Lecture Notes on Structure of Atom - Lecture 8

Key Topics Covered

• Dual Nature of Matter
• De Broglie Wavelength
• Heisenberg Uncertainty Principle
• Quantum Mechanical Model of Atom

Recap & Homework Discussion

• Discussed last session's homework questions:
  • Energy levels and electron transitions in hydrogen and helium atoms.
  • Velocity of electron in different orbits.
  • Ionization energy calculations.
  • Shortest wavelength and energy relationships for hydrogen spectrum.

Dual Nature of Matter

Key Points

• Matter exhibits dual nature: Particle nature & Wave nature.
• Proposed by Louis de Broglie.
• All matter particles exhibit a wave nature; associated wavelength is called de Broglie wavelength.

• Important formula:

  $$ ext{De Broglie Wavelength (λ)} = \frac{h}{{p}} = \frac{h}{{mv}}$$
  • Where, h = Planck's constant, m = mass of the particle, v = velocity of the particle.

Application Formulas

• Wavelength in terms of kinetic energy for charged particle:
  • $$ ext{λ} = \frac{h}{ ext{mv}} = \frac{h}{{ ext{√2mE}_{k}}}$$
• For particle accelerated through potential V:
  • $$ ext{λ} = \frac{h}{{ ext{√2mqV}}}$$
• For electrons:
  • $$ ext{λ} = \frac{12.27}{{ ext{√V}}}$$ (Å)
• For protons and alpha particles:
  • $$λ_{proton} = \frac{286}{{ ext{√V}}}$$ (Å)
  • $$λ_{alpha} = \frac{10}{{ ext{√V}}}$$ (Å)

Significant Implications

• De Broglie wavelength significant for microscopic particles only.
• Macroscopic objects have very small wavelengths, essentially zero.

Heisenberg Uncertainty Principle

Key Points

• States that the position and momentum of a particle cannot both be precisely determined at the same time.
• Formula:
  • $$Δx * Δp ≥ \frac{h}{4π}$$
  • Where, Δx = uncertainty in position, Δp = uncertainty in momentum.
• For velocity:
  • $$ Δx * m * Δv ≥ \frac{h}{4π}$$

Practical Implications

• Significant for microscopic particles like electrons.
• For macroscopic particles, these uncertainties are negligible.
• Contradicts Bohr's model which assumed fixed paths for electrons.

Quantum Mechanical Model

Key Points

• Based on both particle and wave nature of electrons.
• Developed by Schrödinger, introduced wave function ψ (psi).
• Schrödinger wave equation:
  • $$\frac{∂^2ψ}{∂x^2} + \frac{∂^2ψ}{∂y^2} + \frac{∂^2ψ}{∂z^2} + 8π^2m/h^2(E−V)ψ = 0$$
• Simplified form:
  • $$Hψ = Eψ$$
  • Here H is the Hamiltonian operator.
• Quantum numbers arise from solutions of Schrödinger equation, they describe the distribution of electrons in atoms.
• ψ^2 (psi-squared) represents probability density; indicates probability of finding an electron.

Orbital Concept

• Region with high probability of finding an electron is called an orbital, differing from Bohr's orbit.
• Defines spatial distribution of electrons.

Important Results

• Bohr’s model failures due to non-consideration of the wave nature of electrons and uncertain position/momentum.
• Introduced modern quantum mechanical model accommodating electron's dual nature.

Note: Thorough understanding of these principles is crucial for correctly interpreting atomic structure and behavior, especially for JEE and advanced studies.