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- 1. A particle of m moves in the attractive central potential: V(r) = ax6, where a is a constant and the normalized trial wavefunction is (r) = Ae-bx². (a) Compute the normalization constant A. (b) Calculate the ground state energy: E(b) = = (Y(x)|H|Y(x)) (4(x)|4(x)) (c) Minimize (E) to get the value of b (d) Estimate an upper bound to the energy of the lowest state3. Calculate the reflection coefficient R for the finite potential well and confirm that R+T=1. (4. Find the points of maximum and minimum probability density for the nth state of a particle in a one- dimensional box. Check your result for the n=2 state.
- Q2) Consider the potential well 0, |x| < a 00, otherwisw V(x) = Find the corresponding wave functions and eigenvalues.3 Consider a particle, of mass m, in a box (infinite square well) with walls at x = 0 and x = a. Initially the particle had a constant wave function in the left half of the box (I e 0 to a/2). Work out the probability that energy measurement will yield the ground state energy (I e n=1, lowest energy).7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -
- 1. A symmetric infinite well has the potential energy V(x) = { |x| a a. Write the corresponding Schrodinger equation and its solutions b. Write the boundary conditions c. Obtain the eigenfunctions and eigenvalues4. For the potential if 0 a. calculate the product of the uncertainties O„0, for the second excited state V3(x) : sin a a1) Make use of a translation operator and prove Bloch's theorem in the form : y (F+R)=e¹ky (7). An alternative equivalent form for Bloch's theorem is that the wavefunction has the form 7)=eku (F) where u (F) is lattice periodic. By substituting this into the Schrodinger equation explain the origin of energy bands.