1. Intro to Physics Units

Model a hydrogen atom as an electron in a cubical box with side length $L$. Set the value of $L$ so that the volume of the box equals the volume of a sphere of radius $a=5.29\times {10}^{-11}$ m, the Bohr radius. Calculate the energy separation between the ground and first excited levels, and compare the result to this energy separation calculated from the Bohr model.

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.

Consider an electron in the $N$ shell. What is the largest spin 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?

Estimate the energy of the highest-$l$ state for (a) the $L$ shell of Be⁺ and (b) the $N$ shell of Ca⁺.

The energies for an electron in the $K$, $L$, and $M$ shells of the tungsten atom are $-69,500$ eV, $-12,000$ eV, and $-2,200$ eV, respectively. Calculate the wavelengths of the $K_{\alpha }$ and $K_{\beta }$ x rays of tungsten.

For the H₂ molecule the equilibrium spacing of the two protons is $0.074$ nm. The mass of a hydrogen atom is $1.67\times {10}^{-27}$ kg. Calculate the wavelength of the photon emitted in the rotational transition $l=2$ to $l=1$.

During each of these processes, a photon of light is given up. In each process, what wavelength of light is given up, and in what part of the electromagnetic spectrum is that wavelength? A molecule decreases its vibrational energy by $0.198$ eV.

The H₂ molecule has a moment of inertia of $4.6\times {10}^{-48}$ kg-m². What is the wavelength $l$ of the photon absorbed when H₂ makes a transition from the $l=3$ to the $l=4$ rotational level?