{"id":61,"date":"2013-03-28T16:52:42","date_gmt":"2013-03-28T16:52:42","guid":{"rendered":"http:\/\/blogs.nd.edu\/parkhillgroup\/?p=61"},"modified":"2013-03-30T17:22:33","modified_gmt":"2013-03-30T17:22:33","slug":"questions-for-beginners","status":"publish","type":"post","link":"https:\/\/sites.nd.edu\/parkhillgroup\/2013\/03\/28\/questions-for-beginners\/","title":{"rendered":"Questions for beginners."},"content":{"rendered":"<div>These questions show you that three common mathematical tasks at the roots of linear algebra and quantum mechanics are equivalent; the three tasks are:<\/div>\n<ul>\n<li>Diagonalization of a matrix.<\/li>\n<li>Minimization of a linear functional.<\/li>\n<li>Fourier transform of the matrix exponential<\/li>\n<\/ul>\n<div>In what sense equivalent? In the sense that if you have performed any of these tasks you can translate the answer to the other task with little-to-no effort. The &#8220;answer&#8221; if it is a vector can be imagined to be a wavefunction, if it is an scalar, you should imagine it to be an energy or an eigenvalue. Most &#8220;work&#8221; in quantum mechanics falls into one of these three categories. One quick example of their inter-relations coming in handy is <a href=\"http:\/\/en.wikipedia.org\/wiki\/Quantum_phase_estimation_algorithm\">quantum phase estimation which is the (3)-&gt;(1) map<\/a>. A main trick not on the list is Monte-Carlo which can be used to solve (2) and (3) as an integral approximation. It <a href=\"http:\/\/jcp.aip.org\/resource\/1\/jcpsa6\/v131\/i5\/p054106_s1\">can be used for<\/a> (1) as well which was the topic of recent research.<\/div>\n<ol>\n<li>Show how to exponentiate a matrix trivially, assuming it can be diagonalized (ie: assuming you know it&#8217;s eigenvalues and eigenvectors).<\/li>\n<li>Show that the vector v, which minimizes the &#8220;Ritz functional&#8221; of the matrix A,<a href=\"http:\/\/blogs.nd.edu\/parkhillgroup\/files\/2013\/03\/ritz.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-84\" alt=\"ritz\" src=\"http:\/\/blogs.nd.edu\/parkhillgroup\/files\/2013\/03\/ritz-300x32.jpg\" width=\"168\" height=\"25\" \/><\/a> , is the eigenvector of A with smallest eigenvalue. (HINT: assume eigen-decomposition of A). Side question: Show that E{v} is stationary ie: has derivative zero if and only if v is an eigenvector of A.<\/li>\n<li>Show that the Fourier transform: G(\\omega) = \\int_-\\infty^\\infty exp(-iAt+i\\omega t)dt where A is a Hermitian matrix, has poles (singularities etc.) at the eigenvalues of A. (hint use properties of Fourier transform and eigendecomposition of A)<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>These questions show you that three common mathematical tasks at the roots of linear algebra and quantum mechanics are equivalent; the three tasks are: Diagonalization of a matrix. Minimization of a linear functional. Fourier transform of the matrix exponential In &hellip; <a href=\"https:\/\/sites.nd.edu\/parkhillgroup\/2013\/03\/28\/questions-for-beginners\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1293,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[55402],"tags":[],"class_list":["post-61","post","type-post","status-publish","format-standard","hentry","category-for-students"],"_links":{"self":[{"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/posts\/61","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/users\/1293"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/comments?post=61"}],"version-history":[{"count":23,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/posts\/61\/revisions"}],"predecessor-version":[{"id":100,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/posts\/61\/revisions\/100"}],"wp:attachment":[{"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/media?parent=61"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/categories?post=61"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/tags?post=61"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}