Download Physics 3810, Fall 2008 - Problem Set #1 Hints and more Assignments Quantum Mechanics in PDF only on Docsity! Phys 3810, Fall 2008 Problem Set #1, Hint-o-licious Hints 1. Griffiths, 1.3 (a) Change the integration variable to z = x−a. Use the Gaussian integrals from the back of the book or elsewhere. (b) Make the same change of integration variable (you have no choice) but note now that because of the factor of x in front, the integral picks up extra terms. 2. Griffiths, 1.7 Start with 〈p〉 = −ih̄ ∫ Ψ∗ ∂Ψ ∂x dx and evaluate d〈p〉/dt by moving the derivative inside as a partial time derivative. Use the product rule, note that ∂/∂t commutes with ∂/∂x and use the Schrödinger equation and its complex conjugate to substitute. Use integration by parts to see that all the terms cancel except for the one with ∂V ∂x . 3. Griffiths, 1.17 No real tricks here, just some integrals of polynomials to do. If you have an integral of an odd function over an interval symmetric about the origin, it gives zero. 6. Griffiths, 1.18 Once in a while we need to plug some numbers into formulae... the results can teach us something, as they do here. In (a) set d equal to the De Broglie wavelength λ, we can solve for the temperature T which makes them the same. For electrons with a “separation” equal to the lattice spacing, one finds that the temperature is far above room temperature (unless you live inside the Sun) so that electrons must be treated as a quantum system. Of course, they are a many–body quantum system. You’ll find that sodium atoms at the same spacing must be treated with quantum me- chanics when the temperature is very low, about 3 K. (b) Take the atomic spacing in a gas to be the cube root of the volume per atom: d = ( V N )1/3 and you can get V/N from the ideal gas law (assuming it applies). Equating this d to the De Broglie wavelength you can derive the bound on the temperature given in the problem. For Helium at 1 atm the result is also about 3 K. For H2 in outer space at the given intermolecular spacing the “quantum temperature” is extremely low, so we can treat it classically. Show all this. 4. Griffiths, 2.4 Nothing new here, just some integrals to work out. Be sure that your answers are plausible. For 〈p〉 you can use 〈p〉 = d〈x〉 dt The answer for 〈x2〉 is 〈x2〉 = a2 [ 1 3 − 1 2n2π2 ] 1
