A photon is emitted when an electron in a three-dimensional cubical box of side length $8.00\times {10}^{-11}$ m makes a transition from the $n_X=2$, $n_Y=2$, $n_Z=1$ state to the $n_X=1$, $n_Y=1$, $n_Z=1$ state. What is the wavelength of this photon?

Consider an electron in the $N$ shell. What is the smallest orbital angular momentum it could have?

Consider an electron in the $N$ shell. What is the largest orbital angular momentum it could have? Express your answers in terms of $\(\hslash\)$ and in SI units.

Consider an electron in the $N$ shell. What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of $\hslash$ and in SI units.

The orbital angular momentum of an electron has a magnitude of $4.716\times {10}^{-34}$ kg-m²/s. What is the angular momentum quantum number $l$ for this electron?

Calculate, in units of $U$, the magnitude of the maximum orbital angular momentum for an electron in a hydrogen atom for states with a principal quantum number of $2$, $20$, and $200$. Compare each with the value of $nh$ postulated in the Bohr model. What trend do you see?

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

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?