationFun With Carbon Monoxide. p. 1/2
Fun With Carbon Monoxide p. 1/2 p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results C V (J/K-mole) 35 30 25
More informationLet b be the distance of closest approach between the trajectory of the center of the moving ball and the center of the stationary one.
Scattering Classical model As a model for the classical approach to collision, consider the case of a billiard ball colliding with a stationary one. The scattering direction quite clearly depends rather
More informationGroup Representations
Group Representations Alex Alemi November 5, 2012 Group Theory You ve been using it this whole time. Things I hope to cover And Introduction to Groups Representation theory Crystallagraphic Groups Continuous
More informationOn Impossibility of Negative Refraction
On Impossibility of Negative Refraction Vadim A. Markel Radiology/Bioengeneering, UPenn, Philly REFERENCES: V.A.Markel, Correct definition of the Poynting vector in electrically and magnetically polarizable
More informationQuantum Physics III (8.06) Spring 2008 Final Exam Solutions
Quantum Physics III (8.6) Spring 8 Final Exam Solutions May 19, 8 1. Short answer questions (35 points) (a) ( points) α 4 mc (b) ( points) µ B B, where µ B = e m (c) (3 points) In the variational ansatz,
More informationLecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators
Lecture Spin, orbital, and total angular momentum 70.00 Mechanics Very brief background MATH-GA In 9, a famous experiment conducted by Otto Stern and Walther Gerlach, involving particles subject to a nonuniform
More informationMath review. Math review
Math review 1 Math review 3 1 series approximations 3 Taylor s Theorem 3 Binomial approximation 3 sin(x), for x in radians and x close to zero 4 cos(x), for x in radians and x close to zero 5 2 some geometry
More informationEvolution equations with spectral methods: the case of the wave equation
Evolution equations with spectral methods: the case of the wave equation Jerome.Novak@obspm.fr Laboratoire de l Univers et de ses Théories (LUTH) CNRS / Observatoire de Paris, France in collaboration with
More informationECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals
ECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals Laporte Selection Rule Polarization Dependence Spin Selection Rule 1 Laporte Selection Rule We first apply this
More informationIntroduction to Polarization
Phone: Ext 659, E-mail: hcchui@mail.ncku.edu.tw Fall/007 Introduction to Polarization Text Book: A Yariv and P Yeh, Photonics, Oxford (007) 1.6 Polarization States and Representations (Stokes Parameters
More informationClosed-Form Solution Of Absolute Orientation Using Unit Quaternions
Closed-Form Solution Of Absolute Orientation Using Unit Berthold K. P. Horn Department of Computer and Information Sciences November 11, 2004 Outline 1 Introduction 2 3 The Problem Given: two sets of corresponding
More informationQuantization of the open string on exact plane waves and non-commutative wave fronts
Quantization of the open string on exact plane waves and non-commutative wave fronts F. Ruiz Ruiz (UCM Madrid) Miami 2007, December 13-18 arxiv:0711.2991 [hep-th], with G. Horcajada Motivation On-going
More informationAngular Momentum set II
Angular Momentum set II PH - QM II Sem, 7-8 Problem : Using the commutation relations for the angular momentum operators, prove the Jacobi identity Problem : [ˆL x, [ˆL y, ˆL z ]] + [ˆL y, [ˆL z, ˆL x
More informationProblem Solving 1: Line Integrals and Surface Integrals
A. Line Integrals MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics Problem Solving 1: Line Integrals and Surface Integrals The line integral of a scalar function f ( xyz),, along a path C is
More informationPhD in Theoretical Particle Physics Academic Year 2017/2018
July 10, 017 SISSA Entrance Examination PhD in Theoretical Particle Physics Academic Year 017/018 S olve two among the four problems presented. Problem I Consider a quantum harmonic oscillator in one spatial
More informationA Quantum Mechanical Model for the Vibration and Rotation of Molecules. Rigid Rotor
A Quantum Mechanical Model for the Vibration and Rotation of Molecules Harmonic Oscillator Rigid Rotor Degrees of Freedom Translation: quantum mechanical model is particle in box or free particle. A molecule
More informationExact Solutions to the Focusing Discrete Nonlinear Schrödinger Equation
Exact Solutions to the Focusing Discrete Nonlinear Schrödinger Equation Francesco Demontis (based on a joint work with C. van der Mee) Università degli Studi di Cagliari Dipartimento di Matematica e Informatica
More informationAlgebraic Theory of Entanglement
Algebraic Theory of (arxiv: 1205.2882) 1 (in collaboration with T.R. Govindarajan, A. Queiroz and A.F. Reyes-Lega) 1 Physics Department, Syracuse University, Syracuse, N.Y. and The Institute of Mathematical
More informationPhys 622 Problems Chapter 6
1 Problem 1 Elastic scattering Phys 622 Problems Chapter 6 A heavy scatterer interacts with a fast electron with a potential V (r) = V e r/r. (a) Find the differential cross section dσ dω = f(θ) 2 in the
More informationREPRESENTATION THEORY WEEK 7
REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable
More informationPhysics 580: Quantum Mechanics I Department of Physics, UIUC Fall Semester 2017 Professor Eduardo Fradkin
Physics 58: Quantum Mechanics I Department of Physics, UIUC Fall Semester 7 Professor Eduardo Fradkin Problem Set No. 5 Bound States and Scattering Theory Due Date: November 7, 7 Square Well in Three Dimensions
More informationSupporting Information
Supporting Information A: Calculation of radial distribution functions To get an effective propagator in one dimension, we first transform 1) into spherical coordinates: x a = ρ sin θ cos φ, y = ρ sin
More informationSolution Set of Homework # 2. Friday, September 09, 2017
Temple University Department of Physics Quantum Mechanics II Physics 57 Fall Semester 17 Z. Meziani Quantum Mechanics Textboo Volume II Solution Set of Homewor # Friday, September 9, 17 Problem # 1 In
More information( ) ( ) ( ) Invariance Principles & Conservation Laws. = P δx. Summary of key point of argument
Invariance Principles & Conservation Laws Without invariance principles, there would be no laws of physics! We rely on the results of experiments remaining the same from day to day and place to place.
More informationFourier Approach to Wave Propagation
Phys 531 Lecture 15 13 October 005 Fourier Approach to Wave Propagation Last time, reviewed Fourier transform Write any function of space/time = sum of harmonic functions e i(k r ωt) Actual waves: harmonic
More informationSummary: angular momentum derivation
Summary: angular momentum derivation L = r p L x = yp z zp y, etc. [x, p y ] = 0, etc. (-) (-) (-3) Angular momentum commutation relations [L x, L y ] = i hl z (-4) [L i, L j ] = i hɛ ijk L k (-5) Levi-Civita
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 informationPHY752, Fall 2016, Assigned Problems
PHY752, Fall 26, Assigned Problems For clarification or to point out a typo (or worse! please send email to curtright@miami.edu [] Find the URL for the course webpage and email it to curtright@miami.edu
More information3 Angular Momentum and Spin
In this chapter we review the notions surrounding the different forms of angular momenta in quantum mechanics, including the spin angular momentum, which is entirely quantum mechanical in nature. Some
More informationProb (solution by Michael Fisher) 1
Prob 975 (solution by Michael Fisher) We begin by expressing the initial state in a basis of the spherical harmonics, which will allow us to apply the operators ˆL and ˆL z θ, φ φ() = 4π sin θ sin φ =
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationIntro to harmonic analysis on groups Risi Kondor
Risi Kondor Any (sufficiently smooth) function f on the unit circle (equivalently, any 2π periodic f ) can be decomposed into a sum of sinusoidal waves f(x) = k= c n e ikx c n = 1 2π f(x) e ikx dx 2π 0
More informationCoupling of Angular Momenta Isospin Nucleon-Nucleon Interaction
Lecture 5 Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction WS0/3: Introduction to Nuclear and Particle Physics,, Part I I. Angular Momentum Operator Rotation R(θ): in polar coordinates the
More informationRadiative transfer equation in spherically symmetric NLTE model stellar atmospheres
Radiative transfer equation in spherically symmetric NLTE model stellar atmospheres Jiří Kubát Astronomický ústav AV ČR Ondřejov Zářivě (magneto)hydrodynamický seminář Ondřejov 20.03.2008 p. Outline 1.
More informationReview of paradigms QM. Read McIntyre Ch. 1, 2, 3.1, , , 7, 8
Review of paradigms QM Read McIntyre Ch. 1, 2, 3.1, 5.1-5.7, 6.1-6.5, 7, 8 QM Postulates 1 The state of a quantum mechanical system, including all the informaion you can know about it, is represented mathemaically
More informationAngular Momentum - set 1
Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x ] =
More information< E >= 9/10E 1 +1/10E 3 = 9/10E 1 +9/10E 1 = 1.8E 1. no measurement can obtain this as its not an eigenvalue [U:1 mark]
QM1) (a) normalisation ψ ψdx = 1 A (3ψ 1 +ψ 3 ) dx = A (9ψ 1 +ψ 3 +6ψ 1 ψ 3 )dx = 1 wavefunctions are orthonormal so = A (9 + 1) = 1 so A = 1/ 1 prob E 1 is 9/1. so prob < E 3 > is 1/1 < E >= 9/1E 1 +1/1E
More information1 Coherent-Mode Representation of Optical Fields and Sources
1 Coherent-Mode Representation of Optical Fields and Sources 1.1 Introduction In the 1980s, E. Wolf proposed a new theory of partial coherence formulated in the space-frequency domain. 1,2 The fundamental
More informationAnouncements. Assignment 3 has been posted!
Anouncements Assignment 3 has been posted! FFTs in Graphics and Vision Correlation of Spherical Functions Outline Math Review Spherical Correlation Review Dimensionality: Given a complex n-dimensional
More informationPhysics 506 Winter 2008 Homework Assignment #4 Solutions. Textbook problems: Ch. 9: 9.6, 9.11, 9.16, 9.17
Physics 56 Winter 28 Homework Assignment #4 Solutions Textbook problems: Ch. 9: 9.6, 9., 9.6, 9.7 9.6 a) Starting from the general expression (9.2) for A and the corresponding expression for Φ, expand
More informationThe PWcond code: Complex bands, transmission, and ballistic conductance
The PWcond code: Complex bands, transmission, and ballistic conductance SISSA and IOM-DEMOCRITOS Trieste (Italy) Outline 1 Ballistic transport: a few concepts 2 Complex band structure 3 Current of a Bloch
More informationPhoton-atom scattering
Photon-atom scattering Aussois 005 Part Ohad Assaf and Aharon Gero Technion Photon-atom scattering Complex system: Spin of the photon: dephasing Internal atomic degrees of freedom (Zeeman sublevels): decoherence
More informationGraduate Quantum Mechanics I: Prelims and Solutions (Fall 2015)
Graduate Quantum Mechanics I: Prelims and Solutions (Fall 015 Problem 1 (0 points Suppose A and B are two two-level systems represented by the Pauli-matrices σx A,B σ x = ( 0 1 ;σ 1 0 y = ( ( 0 i 1 0 ;σ
More informationOrdinary Least Squares and its applications
Ordinary Least Squares and its applications Dr. Mauro Zucchelli University Of Verona December 5, 2016 Dr. Mauro Zucchelli Ordinary Least Squares and its applications December 5, 2016 1 / 48 Contents 1
More informationHomework assignment 3: due Thursday, 10/26/2017
Homework assignment 3: due Thursday, 10/6/017 Physics 6315: Quantum Mechanics 1, Fall 017 Problem 1 (0 points The spin Hilbert space is defined by three non-commuting observables, S x, S y, S z. These
More informationRobot Control Basics CS 685
Robot Control Basics CS 685 Control basics Use some concepts from control theory to understand and learn how to control robots Control Theory general field studies control and understanding of behavior
More informationA note on SU(6) spin-flavor symmetry.
A note on SU(6) spin-flavor symmetry. 1 3 j = 0,, 1,, 2,... SU(2) : 2 2 dim = 1, 2, 3, 4, 5 SU(3) : u d s dim = 1,3, 3,8,10,10,27... u u d SU(6) : dim = 1,6,6,35,56,70,... d s s SU(6) irrep 56 + (l=0)
More informationPreliminary Examination - Day 1 Thursday, August 9, 2018
UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, August 9, 8 This test covers the topics of Thermodynamics and Statistical Mechanics (Topic ) and Quantum Mechanics (Topic
More informationPhysics 742 Graduate Quantum Mechanics 2 Midterm Exam, Spring [15 points] A spinless particle in three dimensions has potential 2 2 2
Physics 74 Graduate Quantum Mechanics Midterm Exam, Spring 4 [5 points] A spinless particle in three dimensions has potential V A X Y Z BXYZ Consider the unitary rotation operator R which cyclically permutes
More informationSymmetry and Properties of Crystals (MSE638) Stress and Strain Tensor
Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Somnath Bhowmick Materials Science and Engineering, IIT Kanpur April 6, 2018 Tensile test and Hooke s Law Upto certain strain (0.75),
More informationONE AND MANY ELECTRON ATOMS Chapter 15
See Week 8 lecture notes. This is exactly the same as the Hamiltonian for nonrigid rotation. In Week 8 lecture notes it was shown that this is the operator for Lˆ 2, the square of the angular momentum.
More informationI. Introduction. A New Method for Inverting Integrals Attenuated Radon Transform (SPECT) D to N map for Moving Boundary Value Problems
I. Introduction A New Method for Inverting Integrals Attenuated Radon Transform (SPECT) D to N map for Moving Boundary Value Problems F(k) = T 0 e k2 t+ikl(t) f (t)dt, k C. Integrable Nonlinear PDEs in
More informationMORE NOTES FOR MATH 823, FALL 2007
MORE NOTES FOR MATH 83, FALL 007 Prop 1.1 Prop 1. Lemma 1.3 1. The Siegel upper half space 1.1. The Siegel upper half space and its Bergman kernel. The Siegel upper half space is the domain { U n+1 z C
More informationResearch Institute of Geodesy, Topography and Cartography Zdiby, Prague-East, Czech Republic
Leibniz Society of Science at Berlin, Scientific Colloquium Geodesy - Mathematic - Physics - Geophysics in honour of Erik W. Grafarend on the occasion of his 75th birthday Berlin, Germany, 3 February 05
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More information1 Matrices and matrix algebra
1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. For instance 4 2 0 3 1 A = 5 1.2 0.7 x 3 π 3 4 6 27 is a matrix with 3 rows and 5 columns
More informationAstronomy 9620a / Physics 9302a - 1st Problem List and Assignment
Astronomy 9620a / Physics 9302a - st Problem List and Assignment Your solution to problems 4, 7, 3, and 6 has to be handed in on Thursday, Oct. 7, 203.. Prove the following relations (going from the left-
More informationA VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics
More informationAnalysis of Scattering of Radiation in a Plane-Parallel Atmosphere. Stephanie M. Carney ES 299r May 23, 2007
Analysis of Scattering of Radiation in a Plane-Parallel Atmosphere Stephanie M. Carney ES 299r May 23, 27 TABLE OF CONTENTS. INTRODUCTION... 2. DEFINITION OF PHYSICAL QUANTITIES... 3. DERIVATION OF EQUATION
More informationPhysics 216 Problem Set 4 Spring 2010 DUE: MAY 25, 2010
Physics 216 Problem Set 4 Spring 2010 DUE: MAY 25, 2010 1. (a) Consider the Born approximation as the first term of the Born series. Show that: (i) the Born approximation for the forward scattering amplitude
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationRotor Spectra, Berry Phases, and Monopole Fields: From Antiferromagnets to QCD
Rotor Spectra, Berry Phases, and Monopole Fields: From Antiferromagnets to QCD Uwe-Jens Wiese Bern University LATTICE08, Williamsburg, July 14, 008 S. Chandrasekharan (Duke University) F.-J. Jiang, F.
More informationThe octagon method for finding exceptional points, and application to hydrogen-like systems in parallel electric and magnetic fields
Institute of Theoretical Physics, University of Stuttgart, in collaboration with M. Feldmaier, F. Schweiner, J. Main, and H. Cartarius The octagon method for finding exceptional points, and application
More informationInverse Problems in Quantum Optics
Inverse Problems in Quantum Optics John C Schotland Department of Mathematics University of Michigan Ann Arbor, MI schotland@umich.edu Motivation Inverse problems of optical imaging are based on classical
More informationGROUP THEORY IN PHYSICS
GROUP THEORY IN PHYSICS Wu-Ki Tung World Scientific Philadelphia Singapore CONTENTS CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 PREFACE INTRODUCTION 1.1 Particle on a One-Dimensional Lattice 1.2 Representations
More informationQuantum Field Theory Example Sheet 4 Michelmas Term 2011
Quantum Field Theory Example Sheet 4 Michelmas Term 0 Solutions by: Johannes Hofmann Laurence Perreault Levasseur Dave M. Morris Marcel Schmittfull jbh38@cam.ac.uk L.Perreault-Levasseur@damtp.cam.ac.uk
More information