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# HW6sol - Physics 731 Assignment#6 Solutions 1 For the...

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Physics 731 Assignment #6, Solutions 1. For the Hamiltonian H = ωS z , the Heisenberg equations of motion for the spin operators are ˙ S x = 1 i ¯ h [ S x , H ] = - ωS y , ˙ S y = 1 i ¯ h [ S y , H ] = ωS x , ˙ S z = 1 i ¯ h [ S z , H ] = 0 . Clearly, S z ( t ) = S z (0) . By solving the coupled equations for S x and S y , one easily finds S x = S x (0) cos ωt - S y (0) sin ωt, S y = S y (0) cos ωt + S x (0) sin ωt. 2. For the free particle, H = p 2 / (2 m ) . The x and p operators in the Heisenberg picture obey the equations of motion ˙ x = 1 2 mi ¯ h [ x, p 2 ] = p m , ˙ p = 0 . Therefore, p ( t ) = p (0) , and x ( t ) = x (0) + p (0) t/m , and [ x ( t ) , x (0)] = t m [ p (0) , x (0)] = - i ¯ ht m . 3. We are given | α = e - ip (0) a/ ¯ h | 0 . For the harmonic oscillator, the position operator in the Heisenberg picture is given by x ( t ) = x (0) cos ωt + p (0) sin ωt. The expectation value of x is then given by x ( t ) = 0 | e ip (0) a/ ¯ h x ( t ) e - ip (0) a/ ¯ h | 0 = 0 | e ip (0) a/ ¯ h ( x (0) cos ωt + p (0) sin ωt ) e - ip (0) a/ ¯ h | 0 . Using the relations e ip (0) a/ ¯ h x (0) e - ip (0) a/ ¯ h = x (0) + [ ip (0) a/ ¯ h, x (0)] = x (0) + a, e ip (0) a/ ¯ h p (0) e - ip (0) a/ ¯ h = 0 , we see that x ( t ) = a cos ωt. 4. (a) For | α = c 0 | 0 + c 1 | 1 with | c 0 | 2 + | c 1 | 2 = 1 , we have x = ¯ h 2 2Re( c * 0 c 1 ) , which is clearly maximized for a = b = 1 / 2 .

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HW6sol - Physics 731 Assignment#6 Solutions 1 For the...

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