35. Special Relativity

Very large, hot stars—much hotter than our sun—can be identified by the way in which He⁺ ions in their atmosphere absorb light. What are the three longest wavelengths, in nm, in the Balmer series of He⁺?

INT A beam of electrons is incident upon a gas of hydrogen atoms. What minimum speed must the electrons have to cause the emission of 656 nm light from the 3→2 transition of hydrogen?

FIGURE EX39.14 is a graph of |ψ(x)|² for an electron. What is the value of a?

FIGURE P39.31 shows the wave function of a particle confined between x = 0 nm and x = 1.0 nm. The wave function is zero outside this region. Draw a graph of the probability density P(x)=|ψ(x)|²

FIGURE P39.31 shows the wave function of a particle confined between x = 0 nm and x = 1.0 nm. The wave function is zero outside this region. Calculate the probability of finding the particle in the interval 0 nm ≤ x ≤ 0.25 nm.

Consider the electron wave function where x is in cm. Determine the normalization constant c.

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

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.

A particle is described by the wave function where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

A particle is described by the wave function mm where L = 2.0 mm. Determine the normalization constant c.

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. A ²³⁸U nucleus, which decays by alpha emission, is 15 fm in diameter. Model an alpha particle within a ²³⁸U nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have?

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. The probability that a nucleus will undergo alpha decay is proportional to the frequency with which the alpha particle reflects from the walls of the nucleus. What is that frequency (reflections/s) for a maximum-speed alpha particle within a ²³⁸U nucleus?

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. What kind of photons are they—infrared, visible, or ultraviolet?

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. How long is the box in which the electrons are confined?

A 16-nm-long box has a thin partition that divides the box into a 4-nm-long section and a 12-nm-long section. An electron confined in the shorter section is in the n = 2 state. The partition is briefly withdrawn, then reinserted, leaving the electron in the longer section of the box. What is the electron’s quantum state after the partition is back in place?