5.1 Definition and Properties of Bessel Functions
Bessel functions appear as solutions to wave equations and diffusion equations in cylindrical coordinate systems.
📐 Definition: Bessel Differential Equation
$$x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2 - \nu^2)y = 0$$ Bessel function of the first kind $J_\nu(x)$: $$J_\nu(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!\Gamma(m+\nu+1)} \left(\frac{x}{2}\right)^{2m+\nu}$$
$$x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2 - \nu^2)y = 0$$ Bessel function of the first kind $J_\nu(x)$: $$J_\nu(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!\Gamma(m+\nu+1)} \left(\frac{x}{2}\right)^{2m+\nu}$$
💻 Code Example 1: Calculation and Visualization of Bessel Functions
Python Implementation: Basic Properties of Bessel Functions
# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0
"""
Example: 💻 Code Example 1: Calculation and Visualization of Bessel Fu
Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""
import numpy as np
import matplotlib.pyplot as plt
from scipy.special import jv, yv, jn_zeros
# Visualization of Bessel functions
x = np.linspace(0, 20, 1000)
orders = [0, 1, 2, 3, 4]
fig, axes = plt.subplots(2, 2, figsize=(14, 12))
# Bessel function of the first kind J_ν(x)
ax = axes[0, 0]
for nu in orders:
y = jv(nu, x)
ax.plot(x, y, linewidth=2, label=f'$J_{{{nu}}}(x)$')
ax.legend()
ax.set_title('Bessel Function of the First Kind')
# Visualization omitted (see original code)5.2 Wave Equation in Cylindrical Coordinates
Solutions to wave equations in cylindrical coordinate systems are represented by Bessel functions.
📐 Theorem: Wave Equation in Cylindrical Coordinates
$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} + \frac{\partial^2 u}{\partial z^2} = \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}$$ Separation of variables solution: $$u(r,\theta,z,t) = J_m(k_r r) e^{im\theta} e^{ik_z z} e^{-i\omega t}$$
$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} + \frac{\partial^2 u}{\partial z^2} = \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}$$ Separation of variables solution: $$u(r,\theta,z,t) = J_m(k_r r) e^{im\theta} e^{ik_z z} e^{-i\omega t}$$
5.3 Legendre Polynomials
In boundary value problems in spherical coordinate systems, Legendre polynomials play an important role.
📐 Definition: Legendre Differential Equation
$$\frac{d}{dx}\left[(1-x^2)\frac{dy}{dx}\right] + n(n+1)y = 0$$ Rodrigues' formula: $$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}[(x^2-1)^n]$$
$$\frac{d}{dx}\left[(1-x^2)\frac{dy}{dx}\right] + n(n+1)y = 0$$ Rodrigues' formula: $$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}[(x^2-1)^n]$$
5.4 Spherical Harmonics
For three-dimensional problems in spherical coordinate systems, we use spherical harmonics constructed from associated Legendre functions.
📐 Definition: Spherical Harmonics
$$Y_l^m(\theta, \phi) = \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi}$$ where $P_l^m$ is associated Legendre function
$$Y_l^m(\theta, \phi) = \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) e^{im\phi}$$ where $P_l^m$ is associated Legendre function
📝 Applications to Quantum Mechanics:
- Angular part of hydrogen atom wavefunction
- Eigenstates of angular momentum
- $l$: azimuthal quantum number, $m$: magnetic quantum number ($-l \leq m \leq l$)
5.5 Hermite Polynomials
Learn Hermite polynomials that appear as quantum mechanical solutions of harmonic oscillator.
📐 Definition: Hermite Differential Equation
$$\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + 2ny = 0$$ Rodrigues' formula: $$H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}$$
$$\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + 2ny = 0$$ Rodrigues' formula: $$H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}$$
🔬 Application Example: Energy eigenvalues of harmonic oscillator
$$E_n = \hbar\omega\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \ldots$$ Wavefunction: $$\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{1}{\pi}\right)^{1/4} e^{-x^2/2} H_n(x)$$
$$E_n = \hbar\omega\left(n + \frac{1}{2}\right), \quad n = 0, 1, 2, \ldots$$ Wavefunction: $$\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{1}{\pi}\right)^{1/4} e^{-x^2/2} H_n(x)$$
5.6 Laguerre Polynomials
Learn Laguerre polynomials that appear in radial wavefunctions of hydrogen atom.
📐 Definition: Generalized Laguerre Polynomials
$$L_n^{\alpha}(x) = \frac{e^x x^{-\alpha}}{n!} \frac{d^n}{dx^n}(e^{-x} x^{n+\alpha})$$
$$L_n^{\alpha}(x) = \frac{e^x x^{-\alpha}}{n!} \frac{d^n}{dx^n}(e^{-x} x^{n+\alpha})$$
🔬 Application Example: Energy levels of hydrogen atom
$$E_n = -\frac{13.6 \text{ eV}}{n^2}, \quad n = 1, 2, 3, \ldots$$ Radial wavefunction includes $L_{n-l-1}^{2l+1}$
$$E_n = -\frac{13.6 \text{ eV}}{n^2}, \quad n = 1, 2, 3, \ldots$$ Radial wavefunction includes $L_{n-l-1}^{2l+1}$
5.7 Summary and Relationships of Special Functions
📝 Correspondence between Coordinate Systems and Special Functions:
- Cartesian coordinates: Trigonometric functions (sin, cos)
- Cylindrical coordinates: Bessel functions ($J_n$, $Y_n$)
- Spherical coordinates: Legendre polynomials ($P_n$), Spherical harmonics ($Y_l^m$)
📝 Quantum Mechanics and Special Functions:
- Harmonic oscillator: Hermite polynomials ($H_n$)
- Hydrogen atom (radial): Laguerre polynomials ($L_n^{\alpha}$)
- Hydrogen atom (angular): Spherical harmonics ($Y_l^m$)
- Angular momentum: Spherical harmonics ($Y_l^m$)
5.8 Applications to Materials Science (1): Heat Conduction in Cylindrical Samples
Solve heat conduction equation in cylindrical coordinate system with Bessel functions.
🔬 Physical Significance:
Solution to heat conduction equation: $$T(r,t) = T_{\text{surface}} + \sum_{n} A_n J_0(\lambda_n r) e^{-\alpha \lambda_n^2 t}$$ Characteristic time: $\tau = R^2/\alpha$ (temperature decays to about 37%)
Solution to heat conduction equation: $$T(r,t) = T_{\text{surface}} + \sum_{n} A_n J_0(\lambda_n r) e^{-\alpha \lambda_n^2 t}$$ Characteristic time: $\tau = R^2/\alpha$ (temperature decays to about 37%)
5.9 Applications to Materials Science (2): Diffusion Problems in Spherical Particles
Solutions to diffusion equations in spherical coordinate systems and applications to drug release systems.
🔬 Application Examples:
- Drug release capsules (DDS: Drug Delivery System)
- Reactant diffusion from catalyst particles
- Dissolution of nanoparticles
📝 Chapter Exercises
✏️ Exercises
- Numerically find the first 3 positive zeros of $J_0(x)$ and calculate eigenfrequencies of circular membrane.
- Verify orthogonality of Legendre polynomials $P_2(x)$ and $P_3(x)$ by numerical integration.
- Visualize spherical harmonic $Y_2^1(\theta, \phi)$ and confirm its nodal lines.
- Confirm number of nodes in wavefunction $\psi_3(x)$ of harmonic oscillator and find positions of classical turning points.
Summary
- Special functions are solutions to differential equations according to coordinate systems and boundary conditions
- Bessel functions: Wave and diffusion problems in cylindrical coordinates
- Legendre polynomials: Laplace equation in spherical coordinates
- Spherical harmonics: Angular dependence of 3D spherically symmetric problems
- Hermite polynomials: Harmonic oscillator (quantum mechanics)
- Laguerre polynomials: Radial wavefunction of hydrogen atom
- Wide applications in materials science (heat conduction, diffusion) and quantum mechanics