Problem Set #3 Hints for Phys 3810, Spring 2009, Assignments of Quantum Mechanics

Download Problem Set #3 Hints for Phys 3810, Spring 2009 and more Assignments Quantum Mechanics in PDF only on Docsity! Phys 3810, Spring 2009 Problem Set #3, Hint-o-licious Hints 1. Griffiths, 2.29 Actually, this one isn’t so bad. You just need to go through the finite–well derivation in the book and make the appropriate changes for the asymmetric state. The main results you should get to are the results of applying the boundary conditions: −κ = ` cot(`a) and, using the same definitions of z and z0, the new condition for finding the energies (graphically, perhaps) − cot(z) = √ (z0/z)2 − 1 which is not guaranteed to have a root since the right side is positive and the left side starts off being negative for small z. 2. Griffiths, 3.7 3. Griffiths, 3.12 Some steps: Start with 〈x〉 = ∫ ∞ −∞ Ψ(x, t)∗xΨ(x, t) dx and substitute (3.55), Ψ(x, t) = 1√ 2πh̄ ∫ ∞ −∞ eipx/h̄Φ(p, t) dp Watch for complex conjugates and be sure that the dummy variable of p integration is different in the two integrals. When you do this you have a big integral over three variables: x, p and p′. Do what Griffiths says, use xeipx/h̄ = h̄ i d dp eipx/h̄ and then do an integration by parts. Be careful with with variable names to know what the derivatives are acting on. Use (and review) the result of problem 2.26, ∫ ∞ −∞ dx eiqx = 2πδ(q) Use this for one of the integrations, but as usual be careful with the variables. The next integration collapses the delta function and gives the desired result. 4. Griffiths, 3.23 As discussed in class, from considering the action of this H on the state |1〉 and |2〉, one can show that H = ( E E E −E ) With all of the √ 2’s flying around, the normalization is a little funky. 1