Transcription of Hermite and Laguerre Polynomials
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Hermite and Laguerre Polynomials
chapter 13 Hermite and LaguerrePolynomialsIn this chapter we study two sets of orthogonal Polynomials , Hermite andLaguerre Polynomials . These sets are less common in mathematical physicsthan the Legendre and Bessel functions of Chapters 11 and 12, but Hermitepolynomials occur in solutions of the simple harmonic oscillator of quantummechanics and Laguerre Polynomials in wave functions of the hydrogen the general mathematical techniques are similar to those of thepreceding two chapters, the development of these functions is only detailed proofs, along the lines of Chapters 11 and 12, are left to thereader. We start with Hermite Hermite PolynomialsQuantum Mechanical Simple Harmonic OscillatorFor the physicist, Hermite Polynomials are synonymous with the one-dimensional ( , simple) harmonic oscillator of quantum mechanics. For apotential energyV=12Kz2=12m 2z2,forceFz= V/ z= Kz,the Schr odinger equation of the quantum mechanical system is h22md2dz2 (z)+12Kz2 (z)=E (z).
640 Chapter 13 Hermite and Laguerre Polynomials 0.5 5 x y 2(x) 0.5 5 y 1(x) Figure 13.1 Quantum Mechanical Oscillator Wave Functions. The Heavy Bar on the x-Axis Indicates the Allowed Range of the Classical Oscillator with the Same Total Energy almost factorizes. Using naively a2 −b2 = (a−b)(a+b), the basic commutator [p x, x] =h¯/i of ...
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