1. A nucleus with decay constant λ exists at time t = 0. What is the probability that it disintegrates between t and t + Δt?

2. A particle of mass m and electric charge e is placed in the one-dimensional harmonic oscillator potential of frequency w and the uniform electric field S along the same axis.

a. Find the wave functions and the energy spectrum of the particle.

b. With a particle in the ground state of the problem, at time t = 0 the electric field is suddenly switched off. Find the probability of finding the particle at t > 0 in the nth stationary state of the oscillator.

3. Consider a spherical cloud of radius r, at a distance d >> r from the Earth. Suppose that this cloud has uniform density p, temperature T and thermal emissivity η_v  (power per unit volume per unit frequency interval per unit solid angle), i.e. η_v is a function of T only.

a) Suppose the cloud is optically thick. Consider a small area element on the surface of the cloud, located at distance b from the line of sight passing through the center of the cloud, which subtends a solid angle dΩ as seen from the Earth. Find I_vdΩ the power per unit area per unit frequency interval, received from this area element at the Earth.

b) Find the flux Φ, at the Earth from the center cloud, by using the result of a).

c) What is the effective temperature T_eff of the cloud?

4. Using Eqs. (136) and (4.37), show that Urad, v = 3Prad,v for an isotropic radiation field. Prove Eq. (4.63).

6. Assume a γ-ray detector on board of a satellite. Its efiective area is 1500 cm² and its angular position accuracy is Δθ ≈ 10^o. After observing a Quasar for two weeks, it collects about 3000 photons with energy 1 GeV. Determine (a) I = ∫I_vdv and (b) Φ = ∫Φ_vdv.

7. The activity of a given material decreases by a factor 8 in a time interval of 30 days. What is its half-life, mean lifetime and decay constant.? If the sample initially had 10²⁰ atoms, how many disintegrations have occurred in its second month of life?

8. The theories of grand unification predict that the proton is not a stable particle, although it has a long half-life. For a half-life of 10³³ years, how many proton decays can we expect in one year in a mass of 10³ tons of water?

9. Calculate the average velocity of a Maxwell-Boltzmann distribution.

10. Use the literature as a, source to prove Eq. (4.82).

11. Calculate the fraction of hydrogen atoms in the n = 2 state with respect to the ground state for 3 different temperatures: T₁ = 6000 K, T₂ = 107000 K, T₃ = 30,000 K.

12. Use the Saha equation to calculate the temperature of recombination, which you can efine as the temperature when there is an equal amount of neutral hydrogen and free protons/electrons. You can simplify the hydrogen atom to be a single state at -13.6.

13. An atom with a single electron is in a heat bath at a temperature of T₆ = 1 (T₆ means that the temperature is given in units of 10⁶ K). The atom is high Z, so the electron is bound at this temperature, and only three states have appreciable occupations. The ground state has spin 5/2. The first excited state, at 210 eV, has spin 3/2. The second excited state, at 380 eV, has spin 3/2. What are the occupation probabilities for these three states?

14. A nucleus is in a plasma at thermal equilibrium. Calculate the population probabilities of the ground state (E₀ = 0) and of the first three excited states (E₁ = 0.1 MeV, E₂ = 0.5 MeV, E3 = 1.0 MeV) for two temperatures, T = 1.0 x 109 K and 3.0 x 10⁹ K, and assume that all states have the same spin value.

15. Using Eq. (4.110) (a) calculate the cross section (in fm²) for the absorption of a photon with energy hv = I, where I is the ionization potential of a hydrogen atom at its ground state; (b) find the ration of the value obtained in (a) and Πa²H where A_H is the Bohr radius.

16. As a mechanism for downward transitions (final energy smaller than initial energy), spontaneous emission competes with thermally stimulated emission (stimulated emission for which Planck radiation is the source). Show that at room temperature (T = 300 K) thermal stimulation dominates for frequencies much smaller than 5 x 10¹² Hz, whereas spontaneous emission dominates for much larger frequencies. Which mechanism dominates for visible light.?

17. Generalize the Saha equation to consider the temperature for a process in the early universe where two protons and two neutrons combine directly to form ⁴He. That is, find the temperature where half of the neutrons are bound in 'He. Later we will compare this result with the big-bang nucleosynthesis predictions.

18. Negative hydrogen ions H^-, which have a dissociation energy of 0.754 eV, play and important role in the opacity of the outer layers of the Sun. Assume and electron density is n_e = 10²⁶ M^-3 and estimate the temperature (to one figure accuracy) at which the number densities of J and H^- are equal.