14_580_hw_10 - Physics 580 Quantum Mechanics I Prof P M...

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Physics 580 Handout 14 2 November 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ 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 time-evolution? 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 A -degenerate 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 block-diagonal 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 time-independent hamiltonian H , does the time- evolution operator U ( t ) exp( iHt/ ¯ h ) have the same block structure as H ? f) Consider a state | ψ 0 that happens to be a linear combination of states drawn from the same A -multiplet, i.e. , | ψ 0 = n ( a ) γ =1 C γ | a, γ .

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