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Lecture Content Overview

Lecture 4: Harmonic Oscillation

  • Harmonic Oscillation is fundamental in discussing molecular dynamics.

  • The focus is on electromagnetic (EM) waves and spectroscopy related to oscillation states in molecules.

    • Key Equation for Harmonic Oscillation:

      • Displacement equation: ( L = kx )

      • This illustrates how an object, when displaced, aims to return to its equilibrium.

  • The energy states in oscillation are constant and quantized, leading to standing wave solutions.

    • Wave Function Solutions:

      • ( Y = \frac{B}{T} H_n(q) e^{(-\frac{q^2}{2})} )

        • Where ( H_n ) are Hermite polynomials, representing the energy states.

  • Quantized Energy Levels:

    • The lowest energy state is above zero, termed Zero Point Energy (ZPE).

    • Particle tunneling potential is discussed as particles move beyond classical limits at high energy states.

  • Equal Energy States demonstrate symmetry and quantized harmonic motion.

Lecture 5: Hydrogen Atom

  • The model of the hydrogen atom explores the quantum behavior of particles confined in smaller dimensions.

    • Quantum Mechanical Properties:

      • Wave equation: ( H\Psi = E\Psi )

  • Quantum Numbers and Orbitals:

    • Quantum numbers define the state of electrons in an atom:

      • Principal Quantum Number (n): Determines energy levels.

      • Angular Momentum Quantum Number (l): Defines shape of orbital

      • Magnetic Quantum Number (m): Orientation of the orbital in space.

  • For each principal quantum number (n=1,2,3,...), a series of orbitals (s, p, d) exist:

    • Examples of quantum states:

      • n=2, l=1, m=-1,0,1 (2p orbital)

      • n=3, l=2, m=-2,-1,0,1,2 (3d orbital)

Radial Probability Density Function

  • The radial probability density function for electrons in orbitals is given by:

    • ( 4\pi r^2 [R(r)]^2 )

  • This allows calculation of the likelihood of finding an electron in a spherical volume.

  • Shapes of Orbitals:

    • Spherical harmonics describe spatial arrangements and distributions.

    • Each orbital has distinctive shapes based on quantized angular momentum (l).

Banding & Molecular Geometry

  • Molecular geometry is analyzed through symmetry and group theory concepts.

    • Molecular Geometry Examples:

      • Trigonal Planar (e.g., BCl3) can be represented with coordinate transformations.

  • Discussion includes the implications on electronic configurations and shapes of molecules in relation to bonding.

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