Sections#

Review & Summary#

Properties of Atoms#

Atoms have quantized energy levels. The electron has spin \(s = 1/2\) and orbital angular momentum \(\ell = 0, 1, 2, \ldots\) (with \(m_\ell = -\ell, \ldots, +\ell\)). The combination determines the magnetic properties and the structure of the periodic table.

The Stern-Gerlach Experiment#

A beam of silver atoms passing through an inhomogeneous magnetic field splits into discrete beams. This reveals that the magnetic moment (and thus angular momentum) is quantized. The two beams correspond to the two possible values of the \(z\)-component of electron spin: \(S_z = \pm \hbar/2\).

Magnetic Resonance#

Nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) exploit the precession of magnetic moments in an external magnetic field \(\vec{B}\). The Larmor frequency is \(\omega_L = \gamma B\), where \(\gamma\) is the gyromagnetic ratio. Resonance occurs when an applied RF field matches this frequency, causing transitions between spin states.

Exclusion Principle and Multiple Electrons#

The Pauli exclusion principle: no two electrons in an atom can have the same set of quantum numbers \((n, \ell, m_\ell, m_s)\). This determines the maximum occupancy of each subshell (e.g., 2 electrons per orbital) and the structure of multi-electron atoms.

Building the Periodic Table#

The periodic table is built by filling orbitals in order of increasing energy: \(1s\), \(2s\), \(2p\), \(3s\), \(3p\), \(4s\), \(3d\), \(\ldots\) Hund’s rule: when filling degenerate orbitals (e.g., the three \(p\) orbitals), maximize the total spin—place electrons in separate orbitals with parallel spins before pairing.

X Rays and the Ordering of the Elements#

Characteristic x-rays are emitted when a high-energy electron knocks out an inner-shell electron and an outer electron falls into the vacancy. The frequency of the \(K_\alpha\) line obeys Moseley’s law: \(\sqrt{f} \propto Z - 1\), where \(Z\) is the atomic number. This established the ordering of elements in the periodic table.

Lasers#

A laser (Light Amplification by Stimulated Emission of Radiation) produces coherent, monochromatic, highly directional light. Stimulated emission occurs when an incident photon induces an excited atom to emit a photon in phase with the incident one. Population inversion (more atoms in the upper state than the lower) is required for net amplification. Applications include spectroscopy, communications, medicine, and manufacturing.