35. Special Relativity

Which of the following is a reasonable approximation of an inertial reference frame?

Leroy is standing 900 m from firecracker A and 600 m from firecracker B. After exploding, the light from both firecrackers arrives at Leroy at the same time. In Leroy's frame of reference, which firecracker exploded first, or were they simultaneous?

How many electrons, protons, and neutrons are contained in the following atoms or ions: (a) ¹⁰B , (b) ¹³N⁺ , and (c) ¹⁷O⁺⁺⁺ ?

A horizontal beam of laser light of wavelength 585 nm passes through a narrow slit that has width 0.0620 mm. The intensity of the light is measured on a vertical screen that is 2.00 m from the slit. (a) What is the minimum uncertainty in the vertical component of the momentum of each photon in the beam after the photon has passed through the slit? (b) Use the result of part (a) to estimate the width of the central diffraction maximum that is observed on the screen.

An ultrashort pulse has a duration of 9.00 fs and produces light at a wavelength of 556 nm. What are the momentum and momentum uncertainty of a single photon in the pulse?

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 6.50x10^-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 6.64x10^-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 226Ra 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?

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)?

Calculate the de Broglie wavelength of a 5.00-g bullet that is moving at 340 m/s. Will the bullet exhibit wavelike properties?

An electron is moving with a speed of 8.00 * 10^6 m/s. What is the speed of a proton that has the same de Broglie wavelength as this electron?

An alpha particle (m = 6.64x10^-27 kg) emitted in the radioactive decay of uranium-238 has an energy of 4.20 MeV. What is its de Broglie wavelength?

An electron has a de Broglie wavelength of 2.80x10^-10 m. Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).

For crystal diffraction experiments (discussed in Section 39.1), wavelengths on the order of 0.20 nm are often appropriate. Find the energy in electron volts for a particle with this wavelength if the particle is (a) a photon.

(a) An electron moves with a speed of 4.70x10^6 m/s. What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.

(a) The x-coordinate of an electron is measured with an uncertainty of 0.30 mm. What is the x-component of the electron's velocity, vx , if the minimum percent uncertainty in a simultaneous measurement of vx is 1.0%? (b) Repeat part (a) for a proton.

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 3.0x10^-25 kg-m/s. Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

(a) The uncertainty in the y-component of a proton's position is 2.0x10^-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. (c) 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 um. 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. (c) 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. (a) 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 oneelectron 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. E39.27

. You also observe that it takes 17.50 eV to ionize this atom. (a) 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 oneelectron 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. (b) 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?)

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 (a) x=0.000 mm,

What is the probability of finding a 1s hydrogen electron at distance r > aB from the proton?

A Geiger counter is used to measure the decay of a radioactive isotope produced in a nuclear reactor. Initially, when the sample is first removed from the reactor, the Geiger counter registers 15,000 decays/s. 15 h later the count is down to 5500 decays/s. b. At what time after the sample's removal from the reactor is the count 1200 decays/s?

It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, let's analyze the decay of a free neutron that is at rest in the laboratory. We'll ignore the antineutrino and consider the decay n → p⁺ + e⁻. The analysis requires the use of relativistic energy and momentum, from Chapter 36. b. Write the equation that expresses the conservation of relativistic energy for this decay. Your equation will be in terms of the three masses mₙ, mₚ and mₑ and the relativistic factors yₚ and yₑ.

The starship Enterprise, of television and movie fame, is powered by combining matter and antimatter. If the entire 400-kg antimatter fuel supply of the Enterprise combines with matter, how much energy is released? How does this compare to the U.S. yearly energy use, which is roughly 1.0 * 10^20 J?

FIGURE EX39.14 is a graph of |ψ(x)|^2 for an electron. c. What is the probability that the electron is located between x=1.0 nm and x=2.0 nm?

Consider the electron wave function ψ(x)={√c 1−x^2 |x|≤1 cm 0 |x|≥1 cm where x is in cm. d. If 10^4 electrons are detected, how many will be in the interval 0.00 cm≤x≤0.50 cm?

A particle is described by the wave function ψ(x)={ceˣ/ᴸ x≤0 mm ce−ˣ/ᴸ x≥0 mm where L=2.0 mm. c. Calculate the probability of finding the particle within 1.0 mm of the origin.

For a particle in a finite potential well of width L and depth U0, what is the ratio of the probability Prob(in δx at x=L+η) to the probability Prob(in δx at x=L)?

A particle is described by the wave function ψ(x)={ceˣ/ᴸ x≤0 mm ce−ˣ/ᴸ x≥0 mm where L=2.0 mm. d. Interpret your answer to part b by shading the region representing this probability on the appropriate graph in part a.