Problem 1
Consider the wave function :
Ψ(x,t) = Ae−λ|x|e−iωt (1)
where A, λ and ω are positive real numbers.
1. Normalise Ψ
2. Determine the expectation values of x and x2
3. Find the standard deviation of x.Sketch the graph of Ψ as a function of x, and mark the points (⟨x⟩+σ) and (⟨x⟩−σ) to illustrate the sense in which σ represent the "spread" of x. What is the probability that the particle would be found outside this range?
Problem 2 (15 marks)
At the time t = 0, the particle waver function is represented by :
Ax/a if 0 ≤ x ≤ a
Ψ(x,0) = A(b − x)/(b − a) if a ≤ x ≤ b (2)
0 otherwise
where A, a, and b are constants.
1. Normalise Ψ, that is A in terms of a and b.
2. Sketch Ψ(x,0) as a function of x.
3. Where is the particle most likely to be found at t = 0?
4. what is the probability of finding the particle to the left of a? Check your results in the limiting cases when b = a and b = 2a.
5. What is the expectation value of x?
