Samuel J. Ling; Jeff Sanny; William Moebs

Additional Problems

85.

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 $θ = {cos}^{−1} (\sqrt{\frac{n - 1}{n}}) .$

86.

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

87.

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?

88.

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

89.

Show that the maximum number of orbital angular momentum electron states in the nth 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.)

90.

What is the magnitude of an electron magnetic moment?

91.

What is the maximum number of electron states in the $n = 5$ shell?

92.

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 $168 μ m$ . What is the strength of the magnetic field?

93.

Show that the maximum number of electron states in the nth shell of an atom is $2 n^{2}$ .

94.

The valence electron of chlorine is excited to a 3p 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?

95.

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) $6 h^{20}$

96.

The ion ${Be}^{3+}$ 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?

97.

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?

98.

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

99.

Compare the X-ray photon wavelengths for copper and silver.

100.

(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?

101.

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 $n_{i},$ the principal quantum number of the initial state. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?