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<title>Coulomb wave function</title>

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Coulomb wave function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a Coulomb wave function is a solution of the Coulomb wave equation, named after <a href="Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Charles-Augustin de Coulomb</a>. They are used to describe the behavior of <a href="Charged_particle" title="Charged particle">charged particles</a> in a <a href="Coulomb_potential" class="mw-redirect" title="Coulomb potential">Coulomb potential</a> and can be written in terms of <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">confluent hypergeometric functions</a> or <a href="Whittaker_function" title="Whittaker function">Whittaker functions</a> of imaginary argument.

<div class="mw-heading mw-heading2"><h2 id="Coulomb_wave_equation">Coulomb wave equation</h2></div>

The Coulomb wave equation for a single charged particle of mass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> with <a href="Electric_potential_energy#One_point_charge_q_in_the_presence_of_one_point_charge_Q" title="Electric potential energy">Coulomb potential</a><a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\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}})\,,}</annotation>

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</math><img src="./8c39cbf546b873d20014d2df2b19f30c6addd05b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.068ex; height:6.343ex;" alt="{\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}})\,,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=Z_{1}Z_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle Z=Z_{1}Z_{2}}</annotation>

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</math><img src="./cb4a70492870d23df8f15a3603582372f7768bf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.062ex; height:2.509ex;" alt="{\displaystyle Z=Z_{1}Z_{2}}" loading="lazy"> is the product of the charges of the particle and of the field source (in units of the <a href="Elementary_charge" title="Elementary charge">elementary charge</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=-1}">

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<annotation encoding="application/x-tex">{\displaystyle Z=-1}</annotation>

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</math><img src="./75398fae690ddf15266f2811df2c348cb370662f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.749ex; height:2.343ex;" alt="{\displaystyle Z=-1}" loading="lazy"> for the hydrogen atom), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is the <a href="Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar ^{2}k^{2}/(2m)}">

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<annotation encoding="application/x-tex">{\displaystyle \hbar ^{2}k^{2}/(2m)}</annotation>

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</math><img src="./e3bd39ea4c113cca125156586a3cb69e1ff32c3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.816ex; height:3.176ex;" alt="{\displaystyle \hbar ^{2}k^{2}/(2m)}" loading="lazy"> is the energy of the particle. The solution, which is the Coulomb wave function, can be found by solving this equation in parabolic coordinates

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<annotation encoding="application/x-tex">{\displaystyle \xi =r+{\vec {r}}\cdot {\hat {k}},\quad \zeta =r-{\vec {r}}\cdot {\hat {k}}\qquad ({\hat {k}}={\vec {k}}/k)\,.}</annotation>

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</math><img src="./fb97fa175a831339a70e233bb203d8145d184b01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.067ex; height:3.343ex;" alt="{\displaystyle \xi =r+{\vec {r}}\cdot {\hat {k}},\quad \zeta =r-{\vec {r}}\cdot {\hat {k}}\qquad ({\hat {k}}={\vec {k}}/k)\,.}" loading="lazy"></dd></dl>

Depending on the boundary conditions chosen, the solution has different forms. Two of the solutions are<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a>

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<annotation encoding="application/x-tex">{\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}})\,,}</annotation>

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</math><img src="./7ed25aa50527fd4d2cb8cf38a081094d38882c58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:54.7ex; height:4.676ex;" alt="{\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}})\,,}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle M(a,b,z)\equiv {}_{1}\!F_{1}(a;b;z)}</annotation>

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</math><img src="./c55150f587df7cc6b4dd1b481cb6e825cb5d7b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.142ex; height:2.843ex;" alt="{\displaystyle M(a,b,z)\equiv {}_{1}\!F_{1}(a;b;z)}" loading="lazy"> is the <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">confluent hypergeometric function</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =Zmc\alpha /(\hbar k)}">

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<annotation encoding="application/x-tex">{\displaystyle \eta =Zmc\alpha /(\hbar k)}</annotation>

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</math><img src="./7838877737c408010f1c68e07d7bf87fc9082ecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.973ex; height:2.843ex;" alt="{\displaystyle \eta =Zmc\alpha /(\hbar k)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (z)}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (z)}</annotation>

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</math><img src="./11ca17f880240539116aac7e6326909299e2a080.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.35ex; height:2.843ex;" alt="{\displaystyle \Gamma (z)}" loading="lazy"> is the <a href="Gamma_function" title="Gamma function">gamma function</a>. The two boundary conditions used here are

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\vec {k}}^{(\pm )}({\vec {r}})\rightarrow e^{i{\vec {k}}\cdot {\vec {r}}}\qquad ({\vec {k}}\cdot {\vec {r}}\rightarrow \pm \infty )\,,}</annotation>

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</math><img src="./b267eecc6f420aa1c5d155b5fca2a19e89b5acdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:34.397ex; height:4.676ex;" alt="{\displaystyle \psi _{\vec {k}}^{(\pm )}({\vec {r}})\rightarrow e^{i{\vec {k}}\cdot {\vec {r}}}\qquad ({\vec {k}}\cdot {\vec {r}}\rightarrow \pm \infty )\,,}" loading="lazy"></dd></dl>

which correspond to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {k}}}</annotation>

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</math><img src="./5ccd4b98d198d6538010ae815ee1199baabd3493.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}" loading="lazy">-oriented plane-wave asymptotic states before or after its approach of the field source at the origin, respectively. The functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\vec {k}}^{(\pm )}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\vec {k}}^{(\pm )}}</annotation>

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</math><img src="./16445af9f86f310d7394629ab25f76786de6f8af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:4.303ex; height:4.343ex;" alt="{\displaystyle \psi _{\vec {k}}^{(\pm )}}" loading="lazy"> are related to each other by the formula

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\vec {k}}^{(+)}=\psi _{-{\vec {k}}}^{(-)*}\,.}</annotation>

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</math><img src="./329545c9839c4f81cb5dfa717e1df29ee19231c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.561ex; height:4.343ex;" alt="{\displaystyle \psi _{\vec {k}}^{(+)}=\psi _{-{\vec {k}}}^{(-)*}\,.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Partial_wave_expansion">Partial wave expansion</h3></div>

The wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\vec {k}}({\vec {r}})}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{\vec {k}}({\vec {r}})}</annotation>

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</math><img src="./b2fbfcd6784c9dbdad9d00eed0b3bb935e751a8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.634ex; height:3.176ex;" alt="{\displaystyle \psi _{\vec {k}}({\vec {r}})}" loading="lazy"> can be expanded into partial waves (i.e. with respect to the angular basis) to obtain angle-independent radial functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{\ell }(\eta ,\rho )}">

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<annotation encoding="application/x-tex">{\displaystyle w_{\ell }(\eta ,\rho )}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \rho =kr}</annotation>

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<annotation encoding="application/x-tex">{\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}})\,.}</annotation>

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</math><img src="./81005f3ae84d0908bd1bee60d2ba366d56cea25c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.91ex; height:7.509ex;" alt="{\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}})\,.}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\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.}</annotation>

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</math><img src="./a5c052e9d4477781d4994e93198a8b5068527c8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:74.313ex; height:5.676ex;" alt="{\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.}" loading="lazy"></dd></dl>

The equation for single partial wave <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{\ell }(\eta ,\rho )}">

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<annotation encoding="application/x-tex">{\displaystyle w_{\ell }(\eta ,\rho )}</annotation>

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<annotation encoding="application/x-tex">{\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\,.}</annotation>

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</math><img src="./7d1580f31bcb592c7f31d8017f8374e33287f01b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.815ex; height:6.343ex;" alt="{\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\,.}" loading="lazy"></dd></dl>

The solutions are also called Coulomb (partial) wave functions or spherical Coulomb functions. Putting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-2i\rho }">

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<annotation encoding="application/x-tex">{\displaystyle z=-2i\rho }</annotation>

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</math><img src="./07b508ea35cea87381ef8fef009ab8db0dcb6ca9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.162ex; height:2.676ex;" alt="{\displaystyle z=-2i\rho }" loading="lazy"> changes the Coulomb wave equation into the <a href="Whittaker_equation" class="mw-redirect" title="Whittaker equation">Whittaker equation</a>, so Coulomb wave functions can be expressed in terms of Whittaker functions with imaginary arguments <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{-i\eta ,\ell +1/2}(-2i\rho )}">

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<annotation encoding="application/x-tex">{\displaystyle M_{-i\eta ,\ell +1/2}(-2i\rho )}</annotation>

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</math><img src="./daf0f23d447f314852bf9e31c96563e6d2444e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.831ex; height:3.176ex;" alt="{\displaystyle M_{-i\eta ,\ell +1/2}(-2i\rho )}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{-i\eta ,\ell +1/2}(-2i\rho )}">

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<annotation encoding="application/x-tex">{\displaystyle W_{-i\eta ,\ell +1/2}(-2i\rho )}</annotation>

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</math><img src="./61eebbbbac058e8935b13ddb8b78f29ce8fae905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.771ex; height:3.176ex;" alt="{\displaystyle W_{-i\eta ,\ell +1/2}(-2i\rho )}" loading="lazy">. The latter can be expressed in terms of the <a href="Confluent_hypergeometric_functions" class="mw-redirect" title="Confluent hypergeometric functions">confluent hypergeometric functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell \in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell \in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./33524eeb34f272c052dd61233be6b9e80730a2e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.361ex; height:2.176ex;" alt="{\displaystyle \ell \in \mathbb {Z} }" loading="lazy">, one defines the special solutions <a href="#cite_note-4">[4]</a>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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 )\,,}</annotation>

</semantics>

</math><img src="./9f5ebfdf92fe9022e94236bd3e4486801deeb828.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:68.048ex; height:3.676ex;" alt="{\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 )\,,}" loading="lazy"></dd></dl>

where

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\ell }=\arg \Gamma (\ell +1+i\eta )}</annotation>

</semantics>

</math><img src="./d31e9c4df15109564cbe986d1e23825f183390ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.014ex; height:2.843ex;" alt="{\displaystyle \sigma _{\ell }=\arg \Gamma (\ell +1+i\eta )}" loading="lazy"></dd></dl>

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

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\ell }(\eta ,\rho )={\frac {1}{2i}}\left(H_{\ell }^{(+)}(\eta ,\rho )-H_{\ell }^{(-)}(\eta ,\rho )\right)\,,}</annotation>

</semantics>

</math><img src="./4f346d489ccc911a747193e27e1dce9c9afe18eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.167ex; height:5.176ex;" alt="{\displaystyle F_{\ell }(\eta ,\rho )={\frac {1}{2i}}\left(H_{\ell }^{(+)}(\eta ,\rho )-H_{\ell }^{(-)}(\eta ,\rho )\right)\,,}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\ell }(\eta ,\rho )={\frac {1}{2}}\left(H_{\ell }^{(+)}(\eta ,\rho )+H_{\ell }^{(-)}(\eta ,\rho )\right)\,.}</annotation>

</semantics>

</math><img src="./7377aa772279858d0fa1d03630bedc8365633bc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.697ex; height:5.176ex;" alt="{\displaystyle G_{\ell }(\eta ,\rho )={\frac {1}{2}}\left(H_{\ell }^{(+)}(\eta ,\rho )+H_{\ell }^{(-)}(\eta ,\rho )\right)\,.}" loading="lazy"></dd></dl>

In particular one has

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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 )\,.}</annotation>

</semantics>

</math><img src="./f7b0e4051d2928b3e7c9e747e9491f92eaaa5da1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:69.191ex; height:6.843ex;" alt="{\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 )\,.}" loading="lazy"></dd></dl>

The asymptotic behavior of the spherical Coulomb functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}</annotation>

</semantics>

</math><img src="./dc199b7f2cedf58eedadc2bd9b2745afec9b1159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.108ex; height:3.676ex;" alt="{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\ell }(\eta ,\rho )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\ell }(\eta ,\rho )}</annotation>

</semantics>

</math><img src="./1783fe74393e759b6bd4b40b6e14d073f6820fef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.627ex; height:2.843ex;" alt="{\displaystyle F_{\ell }(\eta ,\rho )}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\ell }(\eta ,\rho )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\ell }(\eta ,\rho )}</annotation>

</semantics>

</math><img src="./5d89356a7c6de51a8fff9489367393dc3d74e145.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.959ex; height:2.843ex;" alt="{\displaystyle G_{\ell }(\eta ,\rho )}" loading="lazy"> at large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is

</mrow>

</mrow>

</msubsup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )\sim e^{\pm i\theta _{\ell }(\rho )}\,,}</annotation>

</semantics>

</math><img src="./b7a78d54507fbacc741763b12099cfa03e2f1eec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.024ex; height:3.676ex;" alt="{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )\sim e^{\pm i\theta _{\ell }(\rho )}\,,}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\ell }(\eta ,\rho )\sim \sin \theta _{\ell }(\rho )\,,}</annotation>

</semantics>

</math><img src="./babb221713b2e0253ac29ccbfc1cef6114f68be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.022ex; height:2.843ex;" alt="{\displaystyle F_{\ell }(\eta ,\rho )\sim \sin \theta _{\ell }(\rho )\,,}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\ell }(\eta ,\rho )\sim \cos \theta _{\ell }(\rho )\,,}</annotation>

</semantics>

</math><img src="./65dba75df98d70e0e6e539175541079e84617a97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.609ex; height:2.843ex;" alt="{\displaystyle G_{\ell }(\eta ,\rho )\sim \cos \theta _{\ell }(\rho )\,,}" loading="lazy"></dd></dl>

where

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta _{\ell }(\rho )=\rho -\eta \log(2\rho )-{\frac {1}{2}}\ell \pi +\sigma _{\ell }\,.}</annotation>

</semantics>

</math><img src="./02db69c27cb0b3ccf98b7c481bc6b9e2904c4f9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.123ex; height:5.176ex;" alt="{\displaystyle \theta _{\ell }(\rho )=\rho -\eta \log(2\rho )-{\frac {1}{2}}\ell \pi +\sigma _{\ell }\,.}" loading="lazy"></dd></dl>

The solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}</annotation>

</semantics>

</math><img src="./dc199b7f2cedf58eedadc2bd9b2745afec9b1159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.108ex; height:3.676ex;" alt="{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}" loading="lazy"> correspond to incoming and outgoing spherical waves. The solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\ell }(\eta ,\rho )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\ell }(\eta ,\rho )}</annotation>

</semantics>

</math><img src="./1783fe74393e759b6bd4b40b6e14d073f6820fef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.627ex; height:2.843ex;" alt="{\displaystyle F_{\ell }(\eta ,\rho )}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\ell }(\eta ,\rho )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\ell }(\eta ,\rho )}</annotation>

</semantics>

In particular one has the following partial wave expansion for the wave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\vec {k}}^{(+)}({\vec {r}})}">

<mover>

</mover>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi _{\vec {k}}^{(+)}({\vec {r}})}</annotation>

</semantics>

</math><img src="./a560ac10763105663e9e7be4cd07460eb6b4966a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:7.336ex; height:4.343ex;" alt="{\displaystyle \psi _{\vec {k}}^{(+)}({\vec {r}})}" loading="lazy"> <a href="#cite_note-5">[5]</a>

<mover>

</mover>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}})\,,}</annotation>

</semantics>

</math><img src="./eba6942d184e9a3f01abc6eac5776eda6e0d348d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.985ex; height:7.509ex;" alt="{\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}})\,,}" loading="lazy"></dd></dl>

In the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \to 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta \to 0}</annotation>

</semantics>

</math><img src="./c8f68906466355cc4f890c5df45f411735fce4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.946ex; height:2.676ex;" alt="{\displaystyle \eta \to 0}" loading="lazy"> regular/irregular Coulomb wave functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\ell }(\eta ,\rho )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\ell }(\eta ,\rho )}</annotation>

</semantics>

</math><img src="./1783fe74393e759b6bd4b40b6e14d073f6820fef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.627ex; height:2.843ex;" alt="{\displaystyle F_{\ell }(\eta ,\rho )}" loading="lazy">,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\ell }(\eta ,\rho )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\ell }(\eta ,\rho )}</annotation>

</semantics>

</math><img src="./5d89356a7c6de51a8fff9489367393dc3d74e145.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.959ex; height:2.843ex;" alt="{\displaystyle G_{\ell }(\eta ,\rho )}" loading="lazy"> are proportional to <a href="Spherical_Bessel_functions" class="mw-redirect" title="Spherical Bessel functions">Spherical Bessel functions</a> and spherical Coulomb functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}</annotation>

</semantics>

</math><img src="./dc199b7f2cedf58eedadc2bd9b2745afec9b1159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.108ex; height:3.676ex;" alt="{\displaystyle H_{\ell }^{(\pm )}(\eta ,\rho )}" loading="lazy"> are proportional to <a href="Spherical_Hankel_functions" class="mw-redirect" title="Spherical Hankel functions">Spherical Hankel functions</a>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{\ell }(0,\rho )/\rho =j_{\ell }(\rho )}</annotation>

</semantics>

</math><img src="./2e28a4a710ec06187c7f2a406e77b6960bbc33d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.97ex; height:2.843ex;" alt="{\displaystyle F_{\ell }(0,\rho )/\rho =j_{\ell }(\rho )}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{\ell }(0,\rho )/\rho =-y_{\ell }(\rho )}</annotation>

</semantics>

</math><img src="./64f81178e6a0331426c50ba5b7e8aa45ae07ec26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.292ex; height:2.843ex;" alt="{\displaystyle G_{\ell }(0,\rho )/\rho =-y_{\ell }(\rho )}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\ell }^{(+)}(0,\rho )/\rho =i\,h_{\ell }^{(1)}(\rho )}</annotation>

</semantics>

</math><img src="./9a61dc2416486eb0c707f0269c2e2bbcb2b917a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.437ex; height:3.676ex;" alt="{\displaystyle H_{\ell }^{(+)}(0,\rho )/\rho =i\,h_{\ell }^{(1)}(\rho )}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{\ell }^{(-)}(0,\rho )/\rho =-i\,h_{\ell }^{(2)}(\rho )}</annotation>

</semantics>

</math><img src="./ea6cdeceedbfddd806b97531fa3fb7a2e40842bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.246ex; height:3.676ex;" alt="{\displaystyle H_{\ell }^{(-)}(0,\rho )/\rho =-i\,h_{\ell }^{(2)}(\rho )}" loading="lazy"></dd></dl>

and are normalized same as <a href="Spherical_Bessel_functions" class="mw-redirect" title="Spherical Bessel functions">Spherical Bessel functions</a>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

</mrow>

</mrow>

<mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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')}</annotation>

</semantics>

</math><img src="./383dfa491297330014cc4b45676fbaab9068e1ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:78.547ex; height:9.676ex;" alt="{\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')}" loading="lazy"></dd></dl>

and similar for other 3.

<div class="mw-heading mw-heading2"><h2 id="Properties_of_the_Coulomb_function">Properties of the Coulomb function</h2></div>

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 <a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr=\delta (k-k')}</annotation>

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</math><img src="./9cb287e825ecc6250de5e6f1a49497f6ad760373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.322ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr=\delta (k-k')}" loading="lazy"></dd></dl>

Other common normalizations of continuum wave functions are on the reduced wave number scale (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k/2\pi }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle k/2\pi }</annotation>

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</math><img src="./8694a7b6c2fa6f1da39e6825f6b1dcb63fd598e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.868ex; height:2.843ex;" alt="{\displaystyle k/2\pi }" loading="lazy">-scale),

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr=2\pi \delta (k-k')\,,}</annotation>

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</math><img src="./0d2199e8cae9131d7e0c156d1978b93cc0945778.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.851ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{k'\ell }(r)r^{2}dr=2\pi \delta (k-k')\,,}" loading="lazy"></dd></dl>

and on the energy scale

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }R_{E\ell }^{\ast }(r)R_{E'\ell }(r)r^{2}dr=\delta (E-E')\,.}</annotation>

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</math><img src="./d196e32fe45418d1c0788649a47410bf116bc618.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.314ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }R_{E\ell }^{\ast }(r)R_{E'\ell }(r)r^{2}dr=\delta (E-E')\,.}" loading="lazy"></dd></dl>

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

</mrow>

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<msub>

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<mfrac>

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<annotation encoding="application/x-tex">{\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')}</annotation>

</semantics>

</math><img src="./f955b865fb39197a4785c6a993951b10982956ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.516ex; height:6.343ex;" alt="{\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')}" loading="lazy"></dd></dl>

as a consequence of the normalization

<mover>

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<annotation encoding="application/x-tex">{\displaystyle \int \psi _{\vec {k}}^{\ast }({\vec {r}})\psi _{{\vec {k}}'}({\vec {r}})d^{3}r=(2\pi )^{3}\delta ({\vec {k}}-{\vec {k}}')\,.}</annotation>

</semantics>

</math><img src="./4f5b38a14283c00e12f7ba272d918faee21d9544.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.998ex; height:5.676ex;" alt="{\displaystyle \int \psi _{\vec {k}}^{\ast }({\vec {r}})\psi _{{\vec {k}}'}({\vec {r}})d^{3}r=(2\pi )^{3}\delta ({\vec {k}}-{\vec {k}}')\,.}" loading="lazy"></dd></dl>

The continuum (or scattering) Coulomb wave functions are also orthogonal to all <a href="Hydrogen_atom#Schrödinger_equation" title="Hydrogen atom">Coulomb bound states</a><a href="#cite_note-8">[8]</a>

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{n\ell }(r)r^{2}dr=0}</annotation>

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</math><img src="./07c1e8189f678d29e42a9505020933103dee4e6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.277ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }R_{k\ell }^{\ast }(r)R_{n\ell }(r)r^{2}dr=0}" loading="lazy"></dd></dl>

due to being eigenstates of the same <a href="Hermitian_operator" class="mw-redirect" title="Hermitian operator">hermitian operator</a> (the <a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">hamiltonian</a>) with different eigenvalues.

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBateman1953" class="citation cs2">Bateman, Harry (1953), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf">Higher transcendental functions</a> (PDF), vol. 1, McGraw-Hill, archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/bateman/Vol1.pdf">the original</a> (PDF) on 2011-08-11, retrieved 2011-07-30</cite>.</li>

<li><cite id="CITEREFJaegerHulme1935" class="citation cs2">Jaeger, J. C.; Hulme, H. R. (1935), "The Internal Conversion of γ -Rays with the Production of Electrons and Positrons", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 148 (865): 708–728, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1935RSPSA.148..708J">1935RSPSA.148..708J</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1935.0043">10.1098/rspa.1935.0043</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0080-4630">0080-4630</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/96298">96298</a></cite></li>

<li><cite id="CITEREFSlater1960" class="citation cs2">Slater, Lucy Joan (1960), Confluent hypergeometric functions, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0107026">0107026</a></cite>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div class="reflist">

<li id="cite_note-1"><a href="#cite_ref-1">^</a> <cite id="CITEREFHill2006" class="citation cs2">Hill, Robert N. (2006), Drake, Gordon (ed.), <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/882343">Handbook of atomic, molecular and optical physics</a>, Springer New York, pp. 153–155, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-26308-3">10.1007/978-0-387-26308-3</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-20802-2</bdi></cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFLandauLifshitz1977" class="citation cs2">Landau, L. D.; Lifshitz, E. M. (1977), Course of theoretical physics III: Quantum mechanics, Non-relativistic theory (3rd ed.), Pergamon Press, p. 569</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFMessiah1961" class="citation cs2">Messiah, Albert (1961), Quantum mechanics, North Holland Publ. Co., p. 485</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFGaspard2018" class="citation cs2">Gaspard, David (2018), "Connection formulas between Coulomb wave functions", J. Math. Phys., 59 (11): 112104, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1804.10976">1804.10976</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.5054368">10.1063/1.5054368</a></cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFMessiah1961" class="citation cs2">Messiah, Albert (1961), Quantum mechanics, North Holland Publ. Co., p. 426</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFFormánek2004" class="citation cs2 cs1-prop-foreign-lang-source">Formánek, Jiří (2004), Introduction to quantum theory I (in Czech) (2nd ed.), Prague: Academia, pp. 128–130</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFLandauLifshitz1977" class="citation cs2">Landau, L. D.; Lifshitz, E. M. (1977), Course of theoretical physics III: Quantum mechanics, Non-relativistic theory (3rd ed.), Pergamon Press, p. 121</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFLandauLifshitz1977" class="citation cs2">Landau, L. D.; Lifshitz, E. M. (1977), Course of theoretical physics III: Quantum mechanics, Non-relativistic theory (3rd ed.), Pergamon Press, pp. 668–669</cite>

</li>

</ol></div></div></div><div><div class="zim-footer">

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