te at energy E0 = 0 meV. The temperature of the system is fixed at 290 K. mificant figures, what is the ratio P3/P, where P is the probability of finding a singli is the probability of finding 3 particles in this state?
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- Calculate the energy separations in joules, kilojoules per mole, and electronvolts, and as a wavenumber, between the levels (a) n = 3 and n = 1, (b) n = 7 and n = 6 of an electron in a one-dimensional nanoparticle modelled by a box of length 1.5 nm.Consider an electron in the spin-state σ = [+ [α + ẞ]. (a) Is σ an eigenfunction of Ŝ²? If yes, what is the eigenvalue and what is the corresponding spin magnitude? If no, what is the expectation value? (b) Is σ an eigenfunction of S.? If yes, what is the eigenvalue? If no, what is the expectation value?The potential energy associated with two charged particles is 34 mJ when they are separated by a distance d. What is the potential energy associated with them (in mJ) when they are separated by 3d?
- (c) The kinetic energy, K₁, of electrons emitted from a metal surface after irradiation with UV light of wavelength λ is given by: hc Kg = -4 where his Planck's constant (6.626 x 10-4 Js), c is the speed of light in a vacuum (2.99 x 108 m s¹), and is the work function of the metal surface. In a specific experiment, light with a wavelength of 266 nm was used to irradiate a cadmium (Cd) metal surface.P21.6 The operator for the square of the total spin of two electrons is Stotal = (S₁ + S₂)² = S² + S² + 2(S1xS2x + S1yS2y + $1$22). Given that iħ S₁α = /B₁ S₁,α = 22²1 2 SxB = ħ 2 α, B, S₂α = = 九↓2 ħ α, iħ ħ S,ß α₁ S₂B = 1/- B₁ a, 2 2 show that a (1) a (2) and ß (1) ß (2) are eigenfunctions of the operator Stotal. What is the eigenvalue in each case?Using the Boltzmann’s law, show that, for a spin-1/2 nucleus at thermal equilibrium at temperature T in a magnetic field Bo, the probabilities of finding it in the a and ß states are: yħBo e kT 1 yhBo 1+е kт Pa ; Pg : yhBo 1+е kT
- The first excited state of Cu is reached by absorption of 327-nm radiation. (a) What is the energy difference (J) between the ground and excited states? (b) The ratio of degeneracies is g*/g0 3 for Cu. Find N*/N0 at 2 400 K. (c) By what percentage will the fraction in (b) be changed by a 15-K rise in temperature? (d) What will the ratio N*/N0 be at 6 000 K?A magnesium-24 ion has a positive charge that is double the charge of a proton, and a mass of 3.98 ✕ 10−26 kg. At a particular instant, it is moving with a speed of 5.00 ✕ 106 m/s through a magnetic field. At this instant, its velocity makes an angle of 58° with the direction of the magnetic field at the ion's location. The magnitude of the field is 0.190 T. (a) What is the magnitude of the magnetic force (in N) on the ion? N (b) What is the magnitude of the ion's acceleration (in m/s2) at this instant? m/s2The first excited state of Ca is reached by absorption of 422.7-nm light. (a) What is the energy difference (J) between the ground and excited states? (b) The degeneracies are g*/g0 3 for Ca. Find N*/N0 at 2 500 K. (c) By what percentage will the fraction in (b) be changed by a 15-K rise in temperature? (d) Find N*/N0 at 6 000 K.
- 2.9 Vibrations in crystals. (a) Calculate the average vibrational energy per mole for Si at 400 K to within the limits of the Einstein theory. Let VẸ = 12 x 1012 s-1, (b) If each atom carried three quanta of vibrational energy, how much vibrational energy would the crystal contain?Suppose the difference in rest energy of the neutron and proton were Qn = (mn − mp)c2 = x MeV instead of Qn = 1.29MeV, with all other physical parameters unchanged. Estimate Ymax, the maximum possible mass fraction in 4He, assuming that all available neutrons are incorporated into 4He nuclei. Hint: Compare the new rest energy of the neutron to the electron and consider how that would impact the decay of neutrons. Will free neutrons decay at all in this scenario? If the neutron decay is negligible, then the ratio of protons to neutrons to protons will remain the same as it was at the time of freeze-out (for a given Qn). Value: x=0.158 MeVThe reaction 9Be (3H, p) 11Be is studied by bombarding 9Be with 5 MeV tritons.Protons with energies of 2.260, 1.967 and 0.620 MeV are observed at 90 ° of the incident beam due to the formation of 11Be in the ground state and in two excited states:(a) Find the atomic mass of 11Be.(b) Find the excitation energy of the excited states.