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:

Consider two coupled identical harmonicoscillators described by the Hamiltonian
H= p1^2+p2^2/2m+1/2mw^2x1^2+1/2mw^2x2^2+gx1x2
1- What is the lowest energy of the system?
2- What is the ground state eigenfunction?
3-What is the energy and the eigenfunction for the first excited state?

If i had two spring coupled to one mass with each end of the springs fixed.
|---spring---{mass}--spring----|
neglecting friction and gravity, if the mass is displaced horizontally. would it oscillate forever? if not how can u get the maximum amplitude of oscillation for a given period of time before it

Undamped coupled oscillators:
Two identical undamped oscillators, A and B, each of mass m and natural frequency ω_0 are coupled in such a way that the coupling force exerted on A is σm(d^2x_b/dt^2), and the coupling force exerted on B is σm(d^2x_a/dt^2), where σ is a coupling constant of magnitude less tha

3 Oscillators of equal mass are coupled such that the potential energy of the system is given by:
U = 1/2[k1*(x1^2+x3^2)+k2*x2^2 +k3(x1*x2+x2*x3)]
and k3=(2*k1*k2)^1/2
Find the eigen frequencies by solving the secular equation. What does the zero -frequency mode mean?

Please see the attached file for the fully formatted problems.
Let h 2 C2(R3) be harmonic (h = 0). Use Green's identity for
....
to show that ...is independent of the value of R. Then deduce the mean value theorem
....
Now what can you say if limx!1 h(x) = 0?

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?

See attached file for full problem description.
1. Driven harmoic oscillators
Suppose that a driven harmoic oscillators with belta = 1/3w0 is driven with force F = F0cos(wt) with driven frequency w = 1/3w0.
Find the amplitude 'D' and phase a of the motino x(t) = Dcos(wt- a). expressing them purely in terms of F0, k and nume

Consider the harmonic oscillator, for which the general solution is
x(t) = A cos(wt) + B sin(wt)
1. Express the energy in terms of A and B and show it is time independent.
2. Choose A and B such that x(0)=x1 and x(T)=x2.
3. Write down the energy in terms of x1, x2 and T.
4. Calculate the action S for the trajectory conn