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- Problem 1: This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving 1-x the formula by long division. Prove it by first multiplying both sides of the equation by (1 – x), and then thinking about the right-hand side of the resulting expression. b) Evaluate the partition function for a single harmonic oscillator. Use the result of (a) to simplify your answer as much as possible. c) Use E = - дz to find an expression for the average energy of a single oscillator. z aB Simplify as much as possible. d) What is the total energy of the system of N oscillators at temperature T?Let the system be one mole of argon gas at room temperature and atmospheric pressure. Compute the total energy (kinetic only, neglecting atomic rest energies), entropy, enthalpy, Helmholtz free energy, and Gibbs free energy. Express all answers in SI units.Consider a mass, m, moving under the influence of an effective potential energy -a b T 7-2 U(r) = + " where a and b are positive constants and r is the radial distance from the origin. In this case, U(r) is a 1D potential energy. (c) Expand The function for U(r) for small displacements about the equilibrium point, To. This will be a Taylor series for U(r) in terms of (r-ro)", where n is an integer. Generate the first three nonzero terms in the series.
- (c) Consider a system of two qubits with canonical basis states {|0) , |1)}. Write down an example for a two- qubit density matrix corresponding to a separable pure state and an example for a two-qubit density matrix corresponding to an entangled pure state.Example of numerical instability: Take y′ = −5y, y(0) = 1. We know that the solution should decay to zero as x grows. Using Euler’s method, start with h= 1 and compute y1, y2, y3, y4 to try to approximate y(4). What happened? Now halve the interval. Keep halving the interval and approximating y(4) until the numbers you are getting start to stabilize (that is, until they start going towards zero).Problem 3: Two-level system and density matrice Suppose a 2 x 2 matrix X (not necessarily Hermitian or unitary) is written as X = a000 + a.σ, where ao and ak, k = 1, 2, 3, are numbers, 0o = 1 is the identity matrix and o are the Pauli matrices. (a) How are ao and a related to tr(X) and tr(OX)? Obtain ao and ak in terms of the matrix elements Xij. Assume that ao, ak ER such that X is Hermitian and could be interpreted as a Hamiltonian, what are the eigenvalues of X?
- The Brachistochrone Problem: Show that if the particle is projected withan initial kinetic energy 1/2 m v02 that the brachistochrone is still a cycloidpassing through the two points with a cusp at a height z above the initialpoint given by v02 = 2gz.Problem 2: Average values Prove that, for any system in equilibrium with a reservoir at temperature T, the average 1 дZ value of the energy is Ē = – z дв In Z, where ß = 1/kT. These formulas can be дв extremely useful when you have an explicit formula for the partition function.The partition funetion for the ensemble characterized by constant V, E, and G = µÑ is given to a very good approximation by ø(V, E, µN)=Q(N,V,E)eBHN, where G = µN is the Gibbs energy (µ is the chemical potential and N is the average number of particles). Find an expression for the characteristic thermodynamic function for this ensemble in terms of the partition function ø(V, E, µN).