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Quantum Mechanics

Special Functions

Special functions enter quantum mechanics because boundary conditions, symmetry, and asymptotic behavior turn differential equations into named eigenfunction families.

Function Map

Section titled “Function Map”
Function Family Typical Equation or Role Quantum-Mechanics Use Toolkit Page
Hermite polynomials Hn(x)H_n(x) oscillator equation after Gaussian factorization harmonic oscillator Hermite Polynomials
Associated Legendre functions Pℓm(x)P_\ell^m(x) angular equation on the sphere spherical harmonics, rotor, central potentials Legendre Polynomials
Associated Laguerre polynomials Lnα(x)L_n^\alpha(x) radial Coulomb equation hydrogenic radial wavefunctions Laguerre Polynomials
Bessel functions Jν(x)J_\nu(x) cylindrical radial equation cylindrical wells, scattering, partial waves Bessel Functions
Airy functions Ai⁡(x)\operatorname{Ai}(x), Bi⁡(x)\operatorname{Bi}(x) linear turning-point equation WKB connection and uniform approximations Airy Functions
Gamma function Γ(z)\Gamma(z) factorial continuation normalization integrals and special-function identities Gamma and Beta Functions
Hypergeometric functions common master family for many ODEs solvable potentials and analytic continuation Hypergeometric Functions

Orthogonality Patterns

Section titled “Orthogonality Patterns”
Family Schematic Orthogonality
Hermite ∫−∞∞e−x2Hm(x)Hn(x) dx∝δmn\int_{-\infty}^{\infty}e^{-x^2}H_m(x)H_n(x)\,dx\propto\delta_{mn}
Legendre ∫−11Pℓ(x)Pℓ′(x) dx∝δℓℓ′\int_{-1}^{1}P_\ell(x)P_{\ell'}(x)\,dx\propto\delta_{\ell\ell'}
Laguerre ∫0∞e−xxαLnα(x)Lmα(x) dx∝δnm\int_0^\infty e^{-x}x^\alpha L_n^\alpha(x)L_m^\alpha(x)\,dx\propto\delta_{nm}
Spherical harmonics ∫dΩ Yℓm∗Yℓ′m′=δℓℓ′δmm′\int d\Omega\,Y_{\ell m}^*Y_{\ell' m'}=\delta_{\ell\ell'}\delta_{mm'}

Common Mistakes

Section titled “Common Mistakes”
  • Forgetting the weight function in an orthogonality relation.
  • Confusing a polynomial with the full normalized wavefunction that includes exponential and measure factors.
  • Using a physics convention for Hermite polynomials where a probability convention was intended.
  • Treating asymptotic formulas as globally valid.
Section titled “Canonical Links”

References

Section titled “References”
  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, 1965.
  • F. W. J. Olver et al., eds., NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
  • G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.