Quantum Mechanics Homework Set #8 for PHY 215A, Assignments of Quantum Mechanics

Download Quantum Mechanics Homework Set #8 for PHY 215A and more Assignments Quantum Mechanics in PDF only on Docsity! PHY 215A, Quantum Mechanics: Homework Set #8 Due: Dec. 5, 2007 1. Uncertainty Relations in H.O. Eigenfunctions. 30 points. Using the “energy basis” |n >, and various operators ξ, Π, X, and P, and the ladder operators A and A†, and relations between them, and their matrix representations, obtain < X >, < X2 >, < P >, < P 2 > and hence ∆X∆P for an arbitrary state |n >. Do not use the real space eigenfunctions; they are not needed or wanted in this problem. Discuss in particular the value ∆X∆P for the ground state, and the behavior of this product for large values of n. {Hint: first calculate < ξ >, < ξ2 >, < Π >, < Π2 > and only later substitute in the constants.} 2. Energies/Units: a “Real” Problem. 15 points. An electron m = m e is in a harmonic potential 1 2 kx2 whose value is 5 eV at a point 3 Å from the origin. [These scales might be representative of some electronic state in a molecule or solid.] (a) Find the separation h̄ω of energy levels for this electron. (b) Evaluate ∆X and ∆P for the ground state. There is no new physics here, this problem is to begin getting you accustomed to typical atomic-scale units. So there is emphasis on the correct numerical answers – not significant figures (aim for two), but conversion units and factors of two etc. should be correct. 3. Mixed States. 15 points. A harmonic oscillator is in a state such that a measurement of the energy would yield only 1 2 h̄ω and 3 2 h̄ω, and these with equal probability. What is the largest possible value of < x > in such a state, and what does the corresponding state look like? Describe your reasoning, and show work, of course.