Quantum Mechanical Model of the Atom

🌟 Hydrogen Atom Energy & Wavelength

When an electron jumps from n=5 to n=2 in hydrogen:

  • Energy change: \[\Delta E = 2.18 \times 10^{-18} \text{J} \left( \frac{1}{5^2} – \frac{1}{2^2} \right) = -4.58 \times 10^{-19} \text{J}\] (Negative = energy released!)
  • Photon frequency: \[\nu = \frac{\Delta E}{h} = \frac{4.58 \times 10^{-19}}{6.626 \times 10^{-34}} = 6.91 \times 10^{14} \text{Hz}\]
  • Wavelength: \[\lambda = \frac{c}{\nu} = \frac{3.0 \times 10^8}{6.91 \times 10^{14}} = 434 \text{ nm}\] (Visible blue light! 🔵)

⚛️ Hydrogen-like Atoms (He⁺)

For He⁺ ion (Z=2) in first orbit (n=1):

  • Energy: \[E_1 = \frac{(2.18 \times 10^{-18} \text{J}) \cdot (2)^2}{1^2} = -8.72 \times 10^{-18} \text{J}\]
  • Orbit radius: \[r_n = \frac{(0.0529 \text{ nm}) \cdot n^2}{Z} = \frac{0.0529 \cdot 1^2}{2} = 0.02645 \text{ nm}\] (Smaller than hydrogen! ⚡)

🚫 Limitations of Bohr’s Model

Bohr’s model couldn’t explain:

  1. Fine details in hydrogen spectrum (doublet lines)
  2. Spectra of other atoms (like helium)
  3. Spectral line splitting in magnetic/electric fields (Zeeman/Stark effects)
  4. Chemical bonding between atoms

👉 Needed a better theory!

🌊 Dual Behavior of Matter (de Broglie)

All moving particles have wave-like properties!

  • de Broglie wavelength: \[\lambda = \frac{h}{mv} = \frac{h}{p}\]

💡 Cool Examples:

Cricket ball (0.1 kg at 10 m/s): \[\lambda = \frac{6.626 \times 10^{-34}}{0.1 \times 10} = 6.626 \times 10^{-34} \text{m}\] (Too tiny to detect! ⚾)

Electron (mass = \(9.1 \times 10^{-31}\) kg, K.E. = \(3.0 \times 10^{-25}\) J): \[v = \sqrt{\frac{2 \cdot \text{K.E.}}{m}} = 812 \text{ m/s}, \quad \lambda = \frac{h}{mv} = 896.7 \text{ nm}\] (Measurable wavelength! 🔍)

🎯 Heisenberg Uncertainty Principle

Can’t know position AND velocity perfectly!

  • Math form: \[\Delta x \cdot \Delta p_x \geq \frac{h}{4\pi}\]
  • If you know position well (\(\Delta x\) small), velocity is uncertain (\(\Delta v\) large) ↔️

💡 Practice Problems:

Electron (position uncertainty = 0.1 Å): \[\Delta v = \frac{h}{4\pi \cdot m \cdot \Delta x} = 5.79 \times 10^6 \text{ m/s}\] (Huge uncertainty! 🌪️)

Golf ball (40 g, speed uncertainty 2% of 45 m/s): \[\Delta v = 0.9 \text{ m/s}, \quad \Delta x = \frac{h}{4\pi m \Delta v} = 1.46 \times 10^{-33} \text{m}\] (Super tiny – no problem! ⛳)

🚀 Quantum Mechanical Model (Schrödinger)

Electrons behave as probability waves 🌊:

  • Wave equation: \[\hat{H} \psi = E \psi\]
  • \(\psi\) = wave function (describes electron’s “cloud”)
  • Electron states defined by 3 quantum numbers: \(n, l, m_l\)
  • Solves Bohr’s limitations ✅

📚 NEET Must-Knows

  1. Calculate wavelength/energy for hydrogen transitions (Balmer series)
  2. Find energy & radius for hydrogen-like ions (He⁺, Li²⁺)
  3. Calculate de Broglie wavelength for electrons/objects
  4. Solve uncertainty principle problems
  5. Explain why Bohr’s model failed (duality + uncertainty)