The Harmonic Oscillator
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Harmonic Oscillator: A system in which a particle experiences a restoring force proportional to its displacement from equilibrium. It is a fundamental model for understanding molecular vibrations 1. Schrödinger Equation: A key equation in quantum mechanics that describes how the quantum state of a physical system changes over time. In this chapter, it is solved for the harmonic oscillator 1. Power-Series Method: A mathematical technique used to solve differential equations by expressing the solution as an infinite sum of terms 1. Force Constant (k): A parameter that measures the stiffness of the bond in a molecule. It is the proportionality constant in the force equation ( F = -kx ) 1. Vibration Frequency (ν): The frequency at which a molecule vibrates. For a harmonic oscillator, it is given by ( \nu = \frac{1}{2\pi} \sqrt{\frac{k}{m}} ) 1. Zero-Point Energy: The lowest possible energy that a quantum mechanical system may have. For a harmonic oscillator, it is ( \frac{1}{2} h \nu ) 1. Eigenvalues and Eigenfunctions: Solutions to the Schrödinger equation that describe the allowed energy levels and corresponding wave functions of a quantum system 1. Recursion Relation: A relation that defines each term of a sequence as a function of preceding terms. It is used in the power-series method to solve the Schrödinger equation 1. Hermite Polynomials: A set of orthogonal polynomials that arise in the solution of the Schrödinger equation for the harmonic oscillator 1. Classically Forbidden Region: Regions where the potential energy exceeds the total energy of the system, making it impossible for a classical particle to be found there 1. Reduced Mass (m): A hypothetical mass used in the analysis of two-body problems, defined as ( m = \frac{m_1 m_2}{m_1 + m_2} ) 1. Anharmonicity: The deviation of a system from the ideal harmonic oscillator model, leading to non-equally spaced energy levels 1. Boltzmann Distribution Law: A statistical law that describes the distribution of particles among various energy states in thermal equilibrium 1. Wavenumber (ν̅): The number of wavelengths per unit distance, often used in spectroscopy. It is defined as ( \nu̅ = \frac{1}{λ} = \frac{ν}{c} ) 1.