INDEPENDENT and NON-INTERACTING harmonic oscillators

Calculate the canonical ensemble partition function for a collection of INDEPENDENT and NON-INTERACTING harmonic oscillators. Assume the energy spacing between adjacent states in an oscillator is LARGE compared to kT. Calculate in detail; do not skip steps and state all assumptions/approximations.

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The energy levels of a harmonic oscillator are given by:

E_k = (k + 1/2)h-bar omega. (1)

where k>= 0 is an integer. In the following I will denote the state of an oscillator by this integer.

Consider the system of N oscillators. To specify the states of this system you have to specify in which state each of the oscillators is. So, you need N integers to do that.

If oscillator number 1 is in state k1, oscillator number 2 is in state k2, ..., oscillator number p is in state kp then the total energy of this state is:

E_{k1,k2,...kN} = E_k1 + E_k2 + ... E_kN (2)

The partition function of a system of N noninteracting oscillators is thus given by:

Suppose we have two blocks composed of 6 harmonicoscillators each and have 3 quanta in each initially. After bringing them together, what is the probability of having all the energy move to one of the blocks?

In Exercises 21?28, consider harmonicoscillators with mass in, spring constant k, and damping coefficient b. (The values of these parameters match up with those in Exercises 13?20). For the values specified,
(a) find the general solution of the second-order equation that models the motion of the oscillator;
(b) find the parti

A system is composed of N one dimensional classical oscillators. Assume that the potential for the oscillators contains a small quartic "anharmonic" term
V(x) = (m*Ω^2)/2 + a*x^4
Where a*(x^4) <<< KB T and (x^4) = average value
Calculate the average energy per oscillator to the first order in a

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As per the attachment, I understand what formula needs to be used, but I am unsure of what to do from this point.
See attached file for full problem description.

5. Let the function ... be analytic in a domain D that does not include the origin ...
13. ... state why the functions ... are harmonic in D and why ... is in face, a harmonic conjugate
11. ... Why must this satisfy Laplace\'s equation?
Please see attachment for complete questions. Thanks.

Consider the model of an ideal gas, where every molecule has an additional vibrational degree of freedom, which can be described by an oscillator potential. Since there is no interaction between the molecules (apart from a very weak interaction which allows a thermal equilibrium to arise) the oscillators can be regarded as decou

A tube of air is open at only one end and has a length of 1.5 m. This tube sustains a standing wave at its third harmonic. What is the distance between one node and the adjacent antinode?