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Plane wave expansion

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In physics, the plane wave expansion expresses a plane wave as a sum of spherical waves,

Contents

e i k ⋅ r = ∑ ℓ = 0 ∞ ( 2 ℓ + 1 ) i ℓ j ℓ ( k r ) P ℓ ( k ^ ⋅ r ^ ) ,

where

  • i is the imaginary unit,
  • k is a wave vector of length k,
  • r is a position vector of length r,
  • jℓ are spherical Bessel functions,
  • Pℓ are Legendre polynomials, and
  • the hat ^ denotes the unit vector.

    • In the special case where k is aligned with the z-axis,

    e i k r cos ⁡ θ = ∑ ℓ = 0 ∞ ( 2 ℓ + 1 ) i ℓ j ℓ ( k r ) P ℓ ( cos ⁡ θ ) ,

    where θ is the spherical polar angle of r.

    Expansion in spherical harmonics

    With the spherical harmonic addition theorem the equation can be rewritten as

    e i k ⋅ r = 4 π ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ i ℓ j ℓ ( k r ) Y ℓ m ∗ ( k ^ ) Y ℓ m ( r ^ ) ,

    where

  • Yℓm are the spherical harmonics and
  • the superscript * denotes complex conjugation.

    • Note that the complex conjugation can be interchanged between the two spherical harmonics due to symmetry.

    Applications

    The plane wave expansion is applied in

  • Acoustics
  • Optics
  • Quantum scattering theory

    • References

    Plane wave expansion Wikipedia
    (Text) CC BY-SA


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