Question

Text: Virial Expansions and Perturbations Consider the discrete energy levels of a quantum harmonic oscillator: En=ho(n+1/2) (3.1) Now, consider adding an oscillating discrete perturbation scaled by a parameter a: Î”E=aw (3.2) where n = 0, 1, 2, ..., h is the angular Planck constant and w is the oscillator frequency. a) 4P) For which states is the perturbation the most important? What are the units of a? b) 6P) Assuming that a < 1, show that the first order perturbation theory correction to the Helmholtz free energy for N non-interacting, distinguishable oscillators is given by: F=Na(1-e^(-Î²h)) cos(Ï‰t) (3.3) You may find the following formula useful: (-1)^j cos(x) = âˆ‘(j=0)^(âˆž) (2j)! (3.4) c) 8P) Derive the first order correction to E, the internal energy for the same system. You may find the following relation useful: Î”E = âˆ«(0)^(Î²) a^2 kT^2 dt (3.5)

          Text: Virial Expansions and Perturbations
Consider the discrete energy levels of a quantum harmonic oscillator:
En=ho(n+1/2)
(3.1)
Now, consider adding an oscillating discrete perturbation scaled by a parameter a:
Î”E=aw
(3.2)
where n = 0, 1, 2, ..., h is the angular Planck constant and w is the oscillator frequency.
a) 4P) For which states is the perturbation the most important? What are the units of a?
b) 6P) Assuming that a < 1, show that the first order perturbation theory correction to the Helmholtz free energy for N non-interacting, distinguishable oscillators is given by:
F=Na(1-e^(-Î²h)) cos(Ï‰t)
(3.3)
You may find the following formula useful:
(-1)^j cos(x) = âˆ‘(j=0)^(âˆž) (2j)!
(3.4)
c) 8P) Derive the first order correction to E, the internal energy for the same system. You may find the following relation useful:
Î”E = âˆ«(0)^(Î²) a^2 kT^2 dt
(3.5)

virial expansions and perturbations consider the discrete energy levels of a quantum harmonic oscillator enhon 31 now consider adding an oscillating discrete perturbation scaled by a paramet 26095

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