Quantum Atomic Structure — Wave Functions, Orbitals & Spin Explained | Chapter 41 of University Physics

Chapter 41 extends quantum mechanics into three dimensions, applying Schrödinger’s equation to atomic systems. Explore how quantum numbers define wave functions and orbitals, and uncover the roles of spin, magnetic fields, and entanglement in shaping atomic behavior.

Be sure to watch the full video summary for animations of atomic orbitals and spectral line splitting.

3D Schrödinger Equation & Atomic Wave Functions

In three dimensions, the time-independent Schrödinger equation

−(ħ²/2m)∇²ψ + U(r)ψ = Eψ governs atomic wave functions ψ(x,y,z). These stationary states, when normalized, satisfy ∫|ψ|² dV = 1, giving the probability density for locating electrons in space.

Particle in a Cubical Box

A particle confined to a cube of side L has energy levels:

E = (ħ²π²/2mL²)(n_x² + n_y² + n_z²), with quantum numbers (n_x,n_y,n_z) = 1,2,3…

Degenerate states share the same E despite different quantum‐number combinations, producing 3D standing‐wave patterns in |ψ|².

The Hydrogen Atom & Quantum Numbers

Solving in spherical coordinates yields energy levels E_n = –13.6 eV/n² and four quantum numbers:

• n (principal): energy shell
• l (orbital): shape (s,p,d,f…)
• m_l (magnetic): orientation
• m_s (spin): ±½

Radial probability distributions reveal the most likely electron distances for each (n,l) orbital.

Zeeman Effect & Fine Structure

An external magnetic field splits degenerate m_l levels by ΔE = m_lμ_BB (normal Zeeman effect). Electron spin adds a magnetic moment, and spin–orbit coupling (J = L + S) produces fine‐structure energy shifts in spectral lines.

Many-Electron Atoms & Pauli Exclusion

The Pauli exclusion principle prohibits identical quantum states for two electrons, shaping atomic electron configurations. Central‐field approximations and effective nuclear charge (Z_eff) explain shell structure and chemical behavior. This underlies the periodic table’s arrangement.

X-Ray Spectra & Moseley’s Law

When inner electrons are ejected, outer electrons transition to fill vacancies, emitting characteristic X-rays (e.g., K_α). Moseley’s law (f ∝ (Z–1)²) accurately predicts these frequencies, providing a tool for elemental identification.

Quantum Entanglement & Qubits

Entangled particles share a single wave function regardless of separation. Measuring one instantly determines the other’s state—a phenomenon harnessed in quantum computing and cryptography. Qubits exploit superposition and entanglement to process information in parallel.

Conclusion

Chapter 41 reveals how three-dimensional quantum mechanics defines atomic structure, from orbital shapes to magnetic interactions and entanglement. Master these concepts to understand spectroscopy, quantum technologies, and the behavior of matter at the smallest scales.

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