hw4246 - principle. 6. (20 points) Consider the Hamiltonian...

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Physics 246, Spring 2007 Homework #4 Due in class, Wednesday, February 28, 2007 Feel free to discuss the problems with me and/or each other. Each student must write up his/her own solutions separately. Each problem is worth 10 points unless otherwise indicated. 1. Libo±, problem 6.1, p. 166. 2. Libo±, problem 6.6, p. 170. 3. Libo±, problem 6.11, p. 170. 4. Libo±, problem 6.13, p. 171. 5. (15 points) Consider a Gaussian wave packet of the form given on page 160±. (a) Compute h x i t and h x 2 i t using the explicit form of ψ ( x, t ) and/or P ( x, t ). (b) Show that h p i t and h p 2 i t are time independent. (c) What are Δ x and Δ p at time t ? Relate the results to the uncertainty
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Unformatted text preview: principle. 6. (20 points) Consider the Hamiltonian H = p 2 1 2 m 1 + p 2 2 2 m 2 + V ( x 1-x 2 ) . (a) Show that in general [ H, p 1 ] and [ H, p 2 ] are nonzero. (b) Show that in terms of P = p 1 + p 2 , X = m 1 x 1 + m 2 x 2 m 1 + m 2 (1) p = m 2 p 1-m 1 p 2 m 1 + m 2 , x = x 1-x 2 (2) [ P, X ] = [ p, x ] =-i h are the only nonvanishing commutators among this set. (c) Find H ( P, p, X, x ). (d) What is a complete set of commuting operators for this system? Assume that V has only bound states (no continuum states)....
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