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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 FamilyTypical Equation or RoleQuantum-Mechanics UseToolkit Page
Hermite polynomials Hn(x)H_n(x)oscillator equation after Gaussian factorizationharmonic oscillatorHermite Polynomials
Associated Legendre functions Pℓm(x)P_\ell^m(x)angular equation on the spherespherical harmonics, rotor, central potentialsLegendre Polynomials
Associated Laguerre polynomials Lnα(x)L_n^\alpha(x)radial Coulomb equationhydrogenic radial wavefunctionsLaguerre Polynomials
Bessel functions Jν(x)J_\nu(x)cylindrical radial equationcylindrical wells, scattering, partial wavesBessel Functions
Airy functions Ai⁡(x)\operatorname{Ai}(x), Bi⁡(x)\operatorname{Bi}(x)linear turning-point equationWKB connection and uniform approximationsAiry Functions
Gamma function Γ(z)\Gamma(z)factorial continuationnormalization integrals and special-function identitiesGamma and Beta Functions
Hypergeometric functionscommon master family for many ODEssolvable potentials and analytic continuationHypergeometric Functions

Orthogonality Patterns

Section titled “Orthogonality Patterns”
FamilySchematic 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.