{"id":75183,"date":"2015-04-01T03:00:00","date_gmt":"2015-04-01T00:00:00","guid":{"rendered":"http:\/\/www.fyysika.ee\/?guid=b58c7671b3aef3fe35f41ad030b24702"},"modified":"2015-04-01T03:00:00","modified_gmt":"2015-04-01T00:00:00","slug":"a-higher-order-finite-difference-approximation-with-richardsons-extrapolation-to-the-energyeigenvalues-of-the-quartic-sextic-and-octic-anharmonic-oscillators-2","status":"publish","type":"post","link":"https:\/\/www.fyysika.ee\/?p=75183","title":{"rendered":"A higher-order finite-difference approximation with Richardson?s extrapolation to the energy\r\neigenvalues of the quartic, sextic and octic anharmonic oscillators"},"content":{"rendered":"<p>In this paper, we present highly accurate numerical results for the lowest four energy eigenvalues<br \/>\nof the quartic, sextic and octic anharmonic oscillators over a wide range of the anharmonicity<br \/>\nparameter ##IMG## [http:\/\/ej.iop.org\/images\/0143-0807\/36\/3\/035025\/ejp510005ieqn1.gif] {$lambda .$}<br \/>\nAlso, we provide illustrative graphs describing the dependence of the eigenvalues on ##IMG##<br \/>\n[http:\/\/ej.iop.org\/images\/0143-0807\/36\/3\/035025\/ejp510005ieqn2.gif] {$lambda .$} Our computation is<br \/>\ncarried out by using higher-order finite-difference approximation, involving the nine-and-ten-point<br \/>\ndifferentiation formulas. In addition, we apply Richardson?s extrapolation method in our calculation<br \/>\nfor the purpose of achieving a maximum numerical precision. The main advantage of utilizing the<br \/>\nfinite-difference approach lies in its simplicity and capability to transform the time-independent<br \/>\nSchr?dinger equation into an eigenvalue matrix equation. This allo&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this paper, we present highly accurate numerical results for the lowest four energy eigenvalues<br \/>\nof the quartic, sextic and octic anharmonic oscillators over a wide range of the anharmonicity<br \/>\nparameter ##IMG## [http:\/\/ej.iop.org\/images\/0143-0807\/36&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[178],"tags":[],"class_list":{"0":"post-75183","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-rss-fuusikaharidus","7":"entry"},"_links":{"self":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/75183","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=75183"}],"version-history":[{"count":0,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/75183\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=75183"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=75183"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=75183"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}