### Additional Problems

For a hydrogen atom in an excited state with principal quantum number *n*, show that the smallest angle that the orbital angular momentum vector can make with respect to the *z*-axis is $\theta ={\text{cos}}^{\mathrm{-1}}\left(\sqrt{\frac{n-1}{n}}\right).$

What is the probability that the 1*s* electron of a hydrogen atom is found between $r=0$ and $r=\infty ?$

Sketch the potential energy function of an electron in a hydrogen atom. (a) What is the value of this function at $r=0$? in the limit that $r=\infty $? (b) What is unreasonable or inconsistent with the former result?

Find the value of $l$, the orbital angular momentum quantum number, for the Moon around Earth.

Show that the maximum number of orbital angular momentum electron states in the *n*th shell of an atom is ${n}^{2}$. (Ignore electron spin.) (*Hint:* Make a table of the total number of orbital angular momentum states for each shell and find the pattern.)

What is the magnitude of an electron magnetic moment?

A ground-state hydrogen atom is placed in a uniform magnetic field, and a photon is emitted in the transition from a spin-up to spin-down state. The wavelength of the photon is $168\phantom{\rule{0.2em}{0ex}}\mu \text{m}$. What is the strength of the magnetic field?

The valence electron of chlorine is excited to a 3*p* state. (a) What is the magnitude of the electron’s orbital angular momentum? (b) What are possible values for the *z*-component of angular measurement?

Which of the following notations are allowed (that is, which violate none of the rules regarding values of quantum numbers)? (a) $1{s}^{1};$ (b) $1{d}^{3};$ (c) $4{s}^{2};$ (d) $3{p}^{7};$ (e) $6{h}^{20}$

The ion ${\text{Be}}^{\text{3+}}$ makes an atomic transition from an $n=3$ state to an $n=2$ state. (a) What is the energy of the photon emitted during the transition? (b) What is the wavelength of the photon?

The maximum characteristic X-ray photon energy comes from the capture of a free electron into a *K* shell vacancy. What is this photon frequency for tungsten, assuming that the free electron has no initial kinetic energy?

Derive an expression for the ratio of X-ray photon frequency for two elements with atomic numbers ${Z}_{1}$ and ${Z}_{2}.$

(a) What voltage must be applied to an X-ray tube to obtain 0.0100-fm-wavelength X-rays for use in exploring the details of nuclei? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

A student in a physics laboratory observes a hydrogen spectrum with a diffraction grating for the purpose of measuring the wavelengths of the emitted radiation. In the spectrum, they observe a yellow line and find its wavelength to be 589 nm. (a) Assuming that this is part of the Balmer series, determine ${n}_{\text{i}},$ the principal quantum number of the initial state. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?