Coulomb wave function

In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument.

Coulomb wave equation

The Coulomb wave equation for a single charged particle of mass m {\displaystyle m} is the Schrödinger equation with Coulomb potential

( − ℏ 2 ∇ 2 2 m + Z ℏ c α r ) ψ k → ( r → ) = ℏ 2 k 2 2 m ψ k → ( r → ) , {\displaystyle \left(-\hbar ^{2}{\frac {\nabla ^{2}}{2m}}+{\frac {Z\hbar c\alpha }{r}}\right)\psi _{\vec {k}}({\vec {r}})={\frac {\hbar ^{2}k^{2}}{2m}}\psi _{\vec {k}}({\vec {r}})\,,}

where Z = Z 1 Z 2 {\displaystyle Z=Z_{1}Z_{2}} is the product of the charges of the particle and of the field source (in units of the elementary charge, Z = − 1 {\displaystyle Z=-1} for the hydrogen atom), α {\displaystyle \alpha } is the fine-structure constant, and ℏ 2 k 2 / ( 2 m ) {\displaystyle \hbar ^{2}k^{2}/(2m)} is the energy of the particle. The solution, which is the Coulomb wave function, can be found by solving this equation in parabolic coordinates

ξ = r + r → ⋅ k ^ , ζ = r − r → ⋅ k ^ ( k ^ = k → / k ) . {\displaystyle \xi =r+{\vec {r}}\cdot {\hat {k}},\quad \zeta =r-{\vec {r}}\cdot {\hat {k}}\qquad ({\hat {k}}={\vec {k}}/k)\,.}

Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are

ψ k → ( ± ) ( r → ) = Γ ( 1 ± i η ) e − π η / 2 e i k → ⋅ r → M ( ∓ i η , 1 , ± i k r − i k → ⋅ r → ) , {\displaystyle \psi _{\vec {k}}^{(\pm )}({\vec {r}})=\Gamma (1\pm i\eta )e^{-\pi \eta /2}e^{i{\vec {k}}\cdot {\vec {r}}}M(\mp i\eta ,1,\pm ikr-i{\vec {k}}\cdot {\vec {r}})\,,}

where M ( a , b , z ) ≡ 1 F 1 ( a ; b ; z ) {\displaystyle M(a,b,z)\equiv {}_{1}\!F_{1}(a;b;z)} is the confluent hypergeometric function, η = Z m c α / ( ℏ k ) {\displaystyle \eta =Zmc\alpha /(\hbar k)} and Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. The two boundary conditions used here are

ψ k → ( ± ) ( r → ) → e i k → ⋅ r → ( k → ⋅ r → → ± ∞ ) , {\displaystyle \psi _{\vec {k}}^{(\pm )}({\vec {r}})\rightarrow e^{i{\vec {k}}\cdot {\vec {r}}}\qquad ({\vec {k}}\cdot {\vec {r}}\rightarrow \pm \infty )\,,}

which correspond to k → {\displaystyle {\vec {k}}}-oriented plane-wave asymptotic states before or after its approach of the field source at the origin, respectively. The functions ψ k → ( ± ) {\displaystyle \psi _{\vec {k}}^{(\pm )}} are related to each other by the formula

ψ k → ( + ) = ψ − k → ( − ) ∗ . {\displaystyle \psi _{\vec {k}}^{(+)}=\psi _{-{\vec {k}}}^{(-)*}\,.}

Partial wave expansion

The wave function ψ k → ( r → ) {\displaystyle \psi _{\vec {k}}({\vec {r}})} can be expanded into partial waves (i.e. with respect to the angular basis) to obtain angle-independent radial functions w ℓ ( η , ρ ) {\displaystyle w_{\ell }(\eta ,\rho )}. Here ρ = k r {\displaystyle \rho =kr}.

ψ k → ( r → ) = 4 π r ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ i ℓ w ℓ ( η , ρ ) Y ℓ m ( r ^ ) Y ℓ m ∗ ( k ^ ) . {\displaystyle \psi _{\vec {k}}({\vec {r}})={\frac {4\pi }{r}}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }i^{\ell }w_{\ell }(\eta ,\rho )Y_{\ell }^{m}({\hat {r}})Y_{\ell }^{m\ast }({\hat {k}})\,.}

A single term of the expansion can be isolated by the scalar product with a specific spherical harmonic

ψ k ℓ m ( r → ) = ∫ ψ k → ( r → ) Y ℓ m ( k ^ ) d k ^ = R k ℓ ( r ) Y ℓ m ( r ^ ) , R k ℓ ( r ) = 4 π i ℓ w ℓ ( η , ρ ) / r . {\displaystyle \psi _{k\ell m}({\vec {r}})=\int \psi _{\vec {k}}({\vec {r}})Y_{\ell }^{m}({\hat {k}})d{\hat {k}}=R_{k\ell }(r)Y_{\ell }^{m}({\hat {r}}),\qquad R_{k\ell }(r)=4\pi i^{\ell }w_{\ell }(\eta ,\rho )/r.}

The equation for single partial wave w ℓ ( η , ρ ) {\displaystyle w_{\ell }(\eta ,\rho )} can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic Y ℓ m ( r ^ ) {\displaystyle Y_{\ell }^{m}({\hat {r}})}

d 2 w ℓ d ρ 2 + ( 1 − 2 η ρ − ℓ ( ℓ + 1 ) ρ 2 ) w ℓ = 0 . {\displaystyle {\frac {d^{2}w_{\ell }}{d\rho ^{2}}}+\left(1-{\frac {2\eta }{\rho }}-{\frac {\ell (\ell +1)}{\rho ^{2}}}\right)w_{\ell }=0\,.}

The solutions are also called Coulomb (partial) wave functions or spherical Coulomb functions. Putting z = − 2 i ρ {\displaystyle z=-2i\rho } changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments M − i η , ℓ + 1 / 2 ( − 2 i ρ ) {\displaystyle M_{-i\eta ,\ell +1/2}(-2i\rho )} and W − i η , ℓ + 1 / 2 ( − 2 i ρ ) {\displaystyle W_{-i\eta ,\ell +1/2}(-2i\rho )}. The latter can be expressed in terms of the confluent hypergeometric functions M {\displaystyle M} and U {\displaystyle U}. For ℓ ∈ Z {\displaystyle \ell \in \mathbb {Z} }, one defines the special solutions

H ℓ ( ± ) ( η , ρ ) = ∓ 2 i ( − 2 ) ℓ e π η / 2 e ± i σ ℓ ρ ℓ + 1 e ± i ρ U ( ℓ + 1 ± i η , 2 ℓ + 2 , ∓ 2 i ρ ) , {\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )=\mp 2i(-2)^{\ell }e^{\pi \eta /2}e^{\pm i\sigma _{\ell }}\rho ^{\ell +1}e^{\pm i\rho }U(\ell +1\pm i\eta ,2\ell +2,\mp 2i\rho )\,,}

where

σ ℓ = arg ⁡ Γ ( ℓ + 1 + i η ) {\displaystyle \sigma _{\ell }=\arg \Gamma (\ell +1+i\eta )}

is called the Coulomb phase shift. One also defines the real functions

F ℓ ( η , ρ ) = 1 2 i ( H ℓ ( + ) ( η , ρ ) − H ℓ ( − ) ( η , ρ ) ) , {\displaystyle F_{\ell }(\eta ,\rho )={\frac {1}{2i}}\left(H_{\ell }^{(+)}(\eta ,\rho )-H_{\ell }^{(-)}(\eta ,\rho )\right)\,,}

Regular Coulomb wave function F plotted from 0 to 20 with repulsive and attractive interactions in Mathematica 13.1G ℓ ( η , ρ ) = 1 2 ( H ℓ ( + ) ( η , ρ ) + H ℓ ( − ) ( η , ρ ) ) . {\displaystyle G_{\ell }(\eta ,\rho )={\frac {1}{2}}\left(H_{\ell }^{(+)}(\eta ,\rho )+H_{\ell }^{(-)}(\eta ,\rho )\right)\,.}

In particular one has

F ℓ ( η , ρ ) = 2 ℓ e − π η / 2 | Γ ( ℓ + 1 + i η ) | ( 2 ℓ + 1 ) ! ρ ℓ + 1 e i ρ M ( ℓ + 1 + i η , 2 ℓ + 2 , − 2 i ρ ) . {\displaystyle F_{\ell }(\eta ,\rho )={\frac {2^{\ell }e^{-\pi \eta /2}|\Gamma (\ell +1+i\eta )|}{(2\ell +1)!}}\rho ^{\ell +1}e^{i\rho }M(\ell +1+i\eta ,2\ell +2,-2i\rho )\,.}

The asymptotic behavior of the spherical Coulomb functions H ℓ ( ± ) ( η , ρ ) {\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}, F ℓ ( η , ρ ) {\displaystyle F_{\ell }(\eta ,\rho )}, and G ℓ ( η , ρ ) {\displaystyle G_{\ell }(\eta ,\rho )} at large ρ {\displaystyle \rho } is

H ℓ ( ± ) ( η , ρ ) ∼ e ± i θ ℓ ( ρ ) , {\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )\sim e^{\pm i\theta _{\ell }(\rho )}\,,}

F ℓ ( η , ρ ) ∼ sin ⁡ θ ℓ ( ρ ) , {\displaystyle F_{\ell }(\eta ,\rho )\sim \sin \theta _{\ell }(\rho )\,,}

G ℓ ( η , ρ ) ∼ cos ⁡ θ ℓ ( ρ ) , {\displaystyle G_{\ell }(\eta ,\rho )\sim \cos \theta _{\ell }(\rho )\,,}

where

θ ℓ ( ρ ) = ρ − η log ⁡ ( 2 ρ ) − 1 2 ℓ π + σ ℓ . {\displaystyle \theta _{\ell }(\rho )=\rho -\eta \log(2\rho )-{\frac {1}{2}}\ell \pi +\sigma _{\ell }\,.}

The solutions H ℓ ( ± ) ( η , ρ ) {\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )} correspond to incoming and outgoing spherical waves. The solutions F ℓ ( η , ρ ) {\displaystyle F_{\ell }(\eta ,\rho )} and G ℓ ( η , ρ ) {\displaystyle G_{\ell }(\eta ,\rho )} are real and are called the regular and irregular Coulomb wave functions. In particular one has the following partial wave expansion for the wave function ψ k → ( + ) ( r → ) {\displaystyle \psi _{\vec {k}}^{(+)}({\vec {r}})}

ψ k → ( + ) ( r → ) = 4 π ρ ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ i ℓ e i σ ℓ F ℓ ( η , ρ ) Y ℓ m ( r ^ ) Y ℓ m ∗ ( k ^ ) , {\displaystyle \psi _{\vec {k}}^{(+)}({\vec {r}})={\frac {4\pi }{\rho }}\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }i^{\ell }e^{i\sigma _{\ell }}F_{\ell }(\eta ,\rho )Y_{\ell }^{m}({\hat {r}})Y_{\ell }^{m\ast }({\hat {k}})\,,}

In the limit η → 0 {\displaystyle \eta \to 0} regular/irregular Coulomb wave functions F ℓ ( η , ρ ) {\displaystyle F_{\ell }(\eta ,\rho )},G ℓ ( η , ρ ) {\displaystyle G_{\ell }(\eta ,\rho )} are proportional to Spherical Bessel functions and spherical Coulomb functions H ℓ ( ± ) ( η , ρ ) {\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )} are proportional to Spherical Hankel functions

F ℓ ( 0 , ρ ) / ρ = j ℓ ( ρ ) {\displaystyle F_{\ell }(0,\rho )/\rho =j_{\ell }(\rho )}

G ℓ ( 0 , ρ ) / ρ = − y ℓ ( ρ ) {\displaystyle G_{\ell }(0,\rho )/\rho =-y_{\ell }(\rho )}

H ℓ ( + ) ( 0 , ρ ) / ρ = i h ℓ ( 1 ) ( ρ ) {\displaystyle H_{\ell }^{(+)}(0,\rho )/\rho =i\,h_{\ell }^{(1)}(\rho )}

H ℓ ( − ) ( 0 , ρ ) / ρ = − i h ℓ ( 2 ) ( ρ ) {\displaystyle H_{\ell }^{(-)}(0,\rho )/\rho =-i\,h_{\ell }^{(2)}(\rho )}

and are normalized same as Spherical Bessel functions

∫ 0 ∞ j l ( k r ) j l ( k ′ r ) r 2 d r = ∫ 0 ∞ F ℓ ( ± 1 a 0 k , k r ) k r F ℓ ( ± 1 a 0 k ′ , k ′ r ) k ′ r r 2 d r = π 2 k 2 δ ( k − k ′ ) {\displaystyle \int \limits _{0}^{\infty }j_{l}(k\,r)j_{l}(k'r)\,r^{2}dr=\int \limits _{0}^{\infty }{\frac {F_{\ell }\left(\pm {\frac {1}{a_{0}k}},k\,r\right)}{k\,r}}{\frac {F_{\ell }\left(\pm {\frac {1}{a_{0}k'}},k'r\right)}{k'r}}\,r^{2}dr={\frac {\pi }{2k^{2}}}\delta (k-k')}

and similar for other 3.

Properties of the Coulomb function

The radial parts for a given angular momentum are orthonormal. When normalized on the wave number scale (k-scale), the continuum radial wave functions satisfy

∫ 0 ∞ R k ℓ ∗ ( r ) R k ′ ℓ ( r ) r 2 d r = δ ( k − k ′ ) {\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr=\delta (k-k')}

Other common normalizations of continuum wave functions are on the reduced wave number scale (k / 2 π {\displaystyle k/2\pi }-scale),

∫ 0 ∞ R k ℓ ∗ ( r ) R k ′ ℓ ( r ) r 2 d r = 2 π δ ( k − k ′ ) , {\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr=2\pi \delta (k-k')\,,}

and on the energy scale

∫ 0 ∞ R E ℓ ∗ ( r ) R E ′ ℓ ( r ) r 2 d r = δ ( E − E ′ ) . {\displaystyle \int _{0}^{\infty }R_{E\ell }^{\ast }(r)R_{E'\ell }(r)r^{2}dr=\delta (E-E')\,.}

The radial wave functions defined in the previous section are normalized to

∫ 0 ∞ R k ℓ ∗ ( r ) R k ′ ℓ ( r ) r 2 d r = ( 2 π ) 3 k 2 δ ( k − k ′ ) {\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr={\frac {(2\pi )^{3}}{k^{2}}}\delta (k-k')}

as a consequence of the normalization

∫ ψ k → ∗ ( r → ) ψ k → ′ ( r → ) d 3 r = ( 2 π ) 3 δ ( k → − k → ′ ) . {\displaystyle \int \psi _{\vec {k}}^{\ast }({\vec {r}})\psi _{{\vec {k}}'}({\vec {r}})d^{3}r=(2\pi )^{3}\delta ({\vec {k}}-{\vec {k}}')\,.}

The continuum (or scattering) Coulomb wave functions are also orthogonal to all Coulomb bound states

∫ 0 ∞ R k ℓ ∗ ( r ) R n ℓ ( r ) r 2 d r = 0 {\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{n\ell }(r)r^{2}dr=0}

due to being eigenstates of the same hermitian operator (the hamiltonian) with different eigenvalues.

Coulomb wave function

Coulomb wave equation

Partial wave expansion

Properties of the Coulomb function

Further reading