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  PHY 851, HW 5 due Friday 10/10 1. (a) Prove the Baker-Campbell-Hausdorff (BCH) formula where  A  and  B  arelinear operators e A Be − A =  B  + [ A,B ] + 12! [ A, [ A,B ]] + 13! [ A, [ A, [ A,B ]]] + ···  (1)Hint: Consider the quantity  F  ( λ ) =  e λA Be − λA and expand it as a Taylorseries in the parameter  λ .2. In class, we used the BCH formula to evaluate the Heisenberg picture spinoperators given the Hamiltonian  H   =  ωS  z . Another method, which is oftentimes more straightforward, is to evaluate the commutators in the HeisenbergEquations of Motion (EOM) and solve the resulting differential equations. ddtS  i ( t ) = 1 i   [ S  i ( t ) ,H  ] = 1 i   [ S  i ( t ) ,ωS  z ] .  (2)Do this for  i  =  x,y  and obtain a set of 2 coupled first-order differentialequations. Uncouple them and integrate by forming appropriate linear com-binations. Verify that your results for  S  x,y ( t ) agree with the results obtainedin class (or in the book).3. (a) Solve the Schr¨odinger equation for the time-evolution operator in thecase of a time-dependent Hamiltonian i    ddt  U  ( t, 0) =  H  ( t )  U  ( t, 0) ,  (3)where [ H  ( t  ) ,H  ( t )] = 0. Hint: Divide the finite interval into infinitesimalslices  δt  =  tN   where  N   → ∞ . With the notation  t n  =  nδt , use the group com-position property to write  U  ( t, 0) =  U  ( t N  ,t N  − 1 ) ···U  ( t n ,t n − 1 ) ···U  ( t 1 , 0).Note the Hamiltonian is approximately time-independent on each infinites-imal slice, so that the “usual” expression  U  ( t n ,t n − 1 )  ≈  exp[ − ( i/   ) δtH  ( t n )]can be used.1  (b) Now consider a spin-1/2 particle (say, an electron) in a spatially uni-form but time-dependent magnetic field,  B  =  B ( t )    z  . Solve the Heisenbergequations of motion to find the time-dependent spin operators  S  x ( t ) and S  y ( t ).4. In quantum information theory, the work of a quantum computer consistsof transformations through mutually orthogonal quantum states. Then, theminimal time necessary for such a transformation determines the processingrate of the computer. It is clear that a single stationary energy eigenstate | ψ n ( t )   will never become orthogonal to  | ψ n (0)   since they differ only by thephase exp[ − i/   E  n t ].(a) Take the initial state as a superposition of two normalized energy eigen-kets with energies  E  0  = 0 and  E  1  >  0. Find the initial superpositionthat will evolve into an orthogonal state where  C  ( t )  ≡  ψ (0) | ψ ( t )   = 0,and determine the minimal time  τ   required to do that. Express  τ   interms of fundamental constants and the expectation value of   H   in theinitial state.(b) Take the initial state as a superposition with equal weights of threenormalized energy eigenkets with energies  E  0  = 0,  E  2  > E  1  >  0. Findthe sets of energies which will guarantee  C  ( t ) = 0 and determine theminimal time  τ   to do that.5. Sakurai 2.146. Sakurai 2.192
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