8.3 Hermite Functions and Quantum Harmonic Oscillator

Hermite polynomials are special functions crucial in quantum mechanics. They're used to solve the Schrödinger equation for harmonic oscillators, describing particle wavefunctions in various systems. These polynomials form a complete set of orthogonal functions, making them ideal for expanding quantum states.

In physics, Hermite polynomials pop up everywhere from molecular vibrations to quantum field theory. They help us understand energy levels, calculate thermodynamic properties, and model complex quantum systems. Their versatility makes them a cornerstone of mathematical physics and quantum mechanics.

Hermite Polynomials

Definition of Hermite polynomials

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• Hermite polynomials $H_n(x)$ are a set of orthogonal polynomials defined on the interval $(-\infty, \infty)$
• Satisfy the Hermite differential equation $\frac{d^2H_n(x)}{dx^2} - 2x\frac{dH_n(x)}{dx} + 2nH_n(x) = 0$
• First few Hermite polynomials: $H_0(x) = 1$, $H_1(x) = 2x$, $H_2(x) = 4x^2 - 2$, $H_3(x) = 8x^3 - 12x$
• Generate higher-order polynomials using recurrence relation $H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)$
• Orthogonal with respect to weight function $e^{-x^2}$ on $(-\infty, \infty)$
• Orthogonality relation $\int_{-\infty}^{\infty} H_m(x)H_n(x)e^{-x^2}dx = \sqrt{\pi}2^n n! \delta_{mn}$ where $\delta_{mn}$ is Kronecker delta (equals 1 if $m = n$, 0 otherwise)
• Used in various fields of physics and mathematics (quantum mechanics, probability theory)

Eigenfunctions with Hermite functions

• Eigenfunctions of quantum harmonic oscillator $\psi_n(x)$ expressed using Hermite functions $\phi_n(x)$
• Hermite functions related to Hermite polynomials by $\phi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} e^{-\frac{x^2}{2}} H_n(x)$
• Quantum harmonic oscillator eigenfunctions given by $\psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{1}{\sqrt{2^n n!}} e^{-\frac{m\omega x^2}{2\hbar}} H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right)$
  • $m$ is particle mass, $\omega$ is angular frequency, $\hbar$ is reduced Planck's constant
• Eigenfunctions form a complete orthonormal set in Hilbert space
• Used to solve Schrödinger equation for harmonic oscillator potential $V(x) = \frac{1}{2}m\omega^2x^2$

Energy eigenvalues calculation

• Quantum harmonic oscillator energy eigenvalues $E_n$ are quantized $E_n = \hbar\omega\left(n + \frac{1}{2}\right)$, $n = 0, 1, 2, ...$
• Ground state energy $E_0 = \frac{1}{2}\hbar\omega$ is non-zero (zero-point energy)
• Energy level spacing between adjacent eigenstates is constant $\Delta E = \hbar\omega$
• Eigenvalues obtained by solving Schrödinger equation with harmonic oscillator potential using Hermite functions
• Quantized energy levels result in discrete absorption and emission spectra (observed in molecular vibrations, photons in cavity)

Applications in quantum mechanics

• Describe wavefunctions of particles in harmonic potentials (diatomic molecules, quantum dots)
• Model physical systems with harmonic oscillator behavior (phonons in solids, electromagnetic field quantization)
• Calculate partition function and thermodynamic properties of quantum harmonic oscillator in statistical mechanics
  • Partition function $Z = \sum_{n=0}^{\infty} e^{-\beta E_n}$ where $\beta = \frac{1}{k_BT}$, $k_B$ is Boltzmann constant, $T$ is temperature
  • Average energy $\langle E \rangle = -\frac{\partial \ln Z}{\partial \beta}$, heat capacity $C = \frac{\partial \langle E \rangle}{\partial T}$, entropy $S = k_B \ln Z + \beta \langle E \rangle$
• Used in quantum field theory to describe excitations of fields as harmonic oscillators (quantum electrodynamics, quantum chromodynamics)