. Classical harmonic oscillatorA single classical harmonic oscillator,p²2m 2with angular frequency w = √K/m is in thermal contact with a heat bath at temperature T.(a)Calculate its canonical partition function,H(p, q):=Q(³) = 1/2Kq²+1/2 dp dq exp{-BH(p, q)},(1)(2)(b)Obtain the average (E) of the energy, the variance ((SE)2) of the energy, the free energy A(T),the entropy S(T) and the heat capacity C(T).(c)Compare your results with what you would expect from the equipartition theorem.(d)Consider now that you have 3N classical harmonic oscillators, each one of them with the sameangular frequency w, which do not interact with each other. Find the canonical partition function Q(3, N),the average (E) of the energy, the variance ((8E)2) of the energy, the free energy A(T, N), the entropyS(T, N) and the heat capacity C(T, N).

Answered: Image /qna-images/answer/050dfe3e-6c85-4113-aaa4-30e03766a1d6.jpg

Answered: Classical harmonic oscillator A single…

Similar questions

Calculate the energy density in the range 650 nm to 655 nm inside a cavity at (a) 25 °C, (b) 3000 °C. For this relatively small range of wavelength it is acceptable to approximate the integral of the energy spectral density ρ(λ,T) between λ1 and λ2 by ρ(λ,T)×(λ2 − λ1).
Consider a particle with the following wave-function: ,xL and L and A are constants. (a) What is the normalization constant A (in terms of L)? (b) What is (in terms of L)? ? (ie, ) (c) What is the probability that the particle will be found within x=0 and L/2?
An atom has energy levels 0, E, 2E, Calculate the canonical partition function of a system composed of noninteracting N atoms if the atoms are localized.
(WF-1) The wave function for an electron moving in ID is given by: y(x) = C(x - ix²) for 0 ≤ x ≤ 1 and zero everywhere else. (a) Find the value of the normalization constant (C) for the wave function. (b) What is the probability of locating the electron between x = 0 and x = 1/4? (c) What is the probability of locating the electron between x = 1/4 and x = 3/4?
∆E ∆t ≥ ħTime is a parameter, not an observable. ∆t is some timescale over which the expectation value of an operator changes. For example, an electron's angular momentum in a hydrogen atom decays from 2p to 1s. These decays are relativistic, however the uncertainty principle is still valid, and we can use it to estimate uncertainties. ∆E doesn't change in time, so when an excited state decays to the ground state (infinite lifetime, so no energy uncertainty), the energy uncertainty has to go somewhere. Usually, it’s in the frequency of a photon giving a width (through E = hν) to the transition line in an spectroscopy experiment. The linewidth of the 2p state in 9Be+ is 19.4 MHz. What is its lifetime? (Note: in the relativistic atom–photon system, the Hamiltonian is independent of time and both energy and its uncertainty are conserved.)
Calculate the transmission probability for quantum-mechanical tunneling in each of the following cases. (a) An electron with an energy deficit of U - E= 0.010 0 eV is incident on a square barrier of width L = 0.100 nm. (b) An electron with an energy deficit of 1.00 eV is incident on the same barrier. (c) An alpha particle (mass 6.64 × 10-27 kg) with an energy deficit of 1.00 MeV is incident on a square barrier of width 1.00 fm. (d) An 8.00-kg bowling ball withan energy deficit of 1.00 J is incident on a square barrier of width 2.00 cm.
2.6 (a), (b) • (a) Show that, if the kinetic energy of a particle with mass m, momentum p p²/2m, the single particle partition function can be written Z1 = V/23, where 1 = partition function for the ideal gas will then be is E = Vh2/2nmkgT is the thermal wavelength. The canonical VN ZN N! 23N • (b) Use Stirling's approximation to show that in the thermodynamic limit the Helmholtz free energy of an ideal gas is A = -NkgT In ( V +1 N23
Solid metals can be modeled as a set of uncoupled harmonic oscillators of the same frequency with energy levels given by En = ħwn n = 0, 1, 2,... where the zero-point energy (the lowest energy state) of each oscillator has been adjusted to zero for simplicity. In this model, the harmonic oscillators represent the motions of the metal atoms relative to one another. The frequency of these oscillators is low so that ħw = = 224 KB and the system vibrational partition function is given by 3N Z ² = la₁ - (1 1 e-0/T). (a) If the system contains one mole of atoms, find the average energy (in J) of this system at T= 172 K. (You can use = BkB.) T (b) What is the absolute entropy (in J/K) for this system? You can use either the Gibbs expression for S, or the system partition function to make this evaluation (they are equivalent, as your reading assignment indicates).
Suppose that the electron in the Figure, having a total energy E of 5.1 eV, approaches a barrier of height Us=6.8 eV and thickness L= 750 pm. (a) What is the approximate probability that the electron will be transmitted through the barrier, to appear (and be detectable) on the other side of the barrier? Energy Us Electron L
Show that the hydrogen wave functionsφ 200 and φ21-1 are normalized. If the integrals required are not in Appendix 3, consult a table of integrals or use computer integration.
Plz don't use chat gpt
The ground state wave function for the quantum mechanical simple harmonic oscillator is of the form, y(x)= A,e-** mo, a = where A, is the normalization factor and a is a constant that depends on the mass and classical frequency of the oscillator. Find the normalization factor in terms of the mass and classical frequency w, The following definite integral should be helpful: 1 2a

SEE MORE QUESTIONS