Coulomb wave function

Last updated October 24, 2024

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 $m$ is the Schrödinger equation with Coulomb potential ^[1]

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

\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^[2]^[3]

\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 $M(a,b,z)\equiv {}_{1}\!F_{1}(a;b;z)$ is the confluent hypergeometric function, $\eta =Zmc\alpha /(\hbar k)$ and $\Gamma (z)$ is the gamma function. The two boundary conditions used here are

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

\psi _{\vec {k}}^{(+)}=\psi _{-{\vec {k}}}^{(-)*}\,.

Partial wave expansion

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

\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

\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 $w_{\ell }(\eta ,\rho )$ can be obtained by rewriting the laplacian in the Coulomb wave equation in spherical coordinates and projecting the equation on a specific spherical harmonic $Y_{\ell }^{m}({\hat {r}})$

{\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 $z=-2i\rho$ changes the Coulomb wave equation into the Whittaker equation, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments $M_{-i\eta ,\ell +1/2}(-2i\rho )$ and $W_{-i\eta ,\ell +1/2}(-2i\rho )$ . The latter can be expressed in terms of the confluent hypergeometric functions $M$ and $U$ . For $\ell \in \mathbb {Z}$ , one defines the special solutions ^[4]

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

\sigma _{\ell }=\arg \Gamma (\ell +1+i\eta )

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

F_{\ell }(\eta ,\rho )={\frac {1}{2i}}\left(H_{\ell }^{(+)}(\eta ,\rho )-H_{\ell }^{(-)}(\eta ,\rho )\right)\,,

G_{\ell }(\eta ,\rho )={\frac {1}{2}}\left(H_{\ell }^{(+)}(\eta ,\rho )+H_{\ell }^{(-)}(\eta ,\rho )\right)\,.

In particular one has

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 $H_{\ell }^{(\pm )}(\eta ,\rho )$ , $F_{\ell }(\eta ,\rho )$ , and $G_{\ell }(\eta ,\rho )$ at large $\rho$ is

H_{\ell }^{(\pm )}(\eta ,\rho )\sim e^{\pm i\theta _{\ell }(\rho )}\,,

F_{\ell }(\eta ,\rho )\sim \sin \theta _{\ell }(\rho )\,,

G_{\ell }(\eta ,\rho )\sim \cos \theta _{\ell }(\rho )\,,

where

\theta _{\ell }(\rho )=\rho -\eta \log(2\rho )-{\frac {1}{2}}\ell \pi +\sigma _{\ell }\,.

The solutions $H_{\ell }^{(\pm )}(\eta ,\rho )$ correspond to incoming and outgoing spherical waves. The solutions $F_{\ell }(\eta ,\rho )$ and $G_{\ell }(\eta ,\rho )$ 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 $\psi _{\vec {k}}^{(+)}({\vec {r}})$ ^[5]

\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}})\,,

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 ^[6]^[7]

\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 ( $k/2\pi$ -scale),

\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

\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

\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

\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 ^[8]

\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.

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References

↑ Hill, Robert N. (2006), Drake, Gordon (ed.), Handbook of atomic, molecular and optical physics, Springer New York, pp. 153–155, doi:10.1007/978-0-387-26308-3, ISBN 978-0-387-20802-2
↑ Landau, L. D.; Lifshitz, E. M. (1977), Course of theoretical physics III: Quantum mechanics, Non-relativistic theory (3rd ed.), Pergamon Press, p. 569
↑ Messiah, Albert (1961), Quantum mechanics, North Holland Publ. Co., p. 485
↑ Gaspard, David (2018), "Connection formulas between Coulomb wave functions", J. Math. Phys., 59 (11): 112104, arXiv: 1804.10976 , doi:10.1063/1.5054368
↑ Messiah, Albert (1961), Quantum mechanics, North Holland Publ. Co., p. 426
↑ Formánek, Jiří (2004), Introduction to quantum theory I (in Czech) (2nd ed.), Prague: Academia, pp. 128–130
↑ Landau, L. D.; Lifshitz, E. M. (1977), Course of theoretical physics III: Quantum mechanics, Non-relativistic theory (3rd ed.), Pergamon Press, p. 121
↑ Landau, L. D.; Lifshitz, E. M. (1977), Course of theoretical physics III: Quantum mechanics, Non-relativistic theory (3rd ed.), Pergamon Press, pp. 668–669

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[1] Hill, Robert N. (2006), Drake, Gordon (ed.), Handbook of atomic, molecular and optical physics, Springer New York, pp. 153–155, doi:10.1007/978-0-387-26308-3, ISBN 978-0-387-20802-2

[2] Landau, L. D.; Lifshitz, E. M. (1977), Course of theoretical physics III: Quantum mechanics, Non-relativistic theory (3rd ed.), Pergamon Press, p. 569

[3] Messiah, Albert (1961), Quantum mechanics, North Holland Publ. Co., p. 485

[4] Gaspard, David (2018), "Connection formulas between Coulomb wave functions", J. Math. Phys., 59 (11): 112104, arXiv: 1804.10976 , doi:10.1063/1.5054368

[5] Messiah, Albert (1961), Quantum mechanics, North Holland Publ. Co., p. 426

[6] Formánek, Jiří (2004), Introduction to quantum theory I (in Czech) (2nd ed.), Prague: Academia, pp. 128–130

[7] Landau, L. D.; Lifshitz, E. M. (1977), Course of theoretical physics III: Quantum mechanics, Non-relativistic theory (3rd ed.), Pergamon Press, p. 121

[8] Landau, L. D.; Lifshitz, E. M. (1977), Course of theoretical physics III: Quantum mechanics, Non-relativistic theory (3rd ed.), Pergamon Press, pp. 668–669

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