Show that the internal energy, U, of an Einstein crystal is given by: U 3Nhv 3Nhvexp(-ßhv) 2 1-exp(-ßhv) + Hint: The following identity will avoid you having to use the chain rule: In(1 − @7(x)) = ¯¯(ƒ '(x)) e´ f(x) 1-e d (1)(1 dx
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- How to solve this questionSuppose that ak > 0 for all k e N and E ak < 0. For each of the following, prove that the given series converges. ak ( a ) ΣΕΙ 1+ k3ak (b) Lk=1 1+ ak Vak (c) Ek=1 k < 0o.Show that the minimum cnergy of a simple harmonic oscillator is Fw/2 if ArAp = h/2, where (Ap)² = ((p - (p))?). %3D
- Problem 35-28: What is the ground-state energy in electron volts of 10 noninteracting fermions each with a mass of 5.02E-27 kg in a one- dimensional box of length L 3.90E-10 m ? (Because the quantum number associated with spin can have two values, each spatial = state can hold two fermions.) 0.044 eVA quantum system is described by a wave function (r) being a superposition of two states with different energies E1 and E2: (x) = c191(r)e iEit/h+ c292(x)e¯iE2t/h. where ci = 2icz and the real functions p1(x) and p2(r) have the following properties: vile)dz = ile)dz = 1, "0 = rp(x)T#(x)l& p1(x)92(x)dx% D0. Calculate: 1. Probabilities of measurement of energies E1 and E2 2. Expectation valuc of cnergy (E)Suppose that the ground vibrational state of a molecule is modelled by using the particle-in-a-box wavefunction ψ0 = (2/L)1/2 sin(πx/L) for 0 ≤ x ≤ L and 0 elsewhere. Calculate the Franck–Condon factor for a transition to a vibrational state described by the wavefunction ψ′= (2/L)1/2sin{π(x −L/4)/L} for L/4 ≤ x ≤ 5L/4 and 0 elsewhere.
- compute the accepted value of gProblem 2) Consider the following Maxwell Boltzmann distribution of molecular speeds: P(v) = 4( m 27kBT. mp² v²e 2kgT To calculate average values for say f(v) (function of v) one just integrates f(v) with P(v)dv from zero to infinity = P(v)f(v)dv, where signifies average of f(v). Of course, the distribution should be normalized: P(v)dv=1, (is a requirement for any probability distribution). a) Check the last equation. b) Calculate the average of v. c) Calculate the average of v². d) Calculate from c) the RMS value of the speed. e) Calculate the most probable value of v. f) Square the results of b, d and e and rank them from smallest to the largest value.