1. A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. For all parts of this question, suppose we have an initial state vector |(t = 0)) = E₂), the 2nd energy eigenstate. (This is also called "the 1st excited state", since E₁ is the lowest state or "ground state".) a) At t = 0, what is the expectation value of the energy? (Please answer in terms of m, L, and constants of nature such as л and h.) Find the state vector at time t. What are the possible outcome(s) of an energy measurement now, with what probability(ies)? What is the expectation value of energy at time t? Hint: You do not need any wavefunctions for this part. b) Find the position probability density function at time t: [(x, t)|². Use this to compute (x(t)) for the above state. Is it time dependent? Is it physically reasonable? Why? c) Compute (p(t)). Does your answer make physical sense? Why?
1. A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. For all parts of this question, suppose we have an initial state vector |(t = 0)) = E₂), the 2nd energy eigenstate. (This is also called "the 1st excited state", since E₁ is the lowest state or "ground state".) a) At t = 0, what is the expectation value of the energy? (Please answer in terms of m, L, and constants of nature such as л and h.) Find the state vector at time t. What are the possible outcome(s) of an energy measurement now, with what probability(ies)? What is the expectation value of energy at time t? Hint: You do not need any wavefunctions for this part. b) Find the position probability density function at time t: [(x, t)|². Use this to compute (x(t)) for the above state. Is it time dependent? Is it physically reasonable? Why? c) Compute (p(t)). Does your answer make physical sense? Why?
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