Advanced Quantum Mechanics

Size: px
Start display at page:

Download "Advanced Quantum Mechanics"

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

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 information

C/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/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 information

The Feynman path integral

The 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 information

THEOREMS OF QUANTUM MECHANICS

THEOREMS 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 information

763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.

763622S 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 information

A how to guide to second quantization method.

A 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 information

The 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.

The 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 information

Quantum Mechanics for Scientists and Engineers. David Miller

Quantum 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 information

Difference Equations

Difference 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 information

Lorentz Group. Ling Fong Li. 1 Lorentz group Generators Simple representations... 3

Lorentz 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 information

Section 8.3 Polar Form of Complex Numbers

Section 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 information

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

8.4 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 information

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.

A 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 information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner 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 information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - 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 information

Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws

Representation 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 information

APPENDIX A Some Linear Algebra

APPENDIX 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 information

1 Matrix representations of canonical matrices

1 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 information

Quantum Mechanics I - Session 4

Quantum 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 information

12. The Hamilton-Jacobi Equation Michael Fowler

12. 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 information

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: 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 information

7. Products and matrix elements

7. 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 information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX 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 information

PHYS 705: Classical Mechanics. Calculus of Variations II

PHYS 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 information

Linear Approximation with Regularization and Moving Least Squares

Linear 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 information

Homework & Solution. Contributors. Prof. Lee, Hyun Min. Particle Physics Winter School. Park, Ye

Homework & 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 information

n 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

n 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 information

Section 3.6 Complex Zeros

Section 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 information

where 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

where 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 information

Canonical transformations

Canonical 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 information

HW #6, due Oct Toy Dirac Model, Wick s theorem, LSZ reduction formula. Consider the following quantum mechanics Lagrangian,

HW #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 information

1 Vectors over the complex numbers

1 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 information

From Biot-Savart Law to Divergence of B (1)

From 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 information

MEM 255 Introduction to Control Systems Review: Basics of Linear Algebra

MEM 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 information

The Order Relation and Trace Inequalities for. Hermitian Operators

The 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 information

Mathematical Preparations

Mathematical 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 information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC 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 information

9 Characteristic classes

9 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 information

1 (1 + ( )) = 1 8 ( ) = (c) Carrying out the Taylor expansion, in this case, the series truncates at second order:

1 (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 information

2 More examples with details

2 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 information

This model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:

This 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 information

Scattering of two identical particles in the center-of. of-mass frame. (b)

Scattering 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 information

Poisson brackets and canonical transformations

Poisson 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 information

PHYS 705: Classical Mechanics. Canonical Transformation II

PHYS 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 information

Integrals and Invariants of Euler-Lagrange Equations

Integrals 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 information

Lagrangian Field Theory

Lagrangian 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 information

Phys304 Quantum Physics II (2005) Quantum Mechanics Summary. 2. This kind of behaviour can be described in the mathematical language of vectors:

Phys304 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 information

CHAPTER 5: Lie Differentiation and Angular Momentum

CHAPTER 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 information

Robert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations

Robert 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 information

Homework Notes Week 7

Homework 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 information

PHYS 215C: Quantum Mechanics (Spring 2017) Problem Set 3 Solutions

PHYS 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 information

332600_08_1.qxp 4/17/08 11:29 AM Page 481

332600_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 information

Lecture 20: Noether s Theorem

Lecture 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 information

Quantum Field Theory III

Quantum 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 information

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.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 information

The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

The 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 information

SL n (F ) Equals its Own Derived Group

SL 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 information

10. Canonical Transformations Michael Fowler

10. 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.

= = = (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 information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 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 information

5 The Rational Canonical Form

5 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 information

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

Physics 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 information

Lecture 12: Discrete Laplacian

Lecture 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 information

PHYS 705: Classical Mechanics. Newtonian Mechanics

PHYS 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 information

The Dirac Equation. Elementary Particle Physics Strong Interaction Fenomenology. Diego Bettoni Academic year

The 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 information

Moments 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 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 information

The Dirac Monopole and Induced Representations *

The 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

= 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 information

Lecture 14: Forces and Stresses

Lecture 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 information

3.1 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

3.1 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 information

Some Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)

Some 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 information

ψ = i c i u i c i a i b i u i = i b 0 0 b 0 0

ψ = 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 information

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry

Workshop: 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 information

The Symmetries of Kibble s Gauge Theory of Gravitational Field, Conservation Laws of Energy-Momentum Tensor Density and the

The 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 information

Change. 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 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 information

The internal structure of natural numbers and one method for the definition of large prime numbers

The 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 information

Note 2. Ling fong Li. 1 Klein Gordon Equation Probablity interpretation Solutions to Klein-Gordon Equation... 2

Note 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 information

Lecture 3: Probability Distributions

Lecture 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 information

Mechanics Physics 151

Mechanics 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 information

MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS

MATH 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 information

The non-negativity of probabilities and the collapse of state

The 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 information

Lecture 5.8 Flux Vector Splitting

Lecture 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 information

Formulas for the Determinant

Formulas 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 information

Lecture 10 Support Vector Machines II

Lecture 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 information

Note on the Electron EDM

Note 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 information

Prof. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model

Prof. 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 information

Supplemental document

Supplemental 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 information

Yukawa Potential and the Propagator Term

Yukawa 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 information

EPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski

EPR 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 information

2.3 Nilpotent endomorphisms

2.3 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 information

PHYS 705: Classical Mechanics. Hamilton-Jacobi Equation

PHYS 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 information

The Dirac Equation for a One-electron atom. In this section we will derive the Dirac equation for a one-electron atom.

The 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 information

Lecture Notes 7: The Unruh Effect

Lecture 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 information

Math 217 Fall 2013 Homework 2 Solutions

Math 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 information

Mechanics Physics 151

Mechanics 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 information

12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product

12 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 information

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede

Fall 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 information

CHAPTER 14 GENERAL PERTURBATION THEORY

CHAPTER 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 information

14 The Postulates of Quantum mechanics

14 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 information

Tensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q

Tensor 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