{"id":439774,"date":"2017-08-14T02:00:00","date_gmt":"2017-08-13T23:00:00","guid":{"rendered":"http:\/\/www.fyysika.ee\/?guid=0864cb47e6b1daed08734b3cae26b284"},"modified":"2017-08-14T02:00:00","modified_gmt":"2017-08-13T23:00:00","slug":"energy-splitting-in-a-finite-periodic-multiple-well-potential","status":"publish","type":"post","link":"https:\/\/www.fyysika.ee\/?p=439774","title":{"rendered":"Energy splitting in a finite periodic multiple-well potential"},"content":{"rendered":"<p>The low-lying states for a one-dimensional potential consisting of N identical wells are considered,<br \/>\nassuming that the wells are parabolic around the minima. The N localized approximate eigenfunctions,<br \/>\neach of which matches an eigenfunction of the simple harmonic oscillator in one of the wells, are<br \/>\nconstructed, relying on the WKB approximation. Diagonalizing the Hamiltonian in the subspace spanned<br \/>\nby the approximate eigenfunctions, a formula for the energy eigenvalues is obtained. The present<br \/>\nwork will be useful for introducing Bloch wave functions in a periodic potential to undergraduate<br \/>\nand graduate students. It is shown that the formula for the eigenvalues can reproduce a known<br \/>\nrigorous expression on the Mathieu equation.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The low-lying states for a one-dimensional potential consisting of N identical wells are considered,<br \/>\nassuming that the wells are parabolic around the minima. The N localized approximate eigenfunctions,<br \/>\neach of which matches an eigenfunction of the si&#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-439774","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\/439774","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=439774"}],"version-history":[{"count":0,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/439774\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=439774"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=439774"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=439774"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}