Advanced Quantum Mechanics
|
|
- Rodger Scott
- 5 years ago
- Views:
Transcription
1 Advanced Quantum Mechancs Rajdeep Sensarma! ecture #9 QM of Relatvstc Partcles
2 Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton Spn 1/ representatons and Weyl Spnors Transformaton propertes of Weyl spnors
3 Rules for Possble Actons The dynamcs of relatvstc partcles s obtaned from the acton of the system, whch s a functon of the felds (operator valued functons of space-tme) and ther space-tme dervatves.! In partcular transton ampltudes can be wrtten as a functonal ntegral over feld confg. wth each confg contrbutng e S Choose felds transformng accordng to partcular rreps of the orentz group. The transformaton propertes of the felds are frame ndependent, e.g. a scalar feld remans a scalar feld n all frames. Construct possble orentz nvarant quanttes (orentz scalars) out of felds and ther dervatves. These provde possble agrangan Denstes for descrbng the system. For free partcle, the acton should be quadratc and Poncare nvarant. It should nvolve atmost nd order dervatves. Our Am: Construct possble orentz scalars out of Weyl Spnors
4 Weyl Felds and Grassmann Numbers Snce the (H) Weyl Feld transforms accordng to (1/,0) rrep, we should be able to construct a orentz Scalar by takng antsymmetrc products of H Weyl Felds SU() decomposton rule 1/ X 1/ = For ths we would lke to consder two Weyl spnors (x, t) and (x, t) and consder ther products. We would lke to construct products whch are orentz nvarant. Under T: T! T U T ( ) U ( ) = T Snce U T ( ) U ( ) = Snce the scalar s obtaned by antsymmetrzng the product (look up exchange symmetry and SU()), we should get 0 f χ s same as 0 T =( 1, ) 0 1 = = 0 f s a complex no. However, f the feld s a Grassmann number (ant-commutng number), there can be a non-trval scalar representaton formed out of the spn 1/ feld.
5 Grassmann Numbers A bnary representaton of an nteger Representng numbers: Start wth basc objects (generators) n for all non-negatve nteger n Defne combnaton rules: a) m n = m+n b) The objects commute wth each other m n = n m Arbtrary lnear combnatons of these generators wth co-effcents (0,1) represent ntegers Complex Numbers: Start wth basc objects (generators) 1 and Defne combnaton rules: 1 * = * 1 = * = -1 1* 1 =1 A complex no. can be represented as arbtrary lnear combnaton of { 1,} wth real co-effcents Grassmann Numbers: Start wth set of n ant-commutng generators + =0 =0 A Grassmann number s an arbtrary lnear comb. of { 1, α1,. αn, α1 α., α1 α αn } wth complex co-eff. E.g. wth generators 1 and = a 0 + a a + a 1 1
6 Grassmann Numbers Addng Grassmann Numbers: Thnk of { 1, α1,. αn, α1 α., α1 α αn } as ndependent unt vectors and add component-wse coeffcents Multplyng Grassmann Numbers: Multply each component wth every other component, and keep track of + =0 =0 Complex conjugaton of Grassmann Numbers: Select a set of n generators and assocate a conjugate generator to each ( ) = ( ) = ( ) = = The generators commute wth complex numbers = Defne conjugaton as an operaton whch conjugates both generators and co-effcents.
7 Grassmann Numbers Functons of Grassmann numbers Stck to generators and * Any analytc fn. of * f( )=f 0 + f 1 Any analytc fn. of g( ) =g 0 + g 1 Any analytc fn. of and * A(, )=a 0 + a 1 +ā 1 + a 1 Grassmann dervatves: Identcal to complex derv., BUT, for the dervatve to act on the varable, the varable has to be antcommuted tll t s adjacent to the dervatve operator. E.g.: Grassmann Integrals: ( ( )= d 1 = A(, )= a 1 d A(, ) Grassmann dervatves antcommute d =1 ke dervatves, the varable has to be antcommuted tll t sts adjacent to ntegral operator E.g.: d ( )=f 1 d d A(, )= a 1 = d d A(, ) Scalar product of Grassmann Fn.s : g( ) =g 0 + g 1 h( ) =h 0 + h 1 hh g = d d e h ( )g( ) = d d (1 )(h 0 + h 1 )(g = h 0g 0 + h 0 + g 1 ) 1g 1
8 Weyl Felds and orentz Scalars If Weyl felds are represented by Grassmann numbers, we have seen that T and R T R are quadratc orentz scalars that can be formed. These would ndcate the possblty of wrtng down mass terms for Weyl Fermons. However, Consder the transformaton propertes of where s a left-handed Weyl feld! U ( ) = U ( ) = U R ( ) U ( ) = U R( ) Thus * transforms lke a rght handed Weyl Feld Takng h.c. of above equaton T! T U R ( ) T! R T R! Thus the possble mass term has the form m R or m R The mass term mxes left-handed and rght handed Weyl felds. et us frst descrbe massless Fermons wth Weyl felds and we wll come back to massve felds later.
9 orentz vectors from Weyl Felds It s possble to construct four-vectors (transf. acc. to (1/,1/) rrep) from products of Weyl Felds. If we can construct orentz vectors, we can take ts norm, contract wth other orentz vectors lke etc. to form possble orentz scalars Consder the transformaton propertes of Under rotatons, represented by Untary operators, ths s a scalar. Under Boosts, U ( ) = e ~ (~! ~ )! e 1! e 1 ~ ~ ~ ~! e ~ ~ ' ~ ~ = For Infntesmal Boosts! e 1 ~ ~ e 1 ~ ~ ' 1 j ( j + j ) = {, j } = j
10 orentz vectors from Weyl Felds For Infntesmal Boosts! e ~ ~ ' ~ ~ =! e 1 ~ ~ e 1 ~ ~ ' 1 j ( j + j ) = Consder µ wth 0 beng X dentty matrx Under Boosts, ths behaves lke a 4-vector µ = µ wth 0 = Further, behaves as a 3-vector under rotatons So n all µ = (, ) s a orentz 4-vector Smlarly R µ R = ( R R, R R) s a orentz 4-vector Snce ( R R) = R R the above Grassmann blnears are real (n the sense of conjugaton of G No.s)
11 orentz Scalars and Weyl Acton The smplest orentz scalar s formed by contractng the V µ ( ) µ or µ ( ) The secret behnd Drac s magc of 1st order orentz nvarant eqn. Real near combnaton 1 h µ ( µ ( ) µ 1 Quadratc Acton for eft Handed Weyl Spnors S = d 4 x 1 h µ ( µ ( ) µ d 4 x 1 Smlarly for Rght Handed Weyl µ ( ) µ R R or R µ@ µ ( R ) or 1 h R µ@ µ ( R µ ( ) µ R R 1 R R Quadratc Acton for Rght Handed Weyl Spnors S = d 4 x 1 h R µ@ µ ( R µ ( ) µ R R d 4 x 1 R R Gradent Terms > Poncare Invarant
12 Feld Equatons for Weyl Felds Quadratc Acton for eft Handed Weyl Spnors S = d 4 x 1 = d 4 x µ upto boundary terms Feld Equaton: Multplyng by, µ =0 ( t ) =0 [@ t ~ ~p ] =0 Smlarly for RH Weyl Fermons [@ t + ~ ~p ] R =0 Snce the Fermons are massless t > E = p [1 ~ ~p p ] =0 [1 + ~ ~p p ] R =0 The Soluton of Weyl Equaton are egenstates of the helcty operator gven by where s s the spn operator, wth egenvalues ± 1 ~s ~p p Paul ubansk vector W 0 = ~s ~p related to helcty n ths case
13 Party and Weyl Fermons The helcty operator s a pseudoscalar under O(3),.e. t changes sgn under a party transformaton. So a H Weyl spnor wth helcty +1 wll transform to a RH Weyl spnor wth helcty -1 under party transformaton. A party nvarant theory (even for massless Fermons) requres both H and RH Weyl felds.
14 Massless Drac Fermons [ (1/,0) + (0,1/) We have seen that party mxes the left and rght handed Weyl Fermons. et us keep both felds and construct a 4 component spnor. We want to wrte a party nvarant theory Ths transforms as (1/,0) (0,1/) = R Ths s called a Drac spnor n the chral bass How does party act on the Drac spnor? P = P R = R = 0 0 = s X dentty matrx 0 =1 1 s 4 X 4 dentty matrx et us now wrte the theory of a left handed and a rght handed Weyl Fermon together S = d 4 x 1 [ + R R ] = d 4 x 1 [ 0 ~ ] = d 4 x 1 [ ] where = 0 0 = 0 { µ, } =g µ Clfford Algebra Paul conjugate
15 Massless Drac Fermons [ (1/,0) + (0,1/) Check that we have a orentz nvarant acton Check that µ Transforms lke 4 vector To do ths, we need to fnd the form of orentz generators J and K and R spnors transform as spn 1/ under rotn. J = 0 0! K = 0 0! and R spnors transform dfferently under boosts 0J = J 0 0 K = K 0 K nvolves one space and 1 tme co-ord > J s pseudovector under rotn. changes sgn under party.! U( )! U ( ) and 0! U ( ) 0 wth U( ) = e ( J ~! ~ + K ~ ~ ) U ( ) = e ( J ~! ~ K ~ ~ ) U 1 ( ) = e ( ~ J ~! + ~ K ~ ) Note: γ 0 changes form under T, but mantans ts antcommutaton wth K n new bass Snce γ 0 changes sgn of K whle commutng across, but keeps J unchanged, commutng t across U wll convert t to U -1 0! U ( ) 0 = 0U 1 ( ) Thus s a orentz scalar
16 Massless Drac Fermons [ (1/,0) + (0,1/) Consder the matrx µ = 4 [ µ, ]= 4 ( µ { µ, }) = ( µ g µ 1 4 ) [ µ, ]= [ µ, ] = ( µ [, ]+[ µ, ] ) = ( µ [, ] [, µ ] ) = ( µ µ g µ +g µ ) = ([ µ, ] µ g + g µ ) So [ µ, ]= [ µ, ]=(g µ g µ ) [ µ, ]= [ µ, ] = ([ µ, ] + [ µ, ]) = (g µ g µ + g µ g µ ) = (g µ g µ + g µ g µ ) Γ μν satsfes the e Algebra for orentz generators So S µ = 4 [ µ, ] These relatons only use Drac Algebra and not specfc forms for γ matrces
17 Massless Drac Fermons [ (1/,0) + (0,1/) S µ = 4 [ µ, ] [S µ, ]= [ µ and, ]=(g µ g µ ) γ transforms as a orentz 4-vector µ transforms as a orentz 4-vector S = d 4 x 1 [ ] s a orentz scalar These relatons only use Clfford Algebra and not specfc forms for γ matrces 4 x 4 matrces satsfyng Clfford Algebra s not unque In fact any untary transformaton wll keep { µ, } =g µ nvarant. Thus there are many equvalent bass to wrte Drac spnors. The form of the spnors as well as the γ matrces depend on the bass, but the form of the acton s nvarant. Chral bass = R Charge conjugated spnor c = R Majorana spnor M = Real spnor (Real G No.s) γ matrces are purely magnary
18 Massve Drac Felds et us now come to the queston of a mass term Snce s a orentz scalar m = m( R + R ) s a possble real orentz scalar mass term Ths s not the only possble quadratc orentz scalar We had earler shown that R and R are ndvdually orentz scalars An ndependent scalar can be formed out of the dfference of ths two terms m( R R ) Defne projecton operators to obtan Weyl Fermons from Drac Fermons In the chral bass, followng projecton operators project the Drac spnor nto eft(rght) handed Weyl Fermons (R) = 1 (1 ± 5) 5 = = = 4 µ µ γ 5 transforms as a orentz scalar, but changes sgn under party.e. transforms as a pseudo-scalar. Note: Explct for of γ 5 wll be dfferent n dff. bass, but ts relaton wth γ holds n all bass
19 Massve Drac Fermons m( R R )! m 5 So the most general Drac acton for a massve feld has the form S = 1 d 4 x[ + m m 0 5 ] If party s a good symmetry of the system m = 0 S = 1 d 4 x [ + m] = 1 d 4 x [ µ + m] upto surface terms The Saddle pont equaton for ths acton s [ µ m] =0 Multply by γ 0, and use (γ 0 ) =1 [( 0 t + 0 ~ r m 0 ] =0 0 ~ = ~, 0 t =[ ~ r+ m ] Drac Equaton of 1 patcle Rel. QM
20 Global Symmetres of Drac Acton S = 1 d 4 x [ + m] = 1 d 4 x [ µ + m] Global phase rotaton! e Chral Transformaton! e 5 The acton s nvarant under these transformatons The conserved Noether current s gven by j µ = µ j µ 5 = µ 5 and the correspondng conserved charges are Q = d 3 x 0 = d 3 x [ + R R] Q 5 = d 3 x 0 5 = d 3 x [ R R]
21 Thngs we have not touched Constructng creaton/annhlaton operators and Fock space from felds Calculatng Expermentally measureable quanttes Scatterng ampltude as transton ampltude > Calculaton through fn ntegrals. Fnte Chemcal potental > fnte densty of partcles Imagnary tme and fnte temperature calculatons > Correlaton fn. and response fn. Interactng theores Symmetry Consderatons and possble nteracton terms. Perturbaton Theory calculaton of n pont functons > scatterng, correlaton fn. etc. Saddle Ponts, symmetry breakng and effectve theores Renormalzaton the other gudng prncple Wat for QFT-I next semester.
Lecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationA how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
More informationThe Lorentz group. Generate the group SO(3,1) To construct representa;ons a more convenient (non- Hermi;an) basis is. i j ijk k. j ijk k.
Fermons S 1 2 The Lorentz group Rotatons J Boosts K [ J, J ] ε J [ J, K ] ε K [ K, K ] ε J j jk k j jk k j jk k } Generate the group SO(3,1) ( M ( x x ) J M K M ) 1 ρσ ρ σ x σ ρ x 2 ε jk jk 0 To construct
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationLorentz Group. Ling Fong Li. 1 Lorentz group Generators Simple representations... 3
Lorentz Group Lng Fong L ontents Lorentz group. Generators............................................. Smple representatons..................................... 3 Lorentz group In the dervaton of Drac
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationQuantum Mechanics I - Session 4
Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More information7. Products and matrix elements
7. Products and matrx elements 1 7. Products and matrx elements Based on the propertes of group representatons, a number of useful results can be derved. Consder a vector space V wth an nner product ψ
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationHomework & Solution. Contributors. Prof. Lee, Hyun Min. Particle Physics Winter School. Park, Ye
Homework & Soluton Prof. Lee, Hyun Mn Contrbutors Park, Ye J(yej.park@yonse.ac.kr) Lee, Sung Mook(smlngsm0919@gmal.com) Cheong, Dhong Yeon(dhongyeoncheong@gmal.com) Ban, Ka Young(ban94gy@yonse.ac.kr) Ro,
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationSection 3.6 Complex Zeros
04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there
More informationwhere the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt
Physcs 543 Quantum Mechancs II Fall 998 Hartree-Fock and the Self-consstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationHW #6, due Oct Toy Dirac Model, Wick s theorem, LSZ reduction formula. Consider the following quantum mechanics Lagrangian,
HW #6, due Oct 5. Toy Drac Model, Wck s theorem, LSZ reducton formula. Consder the followng quantum mechancs Lagrangan, L ψ(σ 3 t m)ψ, () where σ 3 s a Paul matrx, and ψ s defned by ψ ψ σ 3. ψ s a twocomponent
More information1 Vectors over the complex numbers
Vectors for quantum mechancs 1 D. E. Soper 2 Unversty of Oregon 5 October 2011 I offer here some background for Chapter 1 of J. J. Sakura, Modern Quantum Mechancs. 1 Vectors over the complex numbers What
More informationFrom Biot-Savart Law to Divergence of B (1)
From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to
More informationMEM 255 Introduction to Control Systems Review: Basics of Linear Algebra
MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More information1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:
68A Solutons to Exercses March 05 (a) Usng a Taylor expanson, and notng that n 0 for all n >, ( + ) ( + ( ) + ) We can t nvert / because there s no Taylor expanson around 0 Lets try to calculate the nverse
More information2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
More informationThis model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:
1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors
More informationScattering of two identical particles in the center-of. of-mass frame. (b)
Lecture # November 5 Scatterng of two dentcal partcle Relatvtc Quantum Mechanc: The Klen-Gordon equaton Interpretaton of the Klen-Gordon equaton The Drac equaton Drac repreentaton for the matrce α and
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationLagrangian Field Theory
Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,
More informationPhys304 Quantum Physics II (2005) Quantum Mechanics Summary. 2. This kind of behaviour can be described in the mathematical language of vectors:
MACQUARIE UNIVERSITY Department of Physcs Dvson of ICS Phys304 Quantum Physcs II (2005) Quantum Mechancs Summary The followng defntons and concepts set up the basc mathematcal language used n quantum mechancs,
More informationCHAPTER 5: Lie Differentiation and Angular Momentum
CHAPTER 5: Le Dfferentaton and Angular Momentum Jose G. Vargas 1 Le dfferentaton Kähler s theory of angular momentum s a specalzaton of hs approach to Le dfferentaton. We could deal wth the former drectly,
More informationRobert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations
Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and x-ray exctatons 9-01 By gong through the procedure ndcated n the text, develop the tme-ndependent Schroednger equaton
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationPHYS 215C: Quantum Mechanics (Spring 2017) Problem Set 3 Solutions
PHYS 5C: Quantum Mechancs Sprng 07 Problem Set 3 Solutons Prof. Matthew Fsher Solutons prepared by: Chatanya Murthy and James Sully June 4, 07 Please let me know f you encounter any typos n the solutons.
More information332600_08_1.qxp 4/17/08 11:29 AM Page 481
336_8_.qxp 4/7/8 :9 AM Page 48 8 Complex Vector Spaces 8. Complex Numbers 8. Conjugates and Dvson of Complex Numbers 8.3 Polar Form and DeMovre s Theorem 8.4 Complex Vector Spaces and Inner Products 8.5
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More informationQuantum Field Theory III
Quantum Feld Theory III Prof. Erck Wenberg February 16, 011 1 Lecture 9 Last tme we showed that f we just look at weak nteractons and currents, strong nteracton has very good SU() SU() chral symmetry,
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationThe Dirac Equation. Elementary Particle Physics Strong Interaction Fenomenology. Diego Bettoni Academic year
The Drac Equaton Eleentary artcle hyscs Strong Interacton Fenoenology Dego Betton Acadec year - D Betton Fenoenologa Interazon Fort elatvstc equaton to descrbe the electron (ncludng ts sn) Conservaton
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationThe Dirac Monopole and Induced Representations *
The Drac Monopole and Induced Representatons * In ths note a mathematcally transparent treatment of the Drac monopole s gven from the pont of vew of nduced representatons Among other thngs the queston
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationLecture 14: Forces and Stresses
The Nuts and Bolts of Frst-Prncples Smulaton Lecture 14: Forces and Stresses Durham, 6th-13th December 2001 CASTEP Developers Group wth support from the ESF ψ k Network Overvew of Lecture Why bother? Theoretcal
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More informationψ = i c i u i c i a i b i u i = i b 0 0 b 0 0
Quantum Mechancs, Advanced Course FMFN/FYSN7 Solutons Sheet Soluton. Lets denote the two operators by  and ˆB, the set of egenstates by { u }, and the egenvalues as  u = a u and ˆB u = b u. Snce the
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department
More informationThe Symmetries of Kibble s Gauge Theory of Gravitational Field, Conservation Laws of Energy-Momentum Tensor Density and the
The Symmetres of Kbble s Gauge Theory of Gravtatonal Feld, Conservaton aws of Energy-Momentum Tensor Densty and the Problems about Orgn of Matter Feld Fangpe Chen School of Physcs and Opto-electronc Technology,Dalan
More informationChange. Flamenco Chuck Keyser. Updates 11/26/2017, 11/28/2017, 11/29/2017, 12/05/2017. Most Recent Update 12/22/2017
Change Flamenco Chuck Keyser Updates /6/7, /8/7, /9/7, /5/7 Most Recent Update //7 The Relatvstc Unt Crcle (ncludng proof of Fermat s Theorem) Relatvty Page (n progress, much more to be sad, and revsons
More informationThe internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
More informationNote 2. Ling fong Li. 1 Klein Gordon Equation Probablity interpretation Solutions to Klein-Gordon Equation... 2
Note 2 Lng fong L Contents Ken Gordon Equaton. Probabty nterpretaton......................................2 Soutons to Ken-Gordon Equaton............................... 2 2 Drac Equaton 3 2. Probabty nterpretaton.....................................
More informationLecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( )
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationThe non-negativity of probabilities and the collapse of state
The non-negatvty of probabltes and the collapse of state Slobodan Prvanovć Insttute of Physcs, P.O. Box 57, 11080 Belgrade, Serba Abstract The dynamcal equaton, beng the combnaton of Schrödnger and Louvlle
More informationLecture 5.8 Flux Vector Splitting
Lecture 5.8 Flux Vector Splttng 1 Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationNote on the Electron EDM
Note on the Electron EDM W R Johnson October 25, 2002 Abstract Ths s a note on the setup of an electron EDM calculaton and Schff s Theorem 1 Basc Relatons The well-known relatvstc nteracton of the electron
More informationProf. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model
EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several
More informationSupplemental document
Electronc Supplementary Materal (ESI) for Physcal Chemstry Chemcal Physcs. Ths journal s the Owner Socetes 01 Supplemental document Behnam Nkoobakht School of Chemstry, The Unversty of Sydney, Sydney,
More informationYukawa Potential and the Propagator Term
PHY304 Partcle Physcs 4 Dr C N Booth Yukawa Potental an the Propagator Term Conser the electrostatc potental about a charge pont partcle Ths s gven by φ = 0, e whch has the soluton φ = Ths escrbes the
More informationEPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski
EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More informationThe Dirac Equation for a One-electron atom. In this section we will derive the Dirac equation for a one-electron atom.
The Drac Equaton for a One-electron atom In ths secton we wll derve the Drac equaton for a one-electron atom. Accordng to Ensten the energy of a artcle wth rest mass m movng wth a velocty V s gven by E
More informationLecture Notes 7: The Unruh Effect
Quantum Feld Theory for Leg Spnners 17/1/11 Lecture Notes 7: The Unruh Effect Lecturer: Prakash Panangaden Scrbe: Shane Mansfeld 1 Defnng the Vacuum Recall from the last lecture that choosng a complex
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationFall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede
Fall 0 Analyss of Expermental easurements B. Esensten/rev. S. Errede We now reformulate the lnear Least Squares ethod n more general terms, sutable for (eventually extendng to the non-lnear case, and also
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More information14 The Postulates of Quantum mechanics
14 The Postulates of Quantum mechancs Postulate 1: The state of a system s descrbed completely n terms of a state vector Ψ(r, t), whch s quadratcally ntegrable. Postulate 2: To every physcally observable
More informationTensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q
For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals
More information