Derive an expression for the Helmholtz free energy of a single harmonic oscillator, whose energy levels are εn = (n + 1) hw. En

a simple harmonic oscillator of mass m and angular frequency w is perturbed by additional potential bx3 ,evaluate the second order correction t the ground state energy of the oscillator.

The Helmholtz energy is also called the "work function" because it is equivalent to irreversible work at constant pressure. True False

Suppose an Einstein Solid is in equilibrium with a reservoir at some temperature T. Assume the ground state energy is 0, the solid is composed of N oscillators, and the size of an energy "unit" is e. (a) Find the partition function for a single oscillator in the solid, Z1. Hint: use the general series summation formula 1+ x + x? + x³ + ... = 1/ (1- x) (b) Find an expression for , the average energy per oscillator in the solid, in terms of kT and e. (c) Find the total energy of the solid as a function of T, using the expression from part (b). (d) Suppose e = 2 eV and T = 25°C. What fraction of the oscillators is in the first excited state, compared to the ground state (assuming no degeneracies of energy levels)?

3) Consider the collection of identical harmonic oscillators (as in the Einstein floor). The permitted energies of each oscilator (E = nhf (n=0, 1, 2.0, hf. 2hf and so on. a) Calculate the splitting function of a single harmonic oscitor. What is the splitting function of N oscilator? wwww wwwwww www www b) Obtain the average energy of the T-temperature N oscilator from the split function. c) Calculate the heat capacity of this system and T → 0 ve T → 0 in limits, what is the heat capacity of the system? Are these results in line with the experiment? Why? What's the right theory about that? w w d) Find the Helmholtz free energy of this system. www ww e) which gives the entropy of this system as a function of temperature. ww wd wwww ww

PROBLEM 2. The potential energy of a weakly anharmonic oscillator can be modeled by: m U(x) P²+Bx*, where the last quatric term describes a small anharmonic correction. The energy levels En of the anharmonic oscillator in the first order in the pa- rameter 3 are given by: En = hw 5) + B(n|z*\n). Calculate the energy of the ground state Eo of the anharmonic oscillator.

In the canonical ensemble, we control the variables T, p, and N, and the fundamental function is the Gibbs free energy (G). But if we control T, p, and μ, then we will have a different fundamental function, Z (This is the case for cells that often regulate their temperature, pressure, and chemical potentials to maintain equilibrium). Which of the below options should the Z function equal? H - TS - μN H + TS + μN H + TS - μN G + μN F - pV - μN -H + TS + μN

Let's consider a harmonic oscillator. The total energy of this oscillator is given by E=(p²/2m) +(½)kx?. A) For constant energy E, graph the energies in the range E to E + dE, the allowed region in the classical phase space (p-x plane) of the oscillator. B) For k = 6.0 N / m, m = 3.0 kg and the maximum amplitude of the oscillator xmax =2.3 m For the region with energies equal to or less than E, the oscillator number of states that can be entered D(E).

a) A particle is placed in the well with an energy E< Uo. Sketch the first three energy levels, and make sure to label each one. V(x) U. +2(x) L X X

The radial function of a particle in a central potential is give by wave [ - r R(r) = A-exp where A is the normalization constant and a is positive constant еxp а 2a of suitable dimensions. If ya is the most probable distance of the particle from the force center, the value of y is