Problem #1 (Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and derive the differential form that reveals A as a potential: dA < -SdT – pdV [Eqn 5.20]

Problem 2.21 Suppose a free particle, which is initially localized in the range -a < x < a, is released at time t = 0: А, if -a < х <а, otherwise, (x, 0) = where A and a are positive real constants. 50 Chap. 2 The Time-Independent Schrödinger Equation (a) Determine A, by normalizing V. (b) Determine (k) (Equation 2.86). (c) Comment on the behavior of (k) for very small and very large values of a. How does this relate to the uncertainty principle? *Problem 2.22 A free particle has the initial wave function (x, 0) = Ae ax where A and a are constants (a is real and positive). (a) Normalize (x, 0). (b) Find V(x, t). Hint: Integrals of the form e-(ax?+bx) dx can be handled by "completing the square." Let y = Ja[x+(b/2a)], and note that (ax? + bx) = y? – (b²/4a). Answer: 1/4 e-ax?/[1+(2ihat/m)] 2a Y (x, t) = VI+ (2iħat/m) (c) Find |4(x, t)2. Express your answer in terms of the quantity w Va/[1+ (2hat/m)²]. Sketch |V|? (as a function of x) at t = 0, and again for some very large t.…

6.2 Let the "uniform" ensemble of energy E be defined as the ensemble of all systems of the given type with energy less than E. The equivalence between (6.29) and (6.27) means that we should obtain the same thermodynamic functions from the "uniform" ensemble of energy E as from the microcanonical ensemble of energy E. In particular, the internal energy is E in both ensembles. Explain why this seemingly paradoxical result is true.

For Problem 8.37, how do I find <1/r> within the integral? I think that the exponent function inside of P(r) is actually troublesome to finding what I need to find; however, I am not certain of what's really the correct procedure here.