1. (40 points) Consider the lowest two eigenstates of the infinite square well of width a, V(x) = if x<00 x>a if 0≤x≤a w

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1 40 Points Consider The Lowest Two Eigenstates Of The Infinite Square Well Of Width A V X If X 00 X A If 0 X A W 1 (69.79 KiB) Viewed 37 times

1. (40 points) Consider the lowest two eigenstates of the infinite square well of width a, V(x) = if x<00 x>a if 0≤x≤a whose wave functions are 4₁ = √√² sin(m1/a), 4₂ = = √sin (2nx/a). (a) Sketch the probability densities in space for these two wave functions and indicate maxima/minima, i.e, where the particle is most likely and least likely to be found in each. (b) In the state 2, what is the expected value of the total energy? For the same state, what is the variance of an energy measurement? (c) Write the time evolution y(x, t)of a system that starts at t= 0 in the state (x,0) = = [3₁(x) + 4₂(x)]. (d) What are E₁ and E₂? What are the probabilities for measuring ₁ and ₂ at t=0 for the wave function in c. What are <E> and < E² > at t=0 for the wave function in c? What about at later times?