Resnick & Halliday Solutions for Chapter: All About Atoms, Exercise 1: Problems

$\left(\mathrm{a}\right)$ How many $l$ values are associated with $n=3$? $\left(\mathrm{b}\right)$ How many $m_{\mathrm{l}}$ values are associated with $l=1$?

Assume that in the Stern-Gerlach experiment, as described for neutral silver atoms, the magnetic field $\overrightarrow{B}$ has a magnitude of $0.60 \mathrm{T}$. $\left(\mathrm{a}\right)$ What is the energy difference between the magnetic moment orientations of the silver atoms in the two sub beams? $\left(\mathrm{b}\right)$ What is the frequency of the radiation that would induce a transition between these two states? $\left(\mathrm{c}\right)$ What is the wavelength of this radiation and $\left(\mathrm{d}\right)$ to what part of the electromagnetic spectrum does it belong?

The active medium in a particular laser that generates laser light at a wavelength of $694 \mathrm{nm}$, is $6.00 \mathrm{cm}$ long and $1.00 \mathrm{cm}$ in diameter. $\left(\mathrm{a}\right)$ Treat the medium as an optical resonance cavity analogous to a closed organ pipe. How many standing-wave nodes are there along the laser axis? $\left(\mathrm{b}\right)$ By what amount $\mathrm{\Delta }f$ would the beam frequency have to shift to increase this number by one? $\left(\mathrm{c}\right)$ Show that $\mathrm{\Delta }f$ is just the inverse of the travel time of laser light, for one round trip back and forth along the laser axis. $\left(\mathrm{d}\right)$ What is the corresponding fractional frequency shift $\frac{\mathrm{\Delta }f}{f}$? The appropriate index of refraction of the lasing medium (a ruby crystal) is $1.75$.

A magnetic field is applied to a freely floating uniform iron sphere with radius $R=1.50 \mathrm{mm}$. The sphere initially had no net magnetic moment, but the field aligns $12\%$ of the magnetic moments of the atoms (that is, $12\%$ of the magnetic moments of the loosely bound electrons in the sphere, with one such electron per atom). The magnetic moment of those aligned electrons is the sphere's intrinsic magnetic moment ${\overrightarrow{\mu }}_{\mathrm{s}}$. What is the sphere's resulting angular speed $\omega$?

A recently named element is darmstadtium $\left(\mathrm{Ds}\right),$ which has $110$ electrons. Assume that you can put the $110$ electrons into the atomic shells one by one and can neglect any electron-electron interaction. With the atom in ground state, what is the spectroscopic notation for the quantum number $l$ for the last electron?

How many electron states are in these subshells: $\left(\mathrm{a}\right) n=4, l=3; \left(\mathrm{b}\right) n=3, l=2; \left(\mathrm{c}\right) n=4, l=1; \left(\mathrm{d}\right) n=2, l=0$?

$\left(\mathrm{a}\right)$ What is the magnitude of the orbital angular momentum in a state with $l=3$? $\left(\mathrm{b}\right)$ What is the magnitude of its largest projection on an imposed $z$ axis?

Suppose that a hydrogen atom in its ground state, moves $80 \mathrm{cm}$ through and perpendicular to a vertical magnetic field that has a magnetic field gradient $\frac{\mathrm{d}B}{\mathrm{d}z}=1.6\times {10}^2 \mathrm{T} {\mathrm{m}}^{-1}$. $\left(\mathrm{a}\right)$ What is the magnitude of force exerted by the field gradient on the atom, due to the magnetic moment of the atom's electron, which we take to be $1 \mathrm{Bohr} \mathrm{magneton}$? $\left(\mathrm{b}\right)$ What is the vertical displacement of the atom in the $80 \mathrm{cm}$ of travel, if its speed is $2.5\times {10}^5 \mathrm{m} {\mathrm{s}}^{-1}$?