[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-566554-105":3,"detail-sidebar-cat-1-en-105":81,"doc-detail-566554-en":127},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","schrodinger-wave-equation-quantum-numbers-and-shapes-of-orbitals","Schrödinger Wave Equation - Quantum Numbers and Shapes of Orbitals","","Schrödinger wave equation is introduced as a 1926 formulation describing the dual nature of the electron as a three-dimensional wave around a positively charged nucleus. The document explains the probability interpretation of the wave function and its physical significance, including electron density and the relation to atomic orbitals. It then details quantum numbers (n, l, m, s), their origins, meanings, allowed ranges, degeneracy, and how they determine energy states and orbital shapes and orientations.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/general/","General",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/schrodinger-wave-equation-quantum-numbers-and-shapes-of-orbitals/566554/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/schrodinger-wave-equation-quantum-numbers-and-shapes-of-orbitals/566554.png","ImageObject",442,249,{"name":42,"@type":43},"AfterMidnight","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-29","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",10,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How does the wave function relate to electron probability in the document?","Question",{"text":63,"@type":64},"The document states that the square of the wave function at any point is proportional to the probability of finding the electron at that point. The place of maximum probability is called electron density, electron cloud, or an atomic orbital.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What do the principle quantum number (n) and azimuthal quantum number (l) determine?",{"text":68,"@type":64},"The principle quantum number (n) determines the average distance from the nucleus and the electron’s energy shell, while the azimuthal quantum number (l) determines the number of subshells and the shapes of subshells.",{"name":70,"@type":61,"acceptedAnswer":71},"How are magnetic quantum number (m) and spin quantum number (s) described?",{"text":72,"@type":64},"Magnetic quantum number (m) gives permitted orientations of subshells, varies from −l to +l, and explains splitting of spectral lines (Zeeman effect). Spin quantum number (s) has values +1/2 and −1/2 and represents the electron’s spin direction on its axis.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},566554,1792002285,{"code":4,"msg":82,"data":83},"success",[84,89,94,99,104,109,114,119,124],{"id":85,"doc_module":22,"doc_module_name":25,"category_name":86,"show_sort_weight":87,"slug":88},11,"Presentations",90,"presentations",{"id":90,"doc_module":22,"doc_module_name":25,"category_name":91,"show_sort_weight":92,"slug":93},12,"Resumes",80,"resumes",{"id":95,"doc_module":22,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},14,"Invoices",70,"invoices",{"id":100,"doc_module":22,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},15,"Posters",60,"posters",{"id":105,"doc_module":22,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},16,"Social Media",50,"social-media",{"id":110,"doc_module":22,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},17,"Forms",40,"forms",{"id":115,"doc_module":22,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},18,"Letters",30,"letters",{"id":120,"doc_module":22,"doc_module_name":25,"category_name":121,"show_sort_weight":122,"slug":123},21,"Paper Templates",5,"papers-templates",{"id":125,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":4,"slug":126},158,"general-158",{"code":4,"msg":82,"data":128},{"doc_id":79,"user_id":129,"nickname":42,"user_avatar":130,"doc_module":22,"category_id":125,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":85,"language":136,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":12,"update_tm":140,"read_time":33},962090896061,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Schrödinger wave equation  \n(1) Schrodinger wave equation is given by Erwin Schrödinger in 1926 and based on dual nature of electron.  \n(2) In it electron is described as a three dimensional wave in the electric field of a positively charged nucleus.  \n(3) The probability of finding an electron at any point around the nucleus can be determined by the help of Schrodinger wave equation which is,  \n􀁷 2 􀀼   􀁷 2 􀀼   􀁷 2 􀀼   8􀁓2m  \n􀁷x 2 􀀎 􀁷y 2 􀀎 􀁷z 2 􀀎 h 2 (E 􀀐 V)􀀼 􀀠 0  \nWhere x ,y and z are the 3 space co-ordinates, m = mass of electron, h = Planck’s constant,  \nE = Total energy, V = potential energy of electron, 􀀼 = amplitude of wave also called as wave function.  \n􀁷 = stands for an infinitesimal change.  \n(4) The Schrodinger wave equation can also be written as :  \n2 8􀁓2m  \n􀂒 􀀼 􀀎  h 2  (E 􀀐 V) 􀀼 􀀠 0  \nWhere 􀂒 = laplacian operator.  \n(5) Physical Significance of 􀀼 and 􀀼 2  \n(i) The wave function 􀀼 represents the amplitude of the electron wave. The amplitude 􀀼 is thus a function of space co-ordinates and time i.e. 􀀼 􀀠 􀀼(x ,y, z   times )  \n(ii) For a single particle, the square of the wave function (􀀼2 ) at any point is proportional to the probability of finding the particle at that point.  \n(iii) If 􀀼 2 is maximum than probability of finding e 􀀐 is maximum around nucleus. And the place where probability of finding e 􀀐 is maximum is called electron density, electron cloud or an atomic orbital. It is different from the Bohr’s orbit.  \n(iv)The solution of this equation provides a set of number called quantum numbers which describe specific or definite energy state of the electron in atom and information about the shapes and orientations of the most probable distribution of electrons around the nucleus.  \nQuantum numbers and Shapes of orbitals  \nQuantum numbers  \n(1) Each orbital in an atom is specified by a set of three quantum numbers (n, l, m) and each electron is designated by a set of four quantum numbers (n, l, m and s) .  \n(2) Principle quantum number (n)  \n(i) It was proposed by Bohr’s and denoted by ‘n’.  \n(ii) It determines the average distance between electron and nucleus, means it is denoted the size of atom.  \n\n| r 􀀠 | n 2\u003Cbr>Z | 􀁵 0.529 Å |\n| --- | --- | --- |\n\n(iii) It determine the energy of the electron in an orbit where electron is present.  \nE 􀀠 􀀐 ~~ ~~Zn22~~ ~~ 􀁵 313 . 3 Kcal per mole  \n(iv) The maximum number of an electron in an orbit represented by this quantum number as 2n2 . No energy shell in atoms of known elements possess more than 32 electrons.  \n(v) It gives the information of orbit K, L, M, N------------ .  \n(vi) The value of energy increases with the increasing value of n.  \n(vii) It represents the major energy shell or orbit to which the electron belongs.  \n(viii) Angular momentum can also be calculated using principle quantum number  \n\n| mvr 􀀠 | nh\u003Cbr>2􀁓 |\n| --- | --- |\n\n(3) Azimuthal quantum number (l)  \n(i) Azimuthal quantum number is also known as angular quantum number. Proposed by Sommerfield and denoted by ‘l’.  \n(ii) It determines the number of sub shells or sublevels to which the electron belongs.  \n(iii) It tells about the shape of subshells.  \n(iv) It also expresses the energies of subshells s 􀀟 p 􀀟 d 􀀟 f (increasing energy) .  \n(v) The value of l 􀀠 (n 􀀐 1) always where ‘n’ is the number of principle shell.  \n(vi) Value of l Name of subshell Shape of subshell  \n=  \n=  \n=  \n0  \ns  \nSpherical  \n1  \np  \nDumbbell  \n2  \nd  \nDouble  \ndumbbell  \n3………..(n-1) f  \nComplex  \nh  \n(vii) It represents the orbital angular momentum. Which is equal to? ~~ ~~ ~~ ~~l(l 􀀎 1)  \n2􀁓  \n(viii) The maximum number of electrons in subshell 􀀠 2(2l 􀀎 1) s 􀀐 subshell 􀁯 2 electrons d 􀀐 subshell 􀁯 10 electrons p 􀀐 subshell 􀁯 6 electrons f 􀀐 subshell 􀁯 14 electrons.  \n(ix) For a given value of ‘n’ the total value of ‘l’ is always equal to the value of ‘n’.  \n(x) The energy of any electron is depend on the value of n & l because total energy = (n + l) . The electron enters in that sub orbit whose (n + l) value or the value of","cbCaimJgwzDWeQIK","https://ap.wps.com/l/cbCaimJgwzDWeQIK","pdf",517254,"English","# Schrödinger Wave Equation\n## Physical Significance of Amplitude and Probability\n## Quantum Numbers and Shapes of Orbitals\n### Quantum Numbers (n, l, m, s)\n### Distribution of Electrons among Quantum Levels","[{\"question\":\"How does the wave function relate to electron probability in the document?\",\"answer\":\"The document states that the square of the wave function at any point is proportional to the probability of finding the electron at that point. The place of maximum probability is called electron density, electron cloud, or an atomic orbital.\"},{\"question\":\"What do the principle quantum number (n) and azimuthal quantum number (l) determine?\",\"answer\":\"The principle quantum number (n) determines the average distance from the nucleus and the electron’s energy shell, while the azimuthal quantum number (l) determines the number of subshells and the shapes of subshells.\"},{\"question\":\"How are magnetic quantum number (m) and spin quantum number (s) described?\",\"answer\":\"Magnetic quantum number (m) gives permitted orientations of subshells, varies from −l to +l, and explains splitting of spectral lines (Zeeman effect). Spin quantum number (s) has values +1/2 and −1/2 and represents the electron’s spin direction on its axis.\"}]","Schrödinger Wave Equation - Quantum Numbers and Shapes of Orbitals | PDF",1791862802]