**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, < p _{x} >,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, ∆p _{x} , 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 E _{0} . 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, E**

_{0}. (Your answer should be a function of α and**k.)**

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