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

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Unformatted text preview: Physics 731 Assignment #6, Solutions 1. For the Hamiltonian H = S z , the Heisenberg equations of motion for the spin operators are S x = 1 i h [ S x ,H ] =- S y , S y = 1 i h [ S y ,H ] = S 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 t- S y (0) sin t, S y = S y (0) cos t + S x (0) sin t. 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 | i = e- ip (0) a/ h | i . For the harmonic oscillator, the position operator in the Heisenberg picture is given by x ( t ) = x (0) cos t + p (0) m sin t. The expectation value of x is then given by h x ( t ) i = h | e ip (0) a/ h x ( t ) e- ip (0) a/ h | i = h | e ip (0) a/ h ( x (0) cos t + p (0) m sin t ) e- ip (0) a/ h | i . 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 h x ( t ) i = a cos t....
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HW6sol - Physics 731 Assignment #6, Solutions 1. For the...

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