1. Intro to Physics Units

The hyperfine interaction in a hydrogen atom between the magnetic dipole moment of the proton and the spin magnetic dipole moment of the electron splits the ground level into two levels separated by $5.9\times {10}^{-6}$ eV. Calculate the wavelength and frequency of the photon emitted when the atom makes a transition between these states, and compare your answer to the value given at the end of Section $41.5$. In what part of the electromagnetic spectrum does this lie? Such photons are emitted by cold hydrogen clouds in interstellar space; by detecting these photons, astronomers can learn about the number and density of such clouds.

(a) If you treat an electron as a classical spherical object with a radius of $1.0\times {10}^{-17}$ m, what angular speed is necessary to produce a spin angular momentum of magnitude $\sqrt{\frac{3}{4}}h$?

(b) Use $v=r\omega$ and the result of part (a) to calculate the speed $v$ of a point at the electron's equator. What does your result suggest about the validity of this model?

A hydrogen atom in the $5g$ state is placed in a magnetic field of $0.600$ T that is in the $z$-direction. Into how many levels is this state split by the interaction of the atom's orbital magnetic dipole moment with the magnetic field?

A hydrogen atom in a $3p$ state is placed in a uniform external magnetic field $\overrightarrow{B}$. Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. What field magnitude $B$ is required to split the $3p$ state into multiple levels with an energy difference of $2.71\times {10}^{-5}$ eV between adjacent levels?

Pure germanium has a band gap of $0.67$ eV. The Fermi energy is in the middle of the gap. For temperatures of $250$ K, $300$ K, and $350$ K, calculate the probability $f\left(E\right)$ that a state at the bottom of the conduction band is occupied.

At the Fermi temperature $T_F$, $E_F=k{T}_F$ (see Exercise $42.22$). When $T=T_F$, what is the probability that a state with energy $E=2E_F$ is occupied?

Silver has a Fermi energy of $5.48$ eV. Calculate the electron contribution to the molar heat capacity at constant volume of silver, $C_V$, at $300$ K. Express your result as a multiple of $R$.

Calculate the density of states $g\left(E\right)$ for the free-electron model of a metal if $E=7.0$ eV and $V=1.0$ cm³. Express your answer in units of states per electron volt.

The maximum wavelength of light that a certain silicon photocell can detect is 1.11 mm. (b) Explain why pure silicon is opaque.

The maximum wavelength of light that a certain silicon photocell can detect is $1.11$ mm. What is the energy gap (in electron volts) between the valence and conduction bands for this photocell?

The average kinetic energy of an ideal-gas atom or molecule is $\left(\frac{3}{2}\right)kT$, where $T$ is the Kelvin temperature (Chapter $18$). The rotational inertia of the H₂ molecule is $4.6\times {10}^{-48}$ kg-m². What is the value of $T$ for which $\left(\frac{3}{2}\right)kT$ equals the energy separation between the $l=0$ and $l=1$ energy levels of H₂? What does this tell you about the number of H₂ molecules in the $l=1$ level at room temperature?

The rotational energy levels of CO are calculated in Example $42.2$. If the energy of the rotating molecule is described by the classical expression $K=\left(\frac{1}{2}\right)I{\omega }^2$, for the $l=1$ level, what is the linear speed of each atom?

Measurements on a certain isotope tell you that the decay rate decreases from $8318$ decays/min to $3091$ decays/min in $4.00$ days. What is the half-life of this isotope?

The unstable isotope ${}^{40}K$ is used for dating rock samples. Its half-life is $1.28\times {10}^9$ y.

(a) How many decays occur per second in a sample containing $1.63\times {10}^{-6}$ g of ${}^{40}K$?

(b) What is the activity of the sample in curies?

The common isotope of uranium, ${}^{238}U$, has a half-life of $4.47\times {10}^9$ years, decaying to ${}^{234}Th$ by alpha emission.

(a) What is the decay constant?

(b) What mass of uranium is required for an activity of $1.00$ curie?

(c) How many alpha particles are emitted per second by $10.0$ g of uranium?