35. Special Relativity

In most metals, the atomic ions form a regular arrangement called a crystal lattice. The conduction electrons in the sea of electrons move through this lattice. FIGURE P40.34 is a one-dimensional model of a crystal lattice. The ions have mass m, charge e, and an equilibrium separation b. Suppose this crystal consists of aluminum ions with an equilibrium spacing of 0.30 nm. What are the energies of the four lowest vibrational states of these ions?

A finite potential well has depth U₀ = 2.00 eV. What is the penetration distance for an electron with energy (a) 0.50 eV, (b) 1.00 eV, and (c) 1.50 eV?

(a) An electron with initial kinetic energy $32$ eV encoun­ters a square barrier with height $41$ eV and width $0.25$ nm. What is the probability that the electron will tunnel through the barrier?

(b) A proton with the same kinetic energy encounters the same barrier. What is the probability that the proton will tunnel through the barrier?

You fly 5000 km across the United States on an airliner at 250 m/s. You return two days later at the same speed. Have you aged more or less than your friends at home?

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?

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 $1.0\times {10}^{20}$ J?

An electron with initial kinetic energy $6.0$ eV encounters a barrier with height $11.0$ eV. What is the probability of tunneling if the width of the barrier is (a) $0.80$ nm and (b) $0.40$ nm?

Prove that the normalization constant of the 2p radial wave function of the hydrogen atom is (24πa_B³)^-1/2, as shown in Equations 41.7. Hint: See the hint in Problem 32.

Which of the following best describes the relationship between Galileo's and Copernicus's views on inertial reference frames?

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

An electron is in a box of width 3.0*10^-10 m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the n = 1 level; (b) the n = 2 level; (c) the n = 3 level? In each case how does the wavelength compare to the width of the box?

CALC A particle of mass m has the wave function ψ(x) = Ax exp (−x²/a²) when it is in an allowed energy level with E = 0. Draw a graph of ψ(x) versus x.

What is the speed of a galaxy with z = 0.068?

Consider the electron wave function where x is in cm. Draw a graph of |ψ(x)|² over the interval −2 cm ≤ x ≤ 2 cm. Provide numerical scales.