Question:
At time t=0 a particle is described by the one-dimensional wave function
where k and α are real positive constants.
(a) State Born’s rule in the context of one-dimensional wave mechanics and explain why this rule leads to the requirement for wave functions to be normalized. Verify that the wave function Ψ(x, 0) in Equation 1 is normalized.
(b) Write down the sandwich integral rule for the expectation value of momentum. Hence find the expectation value of the momentum, < px >, in the state described by Ψ(x, 0).
(c) Given that 
in the state described by Ψ(x, 0), what is the uncertainty of the momentum, ∆px, in this state?
(d) Suppose that the particle is in a potential energy well with the normalized ground-state energy eigenfunction
and corresponding energy eigenvalue E0. Use the overlap rule to find the probability that a measurement of the particle’s energy at time t=0 will give the ground-state energy, E0. (Your answer should be a function of α and k.)
————————————————————————————————
This solution in pdf format is available for sale for just 28 USD. Please fill in the following form and submit your order to Detailed Solution.
Detailed Solution 