Atomic structure is the foundation of modern physics. From Rutherford's gold foil experiment to the quantum model, our understanding of atoms has evolved dramatically. These insights explain emission spectra, energy levels, and the behavior of electrons in atoms.

Quantum mechanics takes atomic theory further, describing electron transitions and photon characteristics. The quantum mechanical model, with its orbitals and quantum numbers, provides a more accurate picture of electron behavior. This knowledge is crucial for understanding atomic interactions and chemical properties.

Atomic Structure

Rutherford's gold foil experiment

• Fired positively charged alpha particles at thin gold foil expected most to pass through with minimal deflection
• Observations showed most particles passed through undeflected, some deflected at large angles, and a few deflected back towards source
• Implications revealed atom is mostly empty space with positive charge and mass concentrated in small, dense nucleus and electrons orbiting at relatively large distance

Atomic spectra significance

• Emission spectrum appears as bright lines on dark background when electrons transition from higher to lower energy levels, each element has unique spectrum (hydrogen, helium)
• Absorption spectrum appears as dark lines on bright background when electrons transition from lower to higher energy levels, corresponds to same wavelengths as emission spectrum
• Spectra provide evidence for discrete energy levels in atoms and are used to identify elements and compositions in various substances (stars, gases)

Bohr model vs quantum model

• Bohr model has electrons orbiting nucleus in fixed, circular orbits transitioning between specific energy levels with emission and absorption of photons during transitions
• Quantum model has electrons existing in probability distributions called orbitals described by quantum numbers (n, l, m, s) with quantized energy levels not always in circular orbits
• Both models account for discrete energy levels and explain emission and absorption spectra
• Quantum model is more accurate accounting for sublevels and orbitals while Bohr model is simpler but limited in describing complex atoms

Quantum Mechanics and Atomic Transitions

Electron energy transitions

• Energy change calculated as $\Delta E = E_f - E_i$ where $E_f$ is final energy level and $E_i$ is initial energy level
• Bohr model calculates energy as $E_n = -\frac{13.6 eV}{n^2}$ where $n$ is principal quantum number
• Quantum model calculates energy as $E_n = -\frac{13.6 eV}{n^2} + \text{correction terms}$ accounting for electron-electron interactions and relativistic effects

Photon characteristics in transitions

• Photon energy calculated as $E = h\nu = \frac{hc}{\lambda}$ where $h$ is Planck's constant ($6.626 \times 10^{-34}$ J⋅s), $\nu$ is frequency, $c$ is speed of light ($2.998 \times 10^8$ m/s), and $\lambda$ is wavelength
• Frequency calculated as $\nu = \frac{\Delta E}{h}$ and wavelength calculated as $\lambda = \frac{hc}{\Delta E}$

Quantum mechanical model features

• Electrons exist in probability distributions called orbitals described by four quantum numbers:
  1. Principal quantum number (n) determines main energy level and size of orbital
  2. Angular momentum quantum number (l) determines shape of orbital (s, p, d, f)
  3. Magnetic quantum number (m) determines orientation of orbital in space
  4. Spin quantum number (s) describes intrinsic angular momentum of electron (up or down)
• Pauli exclusion principle states no two electrons in an atom can have same set of quantum numbers
• Aufbau principle states electrons fill orbitals in order of increasing energy (1s, 2s, 2p, 3s)
• Hund's rule states electrons occupy orbitals of same energy singly before pairing up

Wave function and electron behavior

• The wave function (ψ) describes the quantum state of an electron in an atom
• The Schrödinger equation is used to solve for the wave function and determine allowed energy states
• The square of the wave function (|ψ|²) gives the electron probability density, which represents the likelihood of finding an electron in a particular region of space
• Atomic orbitals are three-dimensional representations of electron probability distributions
• Electron configuration describes the arrangement of electrons in atomic orbitals, following the aufbau principle, Pauli exclusion principle, and Hund's rule