How many ways are there of arranging 3 quanta of energy among 3 harmonic oscillators?
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How many ways are there of arranging 3 quanta of energy among 3 harmonic oscillators? 20...
[20 pts] It can be shown that the multiplicity of the macrostate with r quanta of energy in an Einstein solid with n oscillators is (a) For an Einstein solid with each of the following values of n and r, list all of the possible microstates, count them, and verify formula E. 1 above. (a-1) n 3,r 5 (a-2) n 3,r-7 (a-3) n 4, r 4 (a-4) n 1,r anything (a-5) n anything,1 (b) Compute the multiplicity of an Einstein...
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...
(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...
(6) Here is the Hamiltonian for two coupled harmonic oscillators oriented along ti same axis: 22 22 2mozz + 3.2) + -k(zỉ + 2) + Ariza, (a) What are the energy levels for the two oscillators when there is no coupling, or i = 0 (b) Now include the coupling. What change of variables can be made to solve this syste exactly? (C) What are the energy levels for the system? (d) How would one construct the raising and lowering...
The atoms in a nickel crystal vibrate as harmonic oscillators with an angular frequency of 2.3 x 1013 rad/s. The mass of a nickel atom is 9.75 x 10-26 kg. What is the difference in energy between adjacent vibrational energy levels of nickel? (h = 6.626 x 10-34 J·s, 1 eV = 1.60 x 10-19 J).
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....
How many ways are there to distribute 8 identical apples, 6 oranges, and 7 pears among 3 different people.. Without restriction? With each person getting at least one pair? Please explain this well, and don't write it out in chicken scratch.
6. The energy levels of a harmonic oscillator with angular frequency w are given by 2 (a) Suppose that a system of N almost independent oscillators has total energy E^Nhw 2 Mhw. Show that the number of states with exactly this energy equals the number of ways of distributing M identical objects among N compartments and that this number 1S MI(N 1) Hint: Consider the number of distinct arrangements of a set of M objects and N -1 partitions (b)...
You want to distribute 100 gold coins among 20 different people. How many ways can he do this if the gold coins are all... (g) identical but instead are distributed into 20 identical piggy banks such that no piggy bank is empty. (h) different and each person gets exactly five coins. (i) identical and each person gets exactly five coins. (j) different and the Leader gets ten pieces while the other people can get any number of pieces
Consider the word "TUMBARUMBA" a)How many words can be made by arranging all 10 letters? b)How many of these words included TU as a subword?