851HW13_09Solutions

851HW13_09Solutions - PHYS851 Quantum Mechanics I, Fall...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 13: Solutions 1. In this problem you will derive the 2 2 matrix representations of the three spin observables from first principles: (a) In the basis {| z ) , | z )} , the matrix representation of S z is of course S z = parenleftbigg ( z | S z | z ) ( z | S z | z ) ( z | S z | z ) ( z | S z | z ) parenrightbigg . (1) Use Eqs. (1) and (2) to find the four matrix elements of S z in the basis of its own eigenstates. From S z | z ) = planckover2pi1 2 | z ) and the orthonormality of the basis, it follows that ( z | S z | z ) = planckover2pi1 2 and ( z | S z | z ) = 0. From S z | z ) = planckover2pi1 2 | z ) and the orthonormality of the basis, it follows that ( z | S z | z ) = 0 and ( z | S z | z ) = planckover2pi1 2 . This gives us S z = parenleftbigg 1 1 parenrightbigg (2) (b) Invert the definitions S + = S x + iS y and S = S x iS y , to express S x and S y in terms of S + and S . Inverting these equations gives S x = 1 2 ( S + + S z ) (3) S y = 1 2 i ( S + S ) (4) (c) Use the equation S | s,m s ) = planckover2pi1 radicalbig s ( s +1) m s ( m s 1) | s,m s 1 ) (5) to find the matrix elements of S + and S in the basis {| z ) , | z ) . This formula gives S + | z ) = 0, S + | z ) = planckover2pi1 | z ) , so that orthonormality gives S + = planckover2pi1 parenleftbigg 0 1 0 0 parenrightbigg (6) Likewise, S | z ) = planckover2pi1 | z ) and S | z ) = 0, so that S = planckover2pi1 parenleftbigg 0 0 1 0 parenrightbigg (7) 1 (d) From your answers to 13.1.b and 13.1.c, derive the matrix representations of S x and S y for spin-1/2. S x = 1 2 ( S + + S ) = planckover2pi1 2 parenleftbigg 0 1 1 0 parenrightbigg (8) S y = 1 2 i ( S + S ) = planckover2pi1 2 i parenleftbigg 1 1 0 parenrightbigg = planckover2pi1 2 parenleftbigg i i parenrightbigg...
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This note was uploaded on 12/04/2011 for the course PHY 7070 taught by Professor Smith during the Spring '11 term at Wisconsin.

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851HW13_09Solutions - PHYS851 Quantum Mechanics I, Fall...

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