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by: Jamaal McGlynn


Jamaal McGlynn
GPA 3.78


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Class Notes
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This 9 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 19 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 5 Outline More Eigenbits o rewind through Sections 11 to 18 0 an eigen example Section 18 o the delta function Section 110 vector spaces Closure operation ends With a vector also in the space 06gt gt E V Linear dependence 211 aiigt Ogt iff all al 0 ie two orthonormal vectors span 2 D space lti06gt 04gt ZenW Zltilagtligt E Z igtltilo gt 139 vector components via inner product al linear operators Assume linear transformation gt Qagt stays in V simply comes clown to knowing how the basis transforms Qagt Z allom 2 am De ne ME s jzquotgt E llj Project amount of in agt Via Pioagt aiigt Projection operator completeness I IP2 Eigenproblem Each linear operator has eigenbits Qagt wagt search for the singular solution 9 wIagt Ogt Herrnitian operators are our favourites Qdag Q Eigenvalues of a Herrnitian operator are real Eigenvectors of Herrnitian operators are orthogonal Eigenvectors of Herrnitian operators span the space Question how to prove part 3 i reckon diag If S is Herrnitian there exists a Unitary matrix U built from the eigenvectors of S such that U TQU is diagonal ie we change to a new basis O O 1 OT can always oliag rnatrices if normal N l N 0 Illustrative Eigenproblem Shankar example 186 coupled oscillator example Given initial positions 9310 9320 want 93115 93215 d21 d22 k x r x x dt2 m 1 m 2 dt2 m 1 m 2 these classical Newtons laws can be written as and 2k k a a gquot gk 1 5 tgt9xtgt 952 a R 952 Where Q has real symmetric elements thus Hermitian Eigenproblem contd 1 0 Choice of basis as unity displacement of each mass 0 where Q was represented in this basis 1 O 3 31 1gt 7 2gt SO xtgt 9311gt9322gt 0 1 2 0 Want a basis in which 9 is diagonal om w 1gt and mm w IHgt m 0 Solving eigenproblern w Mg and can I m 1 1 d Hgt 1 1 an 1 1 c which span space x11gt x22gt 931Igt 93HIIgt Eigenproblem contd 2 Given x11gtxullgt we can rewrite the original eqn as 2 O 1quot 1 57511 w1 9311 ie it twist O which has solns 93115 9310Cosw1t 9310 oosw1t1gt 93110 ooswutIIgt oosw1t IIgtltIIxOgtCostt Remember Unitary transform preserves inner products 1 1 m0 ltIxlt0gtgt x O Eigenproblem contd 33 0 So really we just solved for the normal modes 0 If the system starts off in either of the modes it stays 1 oosw1t If x0gt1gt then Wt E COSW t If 93OgtHgt then Htgt 0 important for quantum if starts in eigenmode it stays o if in a linear combination of modes population evolves 0 Coming soon


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