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Unformatted text preview: Physics 580 Handout 14 2 November 2010 Quantum Mechanics I webusers.physics.illinois.edu/ ∼ goldbart/ Homework 10 Prof. P. M. Goldbart 3135 (& 2115) ESB University of Illinois 1) Block diagonal matrices, time evolution and conservation laws : Suppose a system is governed by some complicated hamiltonian that we are unable to diagonalise. Can we still make specific statements about its timeevolution? This question will explain how we can. Suppose that you can identify (usually through observing a symmetry) a Hermitian operator A that commutes with the hamiltonian H , i.e. , [ A,H ] = 0. Suppose A to be suﬃciently simple that you are able to find a complete set of kets  a,γ ⟩ that are eigenkets of A with eigenvalue a . The label γ distinguishes Adegenerate kets. a) Show that H is block diagonal in the  a,γ ⟩basis. b) How are the block sizes related to the degeneracies of A ? c) Sketch a typical blockdiagonal matrix ⟨ a ′ ,γ ′  H  a,γ ⟩ . d) Show that the product of two matrices with a common block structure is also a matrix with that block structure. e) Show that the matrix elements of a function f of a matrix R has the same block structure as the matrix R . For a timeindependent hamiltonian H , does the time evolution operator U ( t ) ≡ exp( − iHt/ ¯ h ) have the same block structure as H ? f) Consider a state  ψ ⟩ that happens to be a linear combination of states drawn from the same Amultiplet, i.e. ,  ψ ⟩ = n ( a ) ∑ γ =1 C γ  a,γ ⟩ ....
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This note was uploaded on 01/22/2012 for the course PHYSICS 850 taught by Professor Staff during the Fall '10 term at University of Illinois, Urbana Champaign.
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