5.2 Consider a particle moving in three dimensions. Is it possible for the particle to be in a state of definite px and y, i.e., can both its y-coordinate and its momentum in the x direction be known at the same time?
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- 2.9. (a) Solve the integral ...| (dx .dx3N) 3N and use it to determine the "volume" the relevant region of the phase space of an extreme relativistic gas ( = pc) of 3N particles moving in one dimension. Determine, as well, the number of ways of distributing a given energy E among this system of particles and show that, asymptotically, w0 = h³N. (b) Compare the thermodynamics of this system with that of the system considered in Problem 2.8.A particle with mass m is in the state mr2 tiat ¥(x,t) = Ae where A and a are positive real constants. 2h Normalise Y and calculate A.Question related to Quantum Mechanics : Problem 2.41
- 1.16. Establish thermodynamically the formulae v (7)= = S and v (R), V = N. Express the pressure P of an ideal classical gas in terms of the variables and 7, and verify the μl above formulae.(3.8) This question introduces a rather efficient method for calculating the mean and variance of probability distributions. We define the moment generating function M(t) for a random variable x by M(t) = (etx). Show that this definition implies that (x) = M(n) (0), (3.51) (3.52) where M(n) (t) mean (x) = d" M/dt" and further that the M (¹) (0) and the variance σ = = M(2)(0) [M(¹) (0)] 2. Hence show that: - (a) for a single Bernoulli trial, = M(t) pe 1-p; (3.53) (b) for the binomial distribution, M(t) = (pe +1 - p)"; (3.54) (c) for the Poisson distribution, M(t) = em(et-1); (3.55) (d) for the exponential distribution, λ M(t) (3.56) Hence derive the mean and variance in each case and show that they agree with the results derived earlier.3.3 The wave function for a particle is V(x,t) = sin(kx)[i cos(wt/2) + sin(wt/2)], where k and w are constants. (a) Is this particle in a state of definite momentum? If so, the momentum. determine (b) Is this particle in a state of definite energy? If so, determine the energy.
- Needs Complete typed solution with 100 % accuracy.In the pair production process, photon energy gets converted to particle/antiparticle pairs. Imagine a single photon in free space, turning into one electron and one positron (antielectron), each with mass mec?. Assume for simplicity both particles move together with equal momenta in the same direction as the original photon as shown, and the photon disappears. Prove this can't happen!Classical mechanics
- 4.12 A particle of mass m moves in a harmonic oscillator potential. The particle is in the first excited state. (a) Calculate (x) for this particle. (b) Calculate (p) for this particle. (c) Calculate (p²) for this particle. (d) At what positions are you most likely to find the particle? At what positions are you least likely to find it?A neutron collides elastically with an initially stationary deuteron. Find the fraction of the kinetic energy lost by the neutron in a heads on collision; I. II. in scattering at right angles.Question related to Quantum Mechanics : Problem 2.52