852HW2_Solutions - PHYS852 Quantum Mechanics II, Spring...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK ASSIGNMENT 2: Solutions 1. Consider the most general normalized spin-1/2 state: | ) = c + | + ) + c- |) , where S z |) = planckover2pi1 2 |) . a.) Compute ( S x ) , ( S y ) and ( S z ) . b.) Compute the variances S x , S y , and S z . c.) Prove that S x = planckover2pi1 2 | c 2 + c 2- | , S y = planckover2pi1 2 | c 2 + + c 2- | and S z = planckover2pi1 | c + || c- | . Hint: Use the fact that ( | c + | 2 + | c- | 2 ) 2 = 1 2 = 1. Answer: a.) ( S x ) = planckover2pi1 2 ( c * + , c *- ) parenleftbigg 0 1 1 0 parenrightbiggparenleftbigg c + c- parenrightbigg = planckover2pi1 2 ( c * + , c *- ) parenleftbigg c- c + parenrightbigg = planckover2pi1 2 ( c * + c- + c *- c + ) ( S y ) = planckover2pi1 2 ( c * + , c *- ) parenleftbigg i i parenrightbiggparenleftbigg c + c- parenrightbigg = planckover2pi1 2 ( c * + , c *- ) parenleftbigg ic- ic + parenrightbigg = planckover2pi1 2 i ( c * + c- c *- c + ) ( S z ) = planckover2pi1 2 ( c * + , c *- ) parenleftbigg 1 1 parenrightbiggparenleftbigg c + c- parenrightbigg = planckover2pi1 2 ( c * + , c *- ) parenleftbigg c + c- parenrightbigg = planckover2pi1 2 ( c * + c + c *- c- ) b.) We know that ( S 2 x ) = ( S 2 y ) = ( S 2 z ) = planckover2pi1 2 4 . So that S x = radicalbig ( S 2 x ) ( S x ) 2 = planckover2pi1 2 radicalBig 1 ( c * + c- + c *- c + ) 2 S y = radicalBig ( S 2 y ) ( S y ) 2 = planckover2pi1 2 radicalBig 1 + ( c * + c- c *- c + ) 2 S z = radicalbig ( S 2 z ) ( S z ) 2 = planckover2pi1 2 radicalBig 1 ( c * + c + c *- c- ) 2 c.) With ( c * + c + + c *- c- ) 2 = 1 2 = 1 we can write S x as S x = planckover2pi1 2 radicalBig ( c * + c + + c *- c- ) 2 ( c * + c- + c *- c + ) 2 = planckover2pi1 2 radicalBig | c + | 4 + 2 | c + | 2 | c- | 2 + | c- | 4 ( c * + ) 2 c 2- 2 | c + | 2 | c- | 2 ( c *- ) 2 c 2 + = planckover2pi1 2 radicalBig ( c * + ) 2 c 2 + ( c * + ) 2 c 2- ( c *- ) 2 c 2 + + ( c *- ) 2 c 2- = planckover2pi1 2 radicalBig (( c * + ) 2 ( c *- ) 2 )( c 2 + c 2- ) = planckover2pi1 2 | c 2 + c 2- | Similarly, we obtain S y = planckover2pi1 2 radicalBig ( c * + ) 2 c 2 + + 2 c * + c *- c + c- + ( c *- ) 2 c 2- + ( c * + ) 2 c 2- 2 c * + c *- c + c- + ( c *- ) 2 c 2 + = planckover2pi1 2 radicalBig ( c * + ) 2 c 2 + + ( c *- ) 2 c 2- + ( c * + ) 2 c 2- + ( c *- ) 2 c 2 + = planckover2pi1 2 | c 2 + + c 2- | 1 and S z = planckover2pi1 2 radicalbig ( | c + | 2 + | c- | 2 ) 2 ( | c + | 2 | c- | 2 ) 2 = planckover2pi1 2 radicalbig 4 | c + | 2 | c- | 2 = planckover2pi1 | c + || c- | 2 2. The unitary rotation operator for spin Hilbert space is U R ( vector ) = e- i planckover2pi1 vector vector S . The component of spin along the direction given by , in spherical polar coordinates is found by taking S z and rotating first by about the y-axis, then by about the z-axis, giving S = U R (...
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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852HW2_Solutions - PHYS852 Quantum Mechanics II, Spring...

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