Energy of emitted/absorbed photon: $E = h\nu = hc/\lambda$

Successfully explained the line spectra of hydrogen.

Limitations: Only works for hydrogen and hydrogen-like ions; doesn't explain multi-electron atoms or spectral line intensities.

4. Quantum Mechanical Model: Electron Clouds (Modern View)

Analogy: Instead of planets, electrons are like buzzing bees around a hive (the nucleus). You can't pinpoint a bee's exact position, but you know it's *somewhere* within the cloud around the hive. The denser parts of the cloud indicate where the bee is *most likely* to be found.

Replaced discrete orbits with probability distributions (orbitals).
Based on wave-particle duality and Heisenberg's Uncertainty Principle.

4.1. Wave-Particle Duality

Analogy: Light and matter are like a coin that can show "heads" (wave) or "tails" (particle) depending on how you look at it. It's both, but you only observe one aspect at a time.

Light: Behaves as both a wave (diffraction, interference) and a particle (photons, photoelectric effect).
Matter (de Broglie Wavelength): Particles like electrons also exhibit wave-like properties.
- $\lambda = h/p = h/(mv)$ where $h$ is Planck's constant, $p$ is momentum, $m$ is mass, $v$ is velocity.

4.2. Heisenberg's Uncertainty Principle

Analogy: Trying to precisely measure both the position and momentum of an electron is like trying to catch a greased watermelon while also measuring its speed. The act of measuring one inevitably disturbs the other. You can know one accurately, but not both simultaneously.

It's impossible to precisely know both the position ($x$) and momentum ($p$) of a particle simultaneously.
$\Delta x \cdot \Delta p \ge \hbar/2$ (where $\hbar = h/(2\pi)$)
This principle means we cannot plot an exact orbit for an electron.

4.3. Schrödinger Equation

Analogy: This is the "rulebook" for the electron bee. Solving it gives you the probability cloud (orbital) where the electron is likely to be found and its allowed energies.

Mathematical equation that describes the wave function ($\Psi$) of a particle.
$H\Psi = E\Psi$ (time-independent Schrödinger equation)
The square of the wave function, $|\Psi|^2$, gives the probability density of finding an electron in a particular region of space.

5. Quantum Numbers: Electron's Address

Analogy: Quantum numbers are like an electron's mailing address, specifying its state and location within the atom.

Principal Quantum Number (n): The street address (main energy level/shell).
Angular Momentum Quantum Number (l): The neighborhood (subshell/orbital shape).
Magnetic Quantum Number (m_l): The house number (orientation of orbital in space).
Spin Quantum Number (m_s): The resident's unique characteristic (electron spin).

5.1. Principal Quantum Number ($n$)

Values: $1, 2, 3, \dots$ (positive integers).
Determines the main energy level and average distance of the electron from the nucleus.
Larger $n$ means higher energy and larger orbital.
Corresponds to Bohr's shells (K, L, M...).

5.2. Angular Momentum (Azimuthal) Quantum Number ($l$)

Values: $0, 1, 2, \dots, n-1$.
Determines the shape of the orbital (subshell).
$l=0 \implies s$ orbital (spherical)
$l=1 \implies p$ orbital (dumbbell-shaped)
$l=2 \implies d$ orbital (more complex, cloverleaf)
$l=3 \impl