35. Special Relativity

Photorefractive keratectomy (PRK) is a laser-based surgical procedure that corrects near- and farsightedness by removing part of the lens of the eye to change its curvature and hence focal length. This procedure can remove layers $0.25$ mm thick using pulses lasting $12.0$ ns from a laser beam of wavelength $193$ nm. Low-intensity beams can be used because each individual photon has enough energy to break the covalent bonds of the tissue. If a $1.50$-mW beam is used, how many photons are delivered to the lens in each pulse?

How many photons per second are emitted by a $7.50$-mW CO₂ laser that has a wavelength of $10.6$ mm?

Using a mixture of CO₂, N₂, and sometimes He, CO₂ lasers emit a wavelength of $10.6$ $\mu$m. At power of $0.100$ kW, such lasers are used for surgery. How many photons per second does a CO₂ laser deliver to the tissue during its use in an operation?

Use Balmer's formula to calculate (a) the wavelength, (b) the frequency, and (c) the photon energy for the Hg line of the Balmer series for hydrogen.

Find the longest and shortest wavelengths in the Lyman and Paschen series for hydrogen. In what region of the electromagnetic spectrum does each series lie?

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?

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 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?

In a set of experiments on a hypothetical one-electron atom, you measure the wavelengths of the photons emitted from transitions ending in the ground level ($n=1$), as shown in the energy-level diagram in Fig. E$39.27$. You also observe that it takes $17.50$ eV to ionize this atom. What is the energy of the atom in each of the levels ($n=1$, $n=2$, etc.) shown in the figure?

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.

$10.0$-g marble is gently placed on a horizontal tabletop that is $1.75$ m wide.

(a) What is the maximum uncertainty in the horizontal position of the marble?

(b) According to the Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity of the marble?

(c) In light of your answer to part (b), what is the longest time the marble could remain on the table? Compare this time to the age of the universe, which is approximately $14$ billion years. (Hint: Can you know that the horizontal velocity of the marble is exactly zero?)

In one experiment, 2000 photons are detected in a 0.10-mm-wide strip where the amplitude of the electromagnetic wave is 10 V/m. How many photons are detected in a nearby 0.10-mm-wide strip where the amplitude is 30 V/m?

When 5×10¹² photons pass through an experimental apparatus, 2.0×10⁹ land in a 0.10-mm-wide strip. What is the probability density at this point?

FIGURE EX39.12 shows the probability density for an electron that has passed through an experimental apparatus. If 1.0×10⁶ electrons are used, what is the expected number that will land in a 0.010-mm-wide strip at 2.000 mm?

FIGURE EX39.13 shows the probability density for an electron that has passed through an experimental apparatus. What is the probability that the electron will land in a 0.010-mm-wide strip at x = 0.000 mm?