5.3 Applications to atomic and molecular systems

Time-independent perturbation theory is a powerful tool for tackling complex quantum systems. It helps us understand how small changes affect atomic and molecular energy levels, giving insight into fine structure, hyperfine structure, and effects like Stark and Zeeman.

This approach is crucial for real-world applications in spectroscopy, atomic clocks, and quantum sensors. By applying perturbation theory to atoms and molecules, we can predict and analyze their behavior in electric and magnetic fields, enhancing our understanding of quantum systems.

Perturbation Theory for Atomic Structure

Fine Structure Corrections

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• Perturbation theory finds approximate solutions to quantum mechanical problems by treating the Hamiltonian as a sum of an unperturbed part and a small perturbation
• Fine structure arises from the coupling between the orbital angular momentum and the spin angular momentum of the electron, splitting energy levels (sodium D lines)
• The first-order correction to the energy levels is the expectation value of the perturbation Hamiltonian in the unperturbed state
• The second-order correction involves a sum over all unperturbed states, weighted by the matrix elements of the perturbation Hamiltonian and the energy denominators (helium fine structure)

Hyperfine Structure Corrections

• Hyperfine structure is caused by the interaction between the magnetic moment of the electron and the magnetic moment of the nucleus, further splitting energy levels (hydrogen 21 cm line)
• Selection rules for the allowed transitions between the split energy levels are determined by the symmetry properties of the perturbation Hamiltonian
• Hyperfine structure can be used to create atomic clocks with extremely high precision (cesium-133)
• The hyperfine interaction is much weaker than the fine structure interaction, leading to smaller energy level splittings

Stark and Zeeman Effects

Stark Effect

• The Stark effect is the splitting and shifting of atomic energy levels in the presence of an external electric field (hydrogen Stark effect)
• In the weak-field limit, the Stark effect can be treated using perturbation theory, with the electric field acting as the perturbation
• The first-order Stark effect is linear in the electric field strength and vanishes for states with definite parity
• The second-order Stark effect is quadratic in the electric field strength and is present for all states (quadratic Stark effect in Rydberg atoms)

Zeeman Effect

• The Zeeman effect is the splitting of atomic energy levels in the presence of an external magnetic field (sodium Zeeman effect)
• The Zeeman effect leads to a splitting of energy levels into multiple sublevels, with the splitting proportional to the magnetic field strength and the magnetic quantum number
• The Landé g-factor characterizes the strength of the Zeeman splitting and depends on the electron's orbital and spin angular momenta
• The Zeeman effect is utilized in various applications, such as atomic clocks, magnetometers, and NMR spectroscopy (proton NMR)

Perturbation Theory in Molecular Systems

Born-Oppenheimer Approximation

• Perturbation theory calculates corrections to the energy levels and wavefunctions of molecules due to various interactions, such as electron-electron repulsion and electron-nuclear attraction
• In the Born-Oppenheimer approximation, the electronic and nuclear motions are separated, and the electronic problem is solved for fixed nuclear positions
• The Born-Oppenheimer approximation is a crucial step in simplifying the molecular Schrödinger equation (vibrational and rotational spectra)
• The Born-Oppenheimer approximation breaks down when there is significant coupling between electronic and nuclear motions (Jahn-Teller effect)

Hydrogen Molecule Ion (H2+)

• The hydrogen molecule ion (H2+) is the simplest molecular system, consisting of two protons and one electron
• The ground state of H2+ can be approximated by a linear combination of atomic orbitals (LCAO) centered on each proton
• The energy levels and wavefunctions of H2+ can be improved by applying perturbation theory to account for the electron-nuclear attraction and the electron-electron repulsion
• The perturbative corrections to the energy levels of H2+ provide information about the bonding and antibonding orbitals, as well as the dissociation energy of the molecule (H2+ potential energy curve)

Applying Perturbation Theory to Real Systems

Atomic Systems

• Apply perturbation theory to calculate the fine and hyperfine structure splittings in alkali atoms, such as sodium and rubidium (rubidium-87 hyperfine structure)
• Determine the Stark shifts and splittings of energy levels in atoms subjected to external electric fields, such as in atomic clocks or quantum sensors (strontium optical lattice clock)
• Calculate the Zeeman splittings of energy levels in atoms in the presence of magnetic fields, such as in magnetometry (atomic magnetometers)

Molecular Systems

• Use perturbation theory to estimate the corrections to the energy levels and wavefunctions of simple molecular systems, such as the hydrogen molecule or the helium atom (helium dimer)
• Analyze the effects of perturbations on the selection rules and transition probabilities for electromagnetic transitions in molecules (carbon monoxide rotational spectrum)
• Combine perturbation theory with other approximation methods, such as the variational method or the WKB approximation, to study more complex molecular systems (nitrogen molecule electronic structure)