Coulomb wave function explained

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\left( - \hbar ^{2 (\nabla ^{2})/(2 m)} + (Z \hbar c \alpha)/(r) \right) \psi _{\vec{k} }(\vec) = \frac \psi_(\vec) \,,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 ^{2k} ^{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} ⋅ \hat{k}, \zeta = r - \vec{r} ⋅ \hat{k} (\hat{k} = \vec{k}/k) . Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are\psi _{\vec{k} }^(\vec) = \Gamma(1\pm i\eta) e^ e^ M(\mp i\eta, 1, \pm ikr - i\vec\cdot\vec) \,,where M(a,b,z) \equiv {} _{1F} _{1(a;b;z)} is the confluent hypergeometric function, η = Zmc\alpha/(\hbar k) and \Gamma(z) is the gamma function. The two boundary conditions used here are\psi _{\vec{k} }^(\vec) \rightarrow e^ \qquad (\vec\cdot\vec \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} }^ are related to each other by the formula\psi _{\vec{k} }^ = \psi_^ \,.

Partial wave expansion

The wave function

\psi _{\vec{k} }(\vec) can be expanded into partial waves (i.e. with respect to the angular basis) to obtain angle-independent radial functions w _{\ell(η,\rho)} . Here \rho = kr .\psi _{\vec{k} }(\vec) = \frac \sum_^\infty \sum_^\ell i^\ell w_(\eta,\rho) Y_\ell^m (\hat) Y_^ (\hat) \,.A single term of the expansion can be isolated by the scalar product with a specific spherical harmonic\psi _{k\ell} (\vec{r}) = \int \psi _{\vec{k} }(\vec) Y_\ell^m (\hat) d\hat = R_(r) Y_\ell^m(\hat), \qquad R_(r) = 4\pi i^\ell w_\ell(\eta,\rho)/r.The equation for single partial wave w _{\ell(η,\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 ^{m(\hat{r})} _{\ell} (d ^{2} w _{\ell})/(+ \left( 1- d\rho ^{2}) (2η)/(- \rho) (\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\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 _{-iη,\ell+1/2} ( -2 i\rho) and W _{-iη,\ell+1/2} ( -2 i\rho) . The latter can be expressed in terms of the confluent hypergeometric functions M and U . For \ell\in Z , one defines the special solutions H ^{(\pm)} _{\ell} ( η ,\rho) = \mp 2 i( -2 ) ^{\ell} e ^{\piη/2} e ^{\pm i \sigma _{\ell}} \rho ^{\ell+1} e ^{\pm} U(\ell +1 \pm i η , 2 \ell +2 ,\mp 2 i\rho) , where\sigma _{\ell} = \arg \Gamma(\ell +1+ i η ) is called the Coulomb phase shift. One also defines the real functionsF _{\ell(η,\rho)} = (1)/(2 i) \left(H ^{( + )} _{\ell} ( η ,\rho) - H ^{( - )} _{\ell} ( η ,\rho) \right) , G _{\ell(η,\rho)} = (1)/(2) \left(H ^{( + )} _{\ell} ( η ,\rho) + H ^{( - )} _{\ell} ( η ,\rho) \right) . In particular one hasF _{\ell(η,\rho)} = (2 ^{\ell} e ^{-\piη/2} |\Gamma(\ell +1+ i η )|)/(( 2 \ell +1 )!) \rho ^{\ell+1} e ^{i\rho} M(\ell +1+ i η , 2 \ell +2 , -2 i\rho) . The asymptotic behavior of the spherical Coulomb functions H ^{(\pm)} _{\ell} ( η ,\rho) , F _{\ell(η,\rho)} , and G _{\ell(η,\rho)} at large \rho isH ^{(\pm)} _{\ell} ( η ,\rho) \sim e ^{\pm i \theta _{\ell(\rho)}} , F _{\ell(η,\rho)} \sim \sin \theta _{\ell(\rho)} , G _{\ell(η,\rho)} \sim \cos \theta _{\ell(\rho)} , where\theta _{\ell(\rho)} = \rho - η log ( 2 \rho) - (1)/(2) \ell \pi + \sigma _{\ell} . The solutions H ^{(\pm)} _{\ell} ( η ,\rho) correspond to incoming and outgoing spherical waves. The solutions F _{\ell(η,\rho)} and G _{\ell(η,\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) \psi _{\vec{k} }^(\vec) = \frac \sum_^\infty \sum_^\ell i^\ell e^ F_\ell(\eta,\rho) Y_\ell^m (\hat) Y_^ (\hat) \,,

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

\int ^{inf ty} _{0} R ^{\ast(r)} _{k\ell} 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 ^{inf ty} _{0} R ^{\ast(r)} _{k\ell} R _{k'\ell} (r) r ^{2} dr = 2 \pi \delta(k - k') , and on the energy scale\int ^{inf ty} _{0} R ^{\ast(r)} _{E\ell} R _{E'\ell} (r) r ^{2} dr = \delta(E - E') . The radial wave functions defined in the previous section are normalized to \int ^{inf ty} _{0} R ^{\ast(r)} _{k\ell} R _{k'\ell} (r) r ^{2} dr = (( 2 \pi) ^{3})/(k ^{2}) \delta(k - k') as a consequence of the normalization\int \psi ^{\ast} _{\vec{k} }(\vec) \psi_(\vec) d^3r = (2\pi)^3 \delta(\vec-\vec') \,.

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

\int ^{inf ty} _{0} R ^{\ast(r)} _{k\ell} R _{n\ell} (r) r ^{2} dr = 0 due to being eigenstates of the same hermitian operator (the hamiltonian) with different eigenvalues.

Further reading

• .
  • .