Introduction

• Proposed by Erwin Schrödinger in 1926.
• Describes the behavior of electrons in atoms using the principles of quantum mechanics.
• Replaced the earlier Bohr-Sommerfeld model by focusing on the wave-like nature of electrons.

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Main Features of the Quantum Mechanical Model

1. Wave-Particle Duality of Electrons
   • Electrons have both wave-like and particle-like behavior, as suggested by de Broglie.
   • Electron paths cannot be precisely determined, only probabilities can be calculated.
2. Schrödinger’s Wave Equation
   • Describes the wave nature of electrons in an atom:
   • H^ψ=Eψ
     • ψ: Wave function representing the probability amplitude of the electron.
     • E: Energy of the electron.
3. Electron Cloud and Orbitals
   • Electrons are found in regions called orbitals, not fixed orbits.
   • Orbitals represent regions where the probability of finding an electron is maximum (electron cloud).
4. Heisenberg’s Uncertainty Principle
   • Proposed by Werner Heisenberg:
     It is impossible to determine both the exact position and momentum of an electron simultaneously. Δx⋅Δp≥h/4π
5. Quantum Numbers
   • To describe an electron’s position and energy, four quantum numbers are used:
     1. Principal Quantum Number (n): Determines the size and energy of the orbital.
     2. Azimuthal Quantum Number (l): Determines the shape of the orbital.
     3. Magnetic Quantum Number (mₗ): Determines the orientation of the orbital.
     4. Spin Quantum Number (msm_s): Represents the spin of the electron ( +½ or -½).
6. Shapes of Orbitals
   • s-orbitals: Spherical in shape.
   • p-orbitals: Dumbbell-shaped.
   • d-orbitals: Complex shapes.

Advantages of the Quantum Mechanical Model

1. Describes the complex behavior of multi-electron atoms.
2. Explains the shapes and orientations of atomic orbitals.
3. Predicts the chemical properties of elements more accurately.
4. Incorporates the wave-like nature and probabilistic location of electrons.

Limitations of Earlier Models Overcome

1. Bohr’s Model Limitations
   • Fixed orbits replaced by probabilistic electron clouds.
   • Applies to all atoms, not just hydrogen.
2. Bohr-Sommerfeld Limitations
   • Relies on wave functions and does not require elliptical orbits.

Key Concepts Explained by Quantum Mechanical Model

1. Electronic Configuration
   • Electrons occupy orbitals in a specific order based on the Aufbau Principle, Pauli Exclusion Principle, and Hund’s Rule.
2. Periodic Table Structure
   • Explains the arrangement of elements in the periodic table based on their electron configurations.
3. Chemical Bonding
   • Helps predict how atoms form covalent or ionic bonds.

Limitations of Quantum Mechanical Model

1. The wave function (ψ\psi) does not have a direct physical meaning.
2. Highly mathematical and abstract, making it difficult to visualize.

Comparison with Bohr’s Model

Feature	Bohr’s Model	Quantum Mechanical Model
Nature of Electron Path	Fixed circular orbits	Probabilistic regions (orbitals)
Applicability	Hydrogen-like atoms only	All atoms
Basis	Classical mechanics + quantum theory	Fully quantum mechanical
Uncertainty Principle	Not addressed	Included

Conclusion

• The quantum mechanical model provides the most accurate and comprehensive description of atomic structure.
• It is the foundation of modern chemistry and physics.