35. Special Relativity

A 1.5-μm-wavelength laser pulse is transmitted through a 2.0-GHz-bandwidth optical fiber. How many oscillations are in the shortest-duration laser pulse that can travel through the fiber?

What minimum bandwidth is needed to transmit a pulse that consists of 100 cycles of a 1.0 MHz oscillation?

What is the minimum uncertainty in position, in nm, of an electron whose velocity is known to be between 3×10⁵ m/s and 4 ×10⁵ m/s? Give your answer to one significant figure.

FIGURE P39.28 shows a pulse train. The period of the pulse train is T = 2 Δt, where Δt is the duration of each pulse. What is the maximum pulse-transmission rate (pulses per second) through an electronics system with a 200 kHz bandwidth? (This is the bandwidth allotted to each FM radio station.)

Consider a single-slit diffraction experiment using electrons. (Single-slit diffraction was described in Section 33.4.) Using Figure 39.5 as a model, draw A graph of |ψ(x)|² for the electrons on the detection screen.

A pulse of light is created by the superposition of many waves that span the frequency range f₀ − (1/2) Δf ≤ f ≤ f₀ + (1/2) Δf, where f₀ = c/λ is called the center frequency of the pulse. Laser technology can generate a pulse of light that has a wavelength of 600 nm and lasts a mere 6.0 fs (1 fs = 1 femtosecond =10⁻¹⁵ s). What is the spatial length of the laser pulse as it travels through space?

For an electron in the 1s state of hydrogen, what is the probability of being in a spherical shell of thickness 0.010a_B at distance (a) ½ a_B, (b) a_B, and (c) 2a_B from the proton?

What is the angular momentum of a hydrogen atom in (a) a 6s state and (b) a 4f state? Give your answers as a multiple of ℏ .

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

A sample of 1.0 x 10¹⁰ atoms that decay by alpha emission has a half-life of 100 min. How many alpha particles are emitted between t = 50 min and t = 200 min?

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. At what time after the sample's removal from the reactor is the count 1200 decays/s?

For those that are not stable, identify both the decay mode and the daughter nucleus.

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. Write the equation that expresses the conservation of relativistic energy for this decay. Your equation will be in terms of the three masses m_n, m_p and m_e and the relativistic factors y_p and y_e.

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. Write the equation that expresses the conservation of relativistic momentum for this decay. Let v represent speed, rather than velocity, then write any minus signs explicitly.

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. What is the total kinetic energy, in MeV, of the proton and electron?