Find the specific heat of a canonical ensemble of classical simple harmonic oscillators.
Find the specific heat of a canonical ensemble of classical simple harmonic oscillators.
1: Consider a system of N classical distinguishable one-dimensional harmonic oscillators with frequency w in the microcanonical ensemble. Determine the phase space volume and the corresponding entropy. Please be sure to include the quantum correction h3N. Please note that the Hamiltonian for the system isN mwqi You may also want to note that the volume of a d-dimensional which can be simplified by using r sphere of radius R is given by 1: Consider a system of N classical distinguishable...
Problem 6 2.20 Displacement distribution of random oscillators The displacement x of a classical simple harmonic oscillator as a func- tion of the time t is given by x A cos (wt + φ) where w is the angular frequency of the oscillator, A is its amplitude of oscilla- tion, and ф is an arbitrary constant which can have any value in the range φ<2π. Suppose that one contemplates an ensemble of such oscillators all of which have the same...
PROBLEM 1 5 points] In classical statistical mechanics, the canonical partition function for a single harmonic oscillator is of the form d dp e Δ ΔΊΔ ) is the regulating spatial and momentum resolution cutoffs, which are often Chosen to be at the scale of the atoms (and n) and are important for making entropy dimensionless but they drop out in parts (b) and (c). Moreover, Z factorizes as Z ZzZp with Z. 3 Calculate the partition function and the...
In the canonical ensemble, a computation of the N-particle partition function in a particular ex- ample reveals that ZN = temperature T. What is Cv, the specific heat at constant volume, for this system? aVNT3N, where a is a constant independent of the volume V and
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.
Question no 6.1, statistical physics by Reif Volume 5 Problems 6.1 Phase space of a classical harmonic oscillator The energy of a one-dimensional harmonic oscillator, whose position coordinate is x and whose momentum is p, is given by where the first term on the right is its kinetic and the second term its potential energy. Here m denotes the mass of the osellating particle and a the spring constant of the restoring force acting on the particle. Consider an ensemble...
Consider a canonical ensemble of ? paramagnetic particles of magnetic moment ? in an external magnetic field B and find a. An expression for the average magnetic moment per particle b. The total magnetization. c. Demonstrate the Curie’s law of paramagnetism that predicts that at high temperature the magnetization of the material is proportional to the ratio between the external field and the temperature Consider a canonical ensemble of N paramagnetic particles of magnetic moment μ in an external magnetic...
1. Imagine you have two simple harmonic oscillators. Oscillator 1 is characterized by k1 10 kg s-2 and oscillator 2 by k25 kg s-2. The oscillators are displaced by the (a) If they each oscillate with a 1 kg mass, what is the period for the oscillators to (b) For some initial displacement d, where do the oscillators first come back into same amount in the same direction and are released together (in phase). come back into phase? phase?
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.
(b) For a system of N independent harmonic oscillators at temperature T, all having a common vibrational unit of energy, the partition function is Z = ZN. For large values of N, the system's internal energy is given by U = Ne %3D eBe For large N, calculate the system's heat capacity C. 3. This problem involves a collection of N independent harmonic oscillators, all having a common angular frequency w (such as in an Einstein solid or in the...