Physics 137A: Quantum Mechanics I, Spring 2007Midterm I 2/22/07Directions: The allotted time is 80 minutes. Two sides of your own notes are allowed. Nobooks or calculators are allowed, and please ask for help only if a question’s meaning is unclear.1. (30 points) Consider the lowest two eigenstates of the infinite square well of width a,V (x) =∞ if x < 0 or x > a0 if 0 ≤ x ≤ a(1)whose wavefunctions areψ1=r2asin(πx/a), ψ2=r2asin(2πx/a). (2)(a) Sketch the probability densities in space from these two wavefunctions and indicate themaxima, i.e., where the particle is most likely to be found in each.(b) What is the expected value of the total energy in the state ψ2? What is the variance of anenergy measurem ent?(c) Write the time evolution ψ(x, t) of a system that starts at t = 0 in the stateψ(x, 0) =1√2(ψ1(x) + iψ2(x)) . (3)2. (40 points) Consider a particle in the infinite square well again. Let the square well potentialbe 0 from −a/2 to a/2 now, and let the particle start at t = 0 in the stateψ(x) = A(a/2 − |x|). (4)(a) Normalize the wavefunction by finding A (choose A real and positive).(b) Compute the inner products hψ1|ψi, hψ2|ψi, wherehf|gi =Z∞−∞f∗(x)g(x) dx. (5)(c) What is the probability that a measurement of energy in this state gives the value E1? (i.e.,the energy of the lowest-energy eigenstate)(d) What is the expectation value of the energy at t = 0? Make sure your answer is physicallyconsistent.(e) Is the expectation value of the energy constant in time in this state? Why or why not?3. (a) (20 points) Find a scattering energy eigenstate ψ with given energy E > 0 of the potentialV (x) = αδ(x), where α > 0, that is even: ψ(x) = ψ(−x). Do not worry about normalization sincethis should not be a normalizable state.(b) (10 points) Prove that any energy eigenstate that is nondegenerate and bound (normalizable)can be chosen to have a real wavefunction (i.e., so that the imaginary part is everywhere zero). Tosay that an energy eigenstate is nondegenerate means that any two normalized wavefunctions withthat energy are related by an overall phase, ψ1= eiθψ2, where θ is