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?

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 undergoes a transition from a $2p$ state to the $1s$ ground state. In the absence of a magnetic field, the energy of the photon emitted is $122$ nm. The atom is then placed in a strong magnetic field in the $z$-direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. How many different photon wavelengths are observed for the $2p\to 1s$ transition? What are the $m_l$ values for the initial and final states for the transition that leads to each photon wavelength?

In a particular state of the hydrogen atom, the angle between the angular momentum vector $\overrightarrow{L}$ and the $z$-axis is $u=26.6$°. If this is the smallest angle for this particular value of the orbital quantum number $l$, what is $l$?

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?

A hydrogen atom is in a $d$ state. In the absence of an external magnetic field, the states with different $m_l$ values have (approximately) the same energy. Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. Calculate the splitting (in electron volts) of the ml levels when the atom is put in a $0.800$ T magnetic field that is in the $+z$-direction