Hermite Polynomials and the Quantum Harmonic Oscillator: An Algebraic Approach

Abstract

This study presents a complete algebraic formulation of the quantum harmonic oscillator, demonstrating that the Hermite polynomials, H_n(x), emerge as the fundamental solution when the power-series method is applied to the Schrödinger equation. The physical requirement of square integrability enforces truncation of the series, leading precisely to these polynomials, which constitute the polynomial part of the energy eigenfunctions, ψ_n(x) = N_nH_n(αx)e−^α2x2/2, where n is the quantum number. This quantum number directly determines the quantized energy levels through E_n = ℏω (n + 1/2). The analysis based on creation and annihilation operators further reveals that the action of the creation operator on the ground state generates the excited states along with their corresponding Hermite functions. The orthogonality of these polynomials, given by ensures the orthonormality of the complete set of eigenfunctions. Therefore, Hermite polynomials encapsulate both the mathematical structure and the physical essence of quantization in this fundamental system.