The answers are given in the images below. Initially I have just defined a partition function. From there I have written the partition function for a single harmonic oscillator. In the second part, I have used a simple equation for finding the internal energy. I have also put in brackets the equation from where this simple equation is derived from.
(b) For a system of N independent harmonic oscillators at temperature T, all having a common...
please solve 2 problems restriction on the total number of particles 4. For photons, there is no (a) Find out the number of photons per quantum state 8T V 2 dv 4 (b) Find out the partition function, Z. cf. g(v)dv 15 e 1 In z n (1 e hulkT) for a (c) Calculate the internal energy, U single oscillator (d) Calculate the pressure, P. 4. For solids Einstein the vibrational levels given energy are as € (j+h, j 0,1,2,....
3. For diatomic ideal gases at room temperature, find out the change in entropy due to mixing using the following partition functions hv expl2kT T V( h2 Ztranslation rotation vibration h2 hv 1 exp 4. For solids, Einstein the vibrational levels given energy are as hv, j-0,1,2,.. Assuming that the N 2 strongly coupled atoms are +=3 equivalent to 3N simple harmonic independent oscillators, find out the followings (a) Equation for the vibrational energy as a function of temperature (b)...
Classically, the heat capacity of a chain of N 1-dimensional harmonic oscillators equals N k, i.e., it is temperature independent. Explain qualitatively why the heat capacity should decrease as T goes to 0 K.
A system is composed of two harmonic oscillators, each of natural frequency w, and having permissible energies "*1/2)w, where" is any non-negative integer. The total energy of the system is = 'hwo, where" is a positive integer. • For a given energy, how many microstates are available to the system? What is the entropy of the system? • A second system is also composed of two harmonic oscillators, each of natural frequency 2w,. The total energy of this system is...
question no 4.22, statistical physics by Reif Volume 5 4.92 Mean energy of a harmonic oscillator A harmonic oscillator has a mass and spring constant which are such that its classical angular frequency of oscllation is equal to w. In a quantum- mechanical description, such an oscillator is characterized by a set of discrete states having energies En given by The quantum number n which labels these states can here assume all the integral values A particular instance of a...
Consider a simple single quantum particle with the energy levels of the harmonic oscillator En = (n + 1/2)ℏω. This particle is in thermal contact with a reservoir with temperature T. a) Calculate the partition function of this particle. b) Calculate the internal energy of the particle as a function of temperature. Deduce and interpret the state of this energy at low and high temperatures. c) Calculate the specific temperature of this particle at constant pressure.
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥ, = îl + Ĥ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... 1....
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥg = îl + Ħ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... a)...
Problem 1 (Harmonic Oscillators) A mass-damper-spring system is a simple harmonic oscillator whose dynamics is governed by the equation of motion where m is the mass, c is the damping coefficient of the damper, k is the stiffness of the spring, F is the net force applied on the mass, and x is the displacement of the mass from its equilibrium point. In this problem, we focus on a mass-damper-spring system with m = 1 kg, c-4 kg/s, k-3 N/m,...
1. (10 pts) Consider a system of N classical, independent harmonic oscil- lators. In the microcanonical ensemble, calculate Ω(E) and Ω(B) exactly. From them, calculate the entropy S(E, N) and temperature T in the large N limit. 2. (10 points) Consider the same system as in problem 1. Calculate the average energy and entropy starting from the canonical ensemble.