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This 12 page Class Notes was uploaded by Jamaal McGlynn on Tuesday October 20, 2015. The Class Notes belongs to PHYS610A at San Diego State University taught by M.Bromley in Fall. Since its upload, it has received 17 views. For similar materials see /class/225319/phys610a-san-diego-state-university in Physics 2 at San Diego State University.
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Date Created: 10/20/15
Lecture 2 Outline Basis Sets SCAMP wednesday 11am s linear vector spaces Section 11 12 Basis sets Sections 13 start on linear operators Sections 15 Linear combinations 0 eg Consider two orthogonal vectorsigt jgt o and a linearly dependent thirdagt aigt bjgt Imagine an Inner Product Space Dot product AB c0s6 Length A V l 2 A 2 cfl U31 Three rules 1 symmetr and 3 linearity c 91 Q1 P Want 3 axioms 04 o4 and 0404 Z O and linearity 04A 04bl cl y 904 C04 Vector norm 04 E x 0404 Z 0 ie kind of a length Unit norm 04i 04404 1 orthogonal 0440474 0 means orthonormal set of vectors has 044043 6m Inner product Via components Dot product AB cos6 axbx ayby azbz Length A V a a5 a In general de ne the inner product of two vectors IO gt EL a igt gt 231 bjjgt lta gt E E afbjltiljgt and An orthonormal set j 6 means Zz39Zj afbj aibl i 3192 i ail It is axiom 1 that gives real 0404 0Li2 Z O Inner product Via dual vector spaces 0 Consider two vectors in the same basis a1 b1 n 002 n b2 3900 Zizl aim I and 211 bilzgt an bn o bra s Via transpose conjugate adjoint inner product 001 lt agtb b 1 an 0 so each ket dgt has an associated dual space bra 04 Orthonormal basis 0 Each basis ket has an associated bra 0 Now we can expand any vector in the basis vectors a1 1 O O a O 1 O agt Zam39 2 a1 a2 an an O O 1 c Find vector components oi via inner product ltj06gt Za ltjligt E Zai5z39j Obj 06gt EdaW Zlti06gtigt E igtlti06gt 139 GramSchmidt procedure 0 eg Consider linearly independent vectors dgt gt o and use inner products to rst orthonormalise Vector expansion 0 which can then expand vector ygt in basis Other useful formulae 0 Schwarz inequality lt0 gt2 S ozoagt gt o Triangle inequality la 5 S lO l WI more on the transpose conjugate 0 Each ket aagt E aagt has a bra aa E aa 0 Each ket ygt ala bl has a bra 7 ala llf 0 Each ket agt aiigt has a bra 04 0 equivalent to the transpose conjugate ltaZltiIaa a2 a1 1 O 0 06 0 1 0 06 0 O 1 Linear operators Assume linear transformation gt Qagt stays in V Linear means Qaagt b gt a9agt b9 gt and that alc ld l 0496 l d Simplest operator Iagt agt leaves the vector alone Consider rotation operator R6 gure 13 Linear operators IMPORTANT 0 Once we now how 9 transforms the basis vectors we know how any vector in space transforms Qagt Z cam Z ailom 2 am 0 eg consider the rotation operator again R i 1e Ri1gt1gt Reigtl2gt 3gt Rem I2gt smooth operators Given two linear operators 9 and A then product operation is A9agt AQagt AQagt note that in general A9agt QAagt De ne commutator as A Q E AS 9A eg R i commutes with R7Ti eg R i does not commute with R j De ning inverse Via 99 1 9 19 I then we must have SUD 1 V194 since we need oAoA 1 oAA lo l oo l I
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