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Atomic and Molecular Physics

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Prerequisites

ArithmeticLogic
GeometryAlgebra
TrigonometryFunctions
VectorsLimits
Physics IMatricesDerivatives
Physics IIGeneral ChemistryEigenvaluesIntegrals
Quantum Physics
Atomic and Molecular Physics

Hydrogen and quantum states

For a hydrogen-like ion with nuclear charge $Z$,

$$ E_n=-\frac{13.6\ \text{eV}\,Z^2}{n^2}, \qquad n=1,2,3,\ldots $$

The principal quantum number $n$ sets the main energy and size scale. The orbital quantum number $\ell$ determines angular momentum magnitude, $m_\ell$ its projection, and $m_s$ the spin projection:

$$ L^2=\ell(\ell+1)\hbar^2, \qquad L_z=m_\ell\hbar. $$

The Pauli exclusion principle prevents two electrons in an atom from sharing all four quantum numbers. Electron configurations fill orbitals according to energy, while shielding and penetration modify the simple hydrogen picture.

Spectra and transitions

Absorption or emission of a photon satisfies

$$ \Delta E = hf = \frac{hc}{\lambda}. $$

For a hydrogen-like ion, the wavelengths of transitions follow the Rydberg relation:

$$ \frac{1}{\lambda}=R_\infty Z^2\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right), \qquad n_i>n_f. $$

Selection rules identify the strongest electric-dipole transitions, commonly $\Delta\ell=\pm1$. Spectral lines broaden because of finite lifetime, collisions, thermal motion, and instrumental resolution.

Molecules

Molecular energy is often approximated as

$$ E \approx E_{\text{electronic}}+E_{\text{vibrational}}+E_{\text{rotational}}. $$

Electronic transitions are typically in the ultraviolet or visible range, vibrations in the infrared, and rotations in the microwave or far-infrared. The Born–Oppenheimer approximation separates fast electronic motion from slower nuclear motion.

For a rigid rotor,

$$ E_J=\frac{\hbar^2}{2I}J(J+1), $$

and for a harmonic vibration,

$$ E_v=\hbar\omega\left(v+\frac12\right). $$

The zero-point energy remains even in the lowest vibrational state. Molecular shape and symmetry determine which transitions are allowed.

Experimental tools and workflow

Spectroscopy measures how matter interacts with electromagnetic radiation. Lasers provide coherent, narrow-band light; fluorescence and absorption reveal energy levels; mass spectrometry separates ions by mass-to-charge ratio.

When analyzing a problem, state whether the hydrogen-like or multi-electron approximation is justified:

  1. Identify whether the system is atomic, molecular, or ionized.

  2. Choose the appropriate energy model and quantum numbers.

  3. Apply conservation of energy and angular momentum.

  4. Check units, selection rules, and limiting cases.

Summary

Atomic and molecular physics connects quantum states to measurable spectra. Energy-level spacing, selection rules, and the separation of electronic, vibrational, and rotational motion provide the main organizing framework.

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