A hydrogen atom is in a state with energy $-1.51$ eV. In the Bohr model, what is the angular momentum of the electron in the atom, with respect to an axis at the nucleus?

Through what potential difference must electrons be accelerated if they are to have:

(a) the same wavelength as an x ray of wavelength $0.220$ nm; and

(b) the same energy as the x ray in part (a)?

The energy-level scheme for the hypothetical one-electron element Searsium is shown in Fig. $E39.25$. The potential energy is taken to be zero for an electron at an infinite distance from the nucleus. An $18$-eV photon is absorbed by a Searsium atom in its ground level. As the atom returns to its ground level, what possible energies can the emitted photons have? Assume that there can be transitions between all pairs of levels.

A beam of alpha particles is incident on a target of lead. A particular alpha particle comes in 'head-on' to a particular lead nucleus and stops $6.50\times {10}^{-14}$ m away from the center of the nucleus. (This point is well outside the nucleus.) Assume that the lead nucleus, which has $82$ protons, remains at rest. The mass of the alpha particle is $6.64\times {10}^{-27}$ kg.

(a) Calculate the electrostatic potential energy at the instant that the alpha particle stops. Express your result in joules and in MeV.

(b) What initial kinetic energy (in joules and in MeV) did the alpha particle have?

(c) What was the initial speed of the alpha particle?

A $4.78$-MeV alpha particle from a $226$Ra decay makes a head-on collision with a uranium nucleus. A uranium nucleus has $92$ protons.

(a) What is the distance of closest approach of the alpha particle to the center of the nucleus? Assume that the uranium nucleus remains at rest and that the distance of closest approach is much greater than the radius of the uranium nucleus.

(b) What is the force on the alpha particle at the instant when it is at the distance of closest approach?

A triply ionized beryllium ion, Be³⁺ (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom except that the nuclear charge is four times as great. What is the ground-level energy of Be³⁺? How does this compare to the ground-level energy of the hydrogen atom?

A triply ionized beryllium ion, Be³⁺ (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom except that the nuclear charge is four times as great. For the hydrogen atom, the wavelength of the photon emitted in the $n=2$ to $n=1$ transition is $122$ nm (see Example $39.6$). What is the wavelength of the photon emitted when a Be³⁺ ion undergoes this transition?