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- 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 -Consider a weakly anharmonic a 1D oscillator with the poten- tial energy m U(x) = w?a² + Ba* 2 Calculate the energy levels in the first order in the small anharmonicity parameter 3 using TIPT and the ladder operators.Two identical particles of spin ½ are moving under the influence of a onedimensional harmonic oscillator potential. Assuming that the two-particle system is in a triplet spin state, find the energy levels, the wave functions, and the degeneracies corresponding to the three lowest states.
- a 4. 00, -Vo, V(z) = 16a 0, Use the WKB approximation to determine the minimum value that V must have in order for this potential to allow for a bound state.Consider an electron trapped in a 20 Å long box whose wavefunction is given by the following linear combination of the particle's n = 2 and n = 3 states: ¥(x,t) =, 2nx - sin ´37x - sin 4 where E, 2ma² a a. Determine if this wavefunction is properly normalized. If not, determine an appropriate value for a normalization constant. b. Show that this is not an eigenfunction to the PitB problem. What are the possible results that could be returned when the energy is measured and what are the probabilities of measuring each of these results?