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Coulomb wave function

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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.

Contents

• Coulomb wave equation

• Partial wave expansion

• Properties of the Coulomb function

• Further reading

• References

Coulomb wave equation

The Coulomb wave equation for a single charged particle of mass m {\displaystyle m} is the Schrödinger equation with Coulomb potentialcite-ref-1[1]

( − − ℏ ℏ 2 ∇ ∇ 2 2 m + Z ℏ ℏ c α α r ) ψ ψ k → → ( r → → ) = ℏ ℏ 2 k 2 2 m ψ 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 α {\displaystyle \alpha } is the fine-structure constant, and ℏ 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 ξ = r + r → → ⋅ ⋅ k ^ ^ , ζ 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 arecite-ref-2[2]cite-ref-3[3]

ψ ψ k → → ( ± ± ) ( r → → ) = Γ Γ ( 1 ± ± i η η ) e − − π π η η / 2 e i k → → ⋅ ⋅ r → → M ( ∓ ∓ i η 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 η = Z m c α α / ( ℏ ℏ k ) {\displaystyle \eta =Zmc\alpha /(\hbar k)} and Γ Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. The two boundary conditions used here are

ψ 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 ψ k → → ( ± ± ) {\displaystyle \psi _{\vec {k}}^{(\pm )}} are related to each other by the formula

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

Partial wave expansion

The wave function ψ 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 ℓ ( η η , ρ ρ ) {\displaystyle w_{\ell }(\eta ,\rho )} . Here ρ ρ = k r {\displaystyle \rho =kr} .

ψ ψ k → → ( r → → ) = 4 π π r ∑ ∑ ℓ ℓ = 0 ∞ ∞ ∑ ∑ m = − − ℓ ℓ ℓ ℓ i ℓ ℓ w ℓ ℓ ( η η , ρ ρ ) Y ℓ ℓ m ( r ^ ^ ) Y ℓ 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 ψ k ℓ ℓ m ( r → → ) = ∫ ∫ ψ ψ k → → ( r → → ) Y ℓ ℓ m ( k ^ ^ ) d k ^ ^ = R k ℓ ℓ ( r ) Y ℓ ℓ m ( r ^ ^ ) , R k ℓ ℓ ( r ) = 4 π π i ℓ ℓ w ℓ ℓ ( η η , ρ 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 ℓ ( η η , ρ 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 ℓ m ( r ^ ^ ) {\displaystyle Y_{\ell }^{m}({\hat {r}})}

d 2 w ℓ ℓ d ρ ρ 2 + ( 1 − − 2 η η ρ ρ − − ℓ ℓ ( ℓ ℓ + 1 ) ρ ρ 2 ) w ℓ 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 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 η , ℓ ℓ + 1 / 2 ( − − 2 i ρ ρ ) {\displaystyle M_{-i\eta ,\ell +1/2}(-2i\rho )} and W − − i η η , ℓ ℓ + 1 / 2 ( − − 2 i ρ 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 ℓ ∈ ∈ Z {\displaystyle \ell \in \mathbb {Z} } , one defines the special solutions cite-ref-4[4]

H ℓ ℓ ( ± ± ) ( η η , ρ ρ ) = ∓ ∓ 2 i ( − − 2 ) ℓ ℓ e π π η η / 2 e ± ± i σ σ ℓ ℓ ρ ρ ℓ ℓ + 1 e ± ± i ρ ρ U ( ℓ ℓ + 1 ± ± i η η , 2 ℓ ℓ + 2 , ∓ ∓ 2 i ρ 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 σ ℓ ℓ = arg ⁡ ⁡ Γ Γ ( ℓ ℓ + 1 + i η η ) {\displaystyle \sigma _{\ell }=\arg \Gamma (\ell +1+i\eta )}

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

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

G ℓ ℓ ( η η , ρ ρ ) = 1 2 ( H ℓ ℓ ( + ) ( η η , ρ ρ ) + H ℓ ℓ ( − − ) ( η η , ρ 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 ℓ ( η η , ρ ρ ) = 2 ℓ ℓ e − − π π η η / 2 | Γ Γ ( ℓ ℓ + 1 + i η η ) | ( 2 ℓ ℓ + 1 ) ! ρ ρ ℓ ℓ + 1 e i ρ ρ M ( ℓ ℓ + 1 + i η η , 2 ℓ ℓ + 2 , − − 2 i ρ 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 ℓ ( ± ± ) ( η η , ρ ρ ) {\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )} , F ℓ ℓ ( η η , ρ ρ ) {\displaystyle F_{\ell }(\eta ,\rho )} , and G ℓ ℓ ( η η , ρ ρ ) {\displaystyle G_{\ell }(\eta ,\rho )} at large ρ ρ {\displaystyle \rho } is

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

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

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

where

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

The solutions H ℓ ℓ ( ± ± ) ( η η , ρ ρ ) {\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )} correspond to incoming and outgoing spherical waves. The solutions F ℓ ℓ ( η η , ρ ρ ) {\displaystyle F_{\ell }(\eta ,\rho )} and G ℓ ℓ ( η η , ρ 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 ψ k → → ( + ) ( r → → ) {\displaystyle \psi _{\vec {k}}^{(+)}({\vec {r}})} cite-ref-5[5]

ψ ψ k → → ( + ) ( r → → ) = 4 π π ρ ρ ∑ ∑ ℓ ℓ = 0 ∞ ∞ ∑ ∑ m = − − ℓ ℓ ℓ ℓ i ℓ ℓ e i σ σ ℓ ℓ F ℓ ℓ ( η η , ρ ρ ) Y ℓ ℓ m ( r ^ ^ ) Y ℓ 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 η → → 0 {\displaystyle \eta \to 0} regular/irregular Coulomb wave functions F ℓ ℓ ( η η , ρ ρ ) {\displaystyle F_{\ell }(\eta ,\rho )} , G ℓ ℓ ( η η , ρ ρ ) {\displaystyle G_{\ell }(\eta ,\rho )} are proportional to Spherical Bessel functions and spherical Coulomb functions H ℓ ℓ ( ± ± ) ( η η , ρ ρ ) {\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )} are proportional to Spherical Hankel functions

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

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

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

H ℓ ℓ ( − − ) ( 0 , ρ ρ ) / ρ ρ = − − i h ℓ ℓ ( 2 ) ( ρ 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 ℓ ( ± ± 1 a 0 k , k r ) k r F ℓ ℓ ( ± ± 1 a 0 k ′ , k ′ r ) k ′ r r 2 d r = π π 2 k 2 δ 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 cite-ref-6[6]cite-ref-7[7]

∫ ∫ 0 ∞ ∞ R k ℓ ℓ ∗ ∗ ( r ) R k ′ ℓ ℓ ( r ) r 2 d r = δ 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 π {\displaystyle k/2\pi } -scale),

∫ ∫ 0 ∞ ∞ R k ℓ ℓ ∗ ∗ ( r ) R k ′ ℓ ℓ ( r ) r 2 d r = 2 π π δ 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 ℓ ∗ ∗ ( r ) R E ′ ℓ ℓ ( r ) r 2 d r = δ 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 ℓ ∗ ∗ ( r ) R k ′ ℓ ℓ ( r ) r 2 d r = ( 2 π π ) 3 k 2 δ 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 ψ k → → ∗ ∗ ( r → → ) ψ ψ k → → ′ ( r → → ) d 3 r = ( 2 π π ) 3 δ 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 statescite-ref-8[8]

∫ ∫ 0 ∞ ∞ R k ℓ ℓ ∗ ∗ ( r ) R n ℓ 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.