Lecture Notes: CHEM 410A - Physical Chemistry

Lecture 5.2: Physical Chemistry - CHEM 410A

Basic Information

• Instructor: Yuezhi Mao
• Course: CHEM 410A (Physical Chemistry)
• Semester: Spring 2026
• Institution: San Diego State University (SDSU)

Recap of 1-D Harmonic Oscillator and Energy Levels

• Harmonic Potential:
  • A potential energy function typically represented in physical chemistry.
  • Configured as a parabolic curve which describes the restorative force experienced by a particle displaced from its equilibrium position.
• Schrödinger Equation (S.E.):
  • Fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time.
• Results from Solving the Schrödinger Equation:
  • Derives the energy levels of a quantum system.
• Energy Levels:
  • Defined by the vibrational quantum number (ν).
  • Levels are evenly spaced.
  • Represented mathematically as:
    $E_v = \bigg( <br />\nu + \frac{1}{2} \bigg) \bar{h}<br />\nu$
• Zero-Point Energy:
  • The lowest energy state (ν = 0) of the quantum oscillator which is not zero; it includes the value given by:
    $E_0 = \frac{1}{2} \bar{h}<br />\nu$

1-D Harmonic Oscillator Wavefunctions

Key Characteristics and Components

• Wavefunctions:
  • Depict the probability amplitude of a particle's position.
• Normalization Constant:
  • Ensure that the total probability across all space equals one.
  • This constant changes with the quantum number ν.
• Hermite Polynomials:
  • A set of orthogonal polynomials crucial for the solutions of the wavefunctions of the harmonic oscillator.
  • Their highest power corresponds directly to ν, and the polynomials will either exclusively represent odd or even powers depending on ν.
• Complete Set of Orthonormal Functions:
  • The H.O. wavefunctions can represent functions such that:
  • They are orthogonal to one another.
  • They can be normalized to 1.
• Ground-State Wavefunction (ν = 0):
  • Expressed as a standard Gaussian function given by:
    $ext{ψ}_0(x) = A e^{-\frac{x^2}{2a^2}}$
  • Where A is the normalization constant, and a is the characteristic length scale of the oscillator.

Features of Harmonic Oscillator Wavefunctions

• Even/Odd Functionality:
  • The wavefunction behaves as an even function when ν is even and as an odd function when ν is odd.
• Number of Nodes:
  • The number of nodes in the wavefunction corresponds to the vibrational quantum number ν.
• Behavior Between Classical Turning Points:
  • The wavefunction oscillates between the classical turning points but decays to 0 beyond those limits.
• Effect of Higher Quantum Numbers (ν):
  • As ν increases, the Hermite polynomial dimensions grow substantially before the Gaussian function rapidly decreases to zero.
  • This phenomenon leads to wavefunctions that spread out over a wider range, resulting in maxima of probability density moving towards classical turning points.
• Classical Turning Points:
  • These points are defined where energy levels intersect with the potential energy curve, given by the equation:
    $E = V, ext{ thus, } x_{tp} = \bigg(2ν + 1 \bigg) \frac{\bar{h}}{<br />\nu}$
  • Expresses the classical turning points of oscillator behavior, where
    $x_{tp} = \bigg( \frac{\bigg( 2ν + 1 \bigg) \bar{h}}{m\bar{ω}} \bigg)^{1/2}$
• Implications of Intrinsic Properties of the Oscillator:
  • Heavier Particles: This leads to smaller energy spacings:
    $\bar{ω} = \frac{k}{m}$
  • Stiffer Spring (Higher k): Results in larger energy spacings and produces narrower wavefunctions.

Evaluating Quantities from H.O. Wavefunctions

1. Normalization:
   • The process of ensuring the total probability of finding a particle in all space is equal to 1.
2. Expectation Values:
   • Measured averages of physical quantities derived from wavefunctions, useful in quantum mechanics to describe particle behavior.
3. Probability in Classically Forbidden Regions (Tunneling Probability):
   • Specifically discussed for the ground state (ν = 0) and includes cases where:
   • Probability beyond classical limits (i.e., for x > x{max} and $x < -x</em>{max}$).
   • Additionally provided a Python program to calculate tunneling probability for states where ν > 0.