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�����9��?���œ/q��̜'ęe �������Q�6����v~�B���|Th�Wc��>*��5�0����!w���s Zm��N~k�;l2d. 3P� By closing this message, you are consenting to our use of cookies. 3099067 >> The standard approach for solving the radial equation requires special consideration of the asymptotic form of the function and the use of the associated Laguerre polynomials. As a rst example of solving the radial Schr odinger equation, consider the spher-ical in nite square well, with V(r) = 0 out to some radius aand V = 1beyond that. The radial equation for the hydrogen atom is solved by the method of Laplace transformation. The function of radial wave of a hydrogen atom contains the principal quantum number ( ) and the orbital quantum number (ℓ). 3 0 obj << 0�;+��K��w$���(��&�*U���6��f����/f5����D��5{C p��#�f?h}�����R�=I�#$�����iiue�
uw(��c4�Ԙ��V�TPB��(���\��!.ϟnY���LK%0A�vO �$H�N�^���E훡=�Y��i�i�8֫v�|��|^�����3�Nt�ة���Cf�k�}\T��|)Θ!T�Ox{H?�� �ӓ��A-|� �o]L&�ħ�b��;��r� NkN First we change to a dimensionless variable, giving the differential equation where the constant Next we look at the equation for large. C*�U�懪�z9,"�/U��>�U
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)� 7�����66p�(!6\Ap�P0�OC�XOȑPY��*n(H"Ϥ�H�ejy. The standard approach for solving the radial equation requires special consideration of the asymptotic form of the function and the use of the associated Laguerre polynomials. From the view point of pedagogy, the hydrogen atom merges many of the concepts and techniques previously developed into one package. Y���XE�䯧{f 䀄$��9qz�=_74��h��:������2�1I�������2FM~*fU1�7�m�MU//���om��\#-JsY�Q�R��/��l�L�-a�����b9'F�,o'��L��#��a�+�R�ɟ�eU"�BN~&���)�p��/s�-g媡�(q�1�.ӆP̀K���5>� r>�� ��������1)��p���_3���by���psQ/ʥ��v�s)'�r��:L��ì��3�>�^����Kֆq�(�j9+�h�Y��&˝�)�&��.��h㿗h�zV-o��VM�YR�j.����U5-�dӠ�z��r��#�"�v��M5�Gf�[P�R���M_�":�]���p�1r�h�p.� yG
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���,V!�IA��7������Cd_���r�X��CHۏv��@�P}�H��/�|f�U��~U��X2���S0WXbw �f'L�V The equation to be solved is thus: h¯ 2 2m du dr 2 + e 4ˇ 0 1 r + h¯2 2m l(l+1) r! hydrogen atom, aren't we. We use cookies to improve your website experience. ���e���qGdЫ���x��*q�_�'V4���T��Fg#�2�[7}�}��3� ��3 � �mU!Z]GH�/ "���E5��˪�=�� To learn about our use of cookies and how you can manage your cookie settings, please see our Cookie Policy. The function of radial wave of a hydrogen atom expresses the movement of electron in a hydrogen atom which propagates spread from the centre of the atom towards all directions and depends on the distance () of origin. It is a particle in a box with spherical, soft walls. ��� 5 Howick Place | London | SW1P 1WG. We point out several important issues: In the conventional series solution method, when two roots of the indicial equation are found to differ by an integer, there can be only one independent solution according to the known theorem; in the recent Laplace transform approach adopted by Shertzer, the asymptotic behaviour of the solution was implicitly assumed. w;�Q|�s'l�0xG���>�XϷ�k��y����L�~yT�s��qTV���fzy�p��I'���q ��-W���r]�!��bR݄���x�6�p~.�X'\� 6.2 Separation of Variables We now discuss the technique for solving our equation for the electron in the hydrogen atom. For l>0 the solutions are harder to nd, but they can still be expressed exactly in terms of sines and cosines. You should be able to write down the solutions for l= 0 almost immediately. Different approaches to the solution of the radial equation for the hydrogen atom are examined. Different approaches to the solution of the radial equation for the hydrogen atom are examined. ���z'/_)����1���DXNQ#_���Y�iB��CY뭔&��Z���\��uJhJ(e�DwqU:C�M�U�$����8�����o� P˘ A�̔��B�M��J��w R���O]d\�`4��^�=$�>�
)(�Yr�sE�j��X%X��c��a�� �>!ǁYE�ʒ�%1��,��H��24pn�e0F4�}m�Wl�u��5��n��/�Ă �Z0#LR � �o�OJ���h~E�[X��4zX-$�Lji��H-"8� ����y1Eҩ�h��>^��l0]gV��%��44 �u(E�X��/%��w�j�6]�Jf6M�i�?V�>�uD_�ż�rX/\�l�����.zx�����lg�2�|S�ϫ҄��_,��M1�ԫu�4�#�g�^�4���t,p�N�+0��T�j��=i����zۼ�0�E�z�(������¢LR;I�S�^#�uGﺘ��y볰&��fu��urk�l����:����0�4Q |���S�h3��'��R�f��n�O���2�Ѡ�2{1> F ���b�&��� �P�����Ч��>����I�O'�N������O��NRNK���z� `�뛈 Inverse transformation of the solution yields the exact radial functions. stream The summation limits are determined by the values of the quantum numbers nand I.A simple recursion relation is obtained for the coefficients of the series. x��[Ks����W�f����a�V���c�R�īr*��"! This can be solved by , so we explicitly include this. We ﬁrst investigate the asymptotic be- the Schrodinger equation is transformed into the Radial equation for the Hydrogen atom: h2 2 r2 d dr r2 dR(r) dr + " h2l(l+1) 2 r2 V(r) E # R(r) = 0 The solutions of the radial equation are the Hydrogen atom radial wave-functions, R(r). The transformed radial equation can be treated with a simple series expansion valid over the entire range. �k�7
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O %PDF-1.5 Registered in England & Wales No. /Length 3234 To learn about our use of cookies and how you can manage your cookie settings, please see our Cookie Policy. The transformed radial equation can be treated with a simple series expansion valid over the entire range. By closing this message, you are consenting to our use of cookies. Finally, the suggestion of how to present this subject at different levels is discussed. II. Gױ#{�4�Ɯ2�:���Ʈ��MF�i1�w�M�nfB,��ǒB5 #>�Q��p)� Solution of the radial equation for hydrogen atom: series solution or Laplace transform? Register to receive personalised research and resources by email, Solution of the radial equation for hydrogen by the Laplace transform method. u=Eu (5) The solution of this equation follows a similar method as was used in solving the harmonic oscillator. We traced back the original treatment of Schrodinger and reviewed his rigorous Laplace transform approach. ;�U 38�.fƧ�= C!P;GH4�>jH��b�!����C*��8���(L��}.�QI �8q�`�������Z��}Z&5��� ��=� 0i���v�O�! Nevertheless, we talk about doing the "hydrogen atom," because our solution will provide us with much of what we need to know about hydrogen.

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