35. Special Relativity

A scientist has devised a new method of isolating individual particles. He claims that this method enables him to detect simultaneously the position of a particle along an axis with a standard deviation of $0.12$ nm and its momentum component along this axis with a standard deviation of $3.0\times {10}^{-25}$ kg-m/s. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

The uncertainty in the y-component of a proton's position is $2.0\times {10}^{-12}$ m. What is the minimum uncertainty in a simultaneous measurement of the $y$-component of the proton's velocity?

A pesky $1.5$-mg mosquito is annoying you as you attempt to study physics in your room, which is $5.0$ m wide and $2.5$ m high. You decide to swat the bothersome insect as it flies toward you, but you need to estimate its speed to make a successful hit.

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

(b) What limit does the Heisenberg uncertainty principle place on your ability to know the horizontal velocity of this mosquito? Is this limitation a serious impediment to your attempt to swat it?

Two stars, both of which behave like ideal blackbodies, radiate the same total energy per second. The cooler one has a surface temperature $T$ and a diameter $3.0$ times that of the hotter star.

(a) What is the temperature of the hotter star in terms of $T$?

(b) What is the ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star?

The shortest visible wavelength is about $400$ nm. What is the temperature of an ideal radiator whose spectral emittance peaks at this wavelength?

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.