Quantum Mechanical Model of the Atom

Energy of Orbit: En = -RH(1/n^2), where R_H = 2.18 \times 10^{-18} \text{ J}.
States: Ground (n=1), Excited (n > 1).
Transitions: Absorption (n{\text{lower}} \rightarrow n{\text{higher}}), Emission (n{\text{higher}} \rightarrow n{\text{lower}}).
Energy Change (\Delta E): \Delta E = -RH(1/nf^2 - 1/n_i^2).
Energy-Wavelength: E = hc/\lambda or \Delta E = h\nu.

Schrödinger Equation: Describes electron wave nature. \Psi^2 is probability density.
Atomic Orbital: 90% probability region for an electron.
Quantum Numbers (Q.N.):
- Principal (n): Energy level, size, distance from nucleus (n = 1, 2, 3, \ldots).
- Angular Momentum (l): Orbital shape (s=0, p=1, d=2, f=3). Values 0 to n-1.
- Magnetic (m_l): Orbital orientation (-l to +l).
- Spin (m_s): Electron spin (+1/2 or -1/2).
Pauli Exclusion Principle: No two electrons have identical four Q.N.
Orbital/Sublevel Characteristics:
- Sublevels per level = n.
- Max electrons per level = 2n^2.
- s: 1 orbital (spherical), p: 3 (dumbbell), d: 5 (complex), f: 7 (more complex).
Nodes: Regions with zero electron probability; increase with n.

Heisenberg Uncertainty Principle: Cannot simultaneously know exact position (\Delta x) and momentum (\Delta p) (\Delta x \cdot \Delta p \ge h/(4\pi)). Implies no exact orbits.
de Broglie Wavelength: Matter has wave-like properties; \lambda = h/(m \cdot v). Explained quantized orbits (2\pi r = n\lambda).

Bright-line Spectra: Unique light patterns from excited atoms, identify elements.
Hydrogen Series (Electron Transitions):
- Lyman: To n=1 (UV).
- Balmer: To n=2 (Visible/UV).
- Paschen: To n=3 (Infrared).
- Brackett: To n=4 (Infrared).
Bohr Model Limitation: Only accurate for single-electron species (e.g., H).