1. Consider a harmonic oscillator of mass m and angular frequency w. At time t =...

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1. Consider a harmonic oscillator of mass m and angular frequency w. At time t = 0, the state of this oscillator is given by: lW(0)) = E Cnlpn) n where the states |Pn) are stationary states with energies (n +1/2)hw. a. What is the probability P that a measurement of the oscillator's energy performed at an arbitrary time t > 0, will yield a result greater than 2hw? When P = 0, what are the non-zero coefficients Cn? b. From now on, assume that only co and cj are different from zero. normalization condition for | (0)) and the mean value (H) of the energy in terms of co and c1. With the additional requirement (H)= hw, calculate |col? and |C12. Write the c. As the normalized state vector |(0)) is defined only to within a global phase factor, we fix this factor by choosing co real and positive. We set: c1 = |cı|e01. We assume that (H) = huw and that: 1 (X) Calculate 01. d. With (0)) so determined, write |(t)) for t > 0 and calculate the value of 01 at t. Deduce the mean value (X)(t) of the position at t. 1. Consider a harmonic oscillator of mass m and angular frequency w. At time t = 0, the state of this oscillator is given by: lW(0)) = E Cnlpn) n where the states |Pn) are stationary states with energies (n +1/2)hw. a. What is the probability P that a measurement of the oscillator's energy performed at an arbitrary time t > 0, will yield a result greater than 2hw? When P = 0, what are the non-zero coefficients Cn? b. From now on, assume that only co and cj are different from zero. normalization condition for | (0)) and the mean value (H) of the energy in terms of co and c1. With the additional requirement (H)= hw, calculate |col? and |C12. Write the c. As the normalized state vector |(0)) is defined only to within a global phase factor, we fix this factor by choosing co real and positive. We set: c1 = |cı|e01. We assume that (H) = huw and that: 1 (X) Calculate 01. d. With (0)) so determined, write |(t)) for t > 0 and calculate the value of 01 at t. Deduce the mean value (X)(t) of the position at t.