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1) Consider the following histogram of values 8 6 4 2 2 3 4 5 6 7 8 a) Calculate (x), (x2), and 02 b) One value is added to the data set and increases O such that 0' = 1.2 o. What possible values could such a point have? 2) A particle has an initial wave function given by P(x, 0) = else a. Normalize the wavefunction. b. Determine (x), (x2), (p), and (p2). Show your work (do not simply write down the answer from e.g. Mathematica). c. Determine ox and op. Does their product obey the Heisenberg Uncertainty Principle? 3) A particle in an infinite square well has an initial wave function given by the one from question 2) a) Determine through the following i. Determine the Cn's; that is, the coefficients of the projection of the initial wave function onto the infinite square well basis eigenfunctions (iswbe). You'll need to write an infinite sum in closed form for this part. ii. Write down P(x, t) as a linear combination of iswbe's, with the appropriate time factor included. b) If you were to measure the energy of the particle, with what probability would you measure E1? E2? E3? (Here, Ei represents the energy of the ith state of the infinite square well) c) What is (E), the expectation value of the energy? 4) Find (x(t)) for a particle in a harmonic oscillator potential that begins at time C = 0 in the superposition state 4

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