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This 14 page Class Notes was uploaded by Citlalli Sauer 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 21 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 4 Outline Eigenbits Q does matrix with orthogonal vectors imply U ze eigenvalue problem Section 18 Hermitian matrix eigenvalues Section 18 Diagonalisation Section 18 an eigen example Section 18 Hermitian matrix fundamentals 1 Eigenvalues of a Hermitian matrix operator are real 2 Eigenvectors of Hermitian matrices operators are orthogonal if they correspond to distinct eigenvalues 3 Eigenvectors of Hermitian matrices operators span the space ie matrix can be diagonalised o Firstly let Qwgt wwgt Where Q QT and dot away w 2w www and taking ajoint comma www o subtracting 0 www wwlw w wwlw 0 thus 0 w w so w of ie real asreqd Hermitian fundamentals 2 Eigenvalues real Eigenvectors orthogonal and span Secondly assuming non degenerate wi s see Shankar Thm 10 for proof including degeneracy let ade and Qwjgt wjwjgt then ltwj9wigt w ltwjlwigt and Wil lw Wjltwilegt adjoint of HHS eqn wj 2wigt w w wi gives 0 Wz W wjlwd For 239 j as before wi w re proved Eigenvalues real For 239 75 j then ltw wjgt 0 Eigenvectors orthogonal Hermitian fundamentals 3 o Eigenvalues real Eigenvectors orthogonal and span a Thirdly again assuming non degenerate wi s o eigenvectors span the space use them as basis a Remembering that de ned S217 j a means that wiltwiwigt wi o and also that 2 wi 2wjgt wjltwiwjgt 0 a Normal matrices Nl7 N 0 have eigenvectors that span the space and are thus diagonalisable Unitary matrices and Diagonalisation 1 Eigenvalues of a Unitary matrix Ul U 1 are complex numbers of modulus 1 ie a E 81 2 Eigenvectors of a Unitary matrix are orthogonal again if they correspond to distinct eigenvalues 3 If 2 is Hermitian there exists a Unitary matrix U built from the eigenvectors of 9 such that U lQU is diagonal with elements eigenvalues 0 Really is a rotation transform U o unitary transform preserves trace and det eg TrUl 2U TrQ Example Diagonalisation Example Diagonalisation contd Example Diagonalisation contd Eg Classical Eigenproblem Sliaiikar example 186 coupled oscillator example Given initial positions 51310 51320 want 51312 51322 2k 0 d21 k d22 k x1 x2 and 511 a2 m dt2 m m dt2 m these Classical Newtons laws can be written as 9 a lttgtgt Blast i1 1e m Q72 m 372 Where Q has real symmetric elements thus Hermitiem Eigenproblem contd 1 a Choice of basis as unity displacement of each mass 0 where Q was represented in this basis 1 0 1 1gt 7 l2gt SO lxtgt 9311gt9322gt 0 1 5132 a Want a basis in which 9 is diagonal om w 1gt and mm w IHgt m 0 Solving eigenproblem w 1 and can m 1 1 d Hgt 1 1 an 1 1 a which span space x11gt x22gt x11gt xHIIgt Eigenproblem contd 2 Given x11gt xHIIgt we can rewrite the original eqn as it w 0 Q7 0 51511 w 51311 ie it a xj 0 which has solns 51312 51310 COS cult 51310 cosw1tlgt 33110 coswutllgt cosw1t IIIgtltII U0gtCOSCUHt Remember Unitary transform preserves inner products ltIx0gt1 1 9310 1 1 2 Eigenproblem contd 33 a 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 0080 15 If la0gtHgt then Htgt 0 important for quantum if starts in eigenmode it stays a if in a linear combination of modes population evolves 0 Coming soon Simultaneous Diagonalisation If 9 and A are two Hermitian operators that commute S2 A 0 then there is a basis of common eigenvectors that diagonalises them both Assuming non degeneracy let w w gt and AQwigt w Aw gt QAwigt Amour w Aw gt So Aw gt is also an eigenvector of Q with eigenvalue wi BUT wi eigenvalue has one eigenvector up to a scale Aw gt Aiwigt thus QAwigt w A wigt So every eigenvector of Q is an eigenvector of A and thus wi diagonalises both 9 and A Function of Operators o In analogy with a power series expansion of functions f 93 Zn anxquot a Can de ne equivalent functions of operators to be 00 0 2 an n Q fQ nE0anQ eg 8 nE0 n 1 2 2 0 makes sense if the sum sensibly converges 0 eg choosing the operator in it s eigenbasis our 0 2 0 m 9m e
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