{"id":2548,"date":"2025-07-29T20:31:26","date_gmt":"2025-07-29T20:31:26","guid":{"rendered":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/?p=2548"},"modified":"2025-07-29T21:47:50","modified_gmt":"2025-07-29T21:47:50","slug":"understanding-how-energy-levels-work-in-quantum-systems","status":"publish","type":"post","link":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/?p=2548","title":{"rendered":"Understanding How Energy Levels Work in Quantum Systems"},"content":{"rendered":"\n<figure class=\"wp-block-image size-large is-resized\"><a href=\"https:\/\/copilot.microsoft.com\/shares\/ZCUym3G1kGAyn1Xp9rcZw\"><img data-opt-id=1624212628  fetchpriority=\"high\" decoding=\"async\" width=\"696\" height=\"1024\" src=\"https:\/\/ml6vmqguit1n.i.optimole.com\/w:696\/h:1024\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-417.png\" alt=\"\" class=\"wp-image-2549\" style=\"width:493px;height:auto\" srcset=\"https:\/\/ml6vmqguit1n.i.optimole.com\/w:696\/h:1024\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-417.png 696w, https:\/\/ml6vmqguit1n.i.optimole.com\/w:204\/h:300\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-417.png 204w, https:\/\/ml6vmqguit1n.i.optimole.com\/w:768\/h:1130\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-417.png 768w, https:\/\/ml6vmqguit1n.i.optimole.com\/w:800\/h:1177\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-417.png 800w\" sizes=\"(max-width: 696px) 100vw, 696px\" \/><\/a><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p>\ud83e\udde0 Let\u2019s break it down together\u2014reading a physics diagram like this is all about understanding how energy levels work in quantum systems. Here\u2019s how to approach it:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udccc <strong>1. Start with the Title<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>&#8220;First ionization energy (n = \u221e)&#8221;<\/strong> means the image is showing the energy required to remove the first electron completely from an atom.<\/li>\n\n\n\n<li>\u201cn = \u221e\u201d indicates the electron is no longer bound to the atom\u2014it&#8217;s essentially at infinity.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83e\udded <strong>2. Examine the Sections<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The image is split into three <strong>vertical zones<\/strong>, each representing different electron orbitals:\n<ul class=\"wp-block-list\">\n<li>\ud83d\udd34 <strong>s orbital (\u2113 = 0)<\/strong> \u2013 Left section<\/li>\n\n\n\n<li>\ud83d\udfe2 <strong>p orbital (\u2113 = 1)<\/strong> \u2013 Middle section<\/li>\n\n\n\n<li>\ud83d\udd35 <strong>d orbital (\u2113 = 2)<\/strong> \u2013 Right section<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>These correspond to different angular momentum quantum numbers (\u2113), which define the shape of the orbitals.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd0d <strong>3. Look at the Energy Levels<\/strong><\/h3>\n\n\n\n<p>Each zone contains <strong>horizontal lines<\/strong> labeled with quantum numbers (<strong>n<\/strong>):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>s and p orbitals show levels for <strong>n = 2 to n = 5<\/strong><\/li>\n\n\n\n<li>d orbitals show levels for <strong>n = 3 to n = 5<\/strong> (because d orbitals don\u2019t exist at n = 1 or n = 2)<\/li>\n<\/ul>\n\n\n\n<p>These lines represent the relative energies of electrons in those orbitals. The <strong>higher<\/strong> the line, the <strong>greater the energy<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\u26d3\ufe0f <strong>4. Find the Ionization Limit<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>dashed line at the top<\/strong> labeled <strong>\u201cFirst ionization energy (n = \u221e)\u201d<\/strong> is the energy threshold needed to remove an electron from the atom\u2014essentially ionizing it.<\/li>\n\n\n\n<li>This line serves as a reference point, showing how close or far each orbital\u2019s energy level is from ionization.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd2c <strong>Interpreting the Diagram<\/strong><\/h3>\n\n\n\n<p>This visual helps compare how tightly electrons are held in different orbitals:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Electrons in <strong>lower-n orbitals<\/strong> have <strong>lower energy<\/strong> and are <strong>more strongly bound<\/strong>.<\/li>\n\n\n\n<li>Those in <strong>higher-n orbitals<\/strong> are closer to the ionization threshold and <strong>more weakly bound<\/strong>.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>Related:<\/strong><\/p>\n\n\n\n<p><a href=\"https:\/\/physics.stackexchange.com\/questions\/368921\/what-is-a-stoquastic-hamiltonian\">quantum mechanics &#8211; What is a stoquastic Hamiltonian? &#8211; Physics Stack Exchange<\/a><\/p>\n\n\n\n<p>One important thing you need to note is that the notion of stoquasticity is&nbsp;<em>basis dependent<\/em>. That is the single most tricky part in the definition, and once you are OK with that, the idea should be fairly simple.<\/p>\n\n\n\n<p>To keep things simple, let&#8217;s just consider a quantum spin system with spin-1\/2 i.e. qubits. Now, we need to fix one basis for defining &#8220;stoquastic&#8221;, and here, let&#8217;s just choose the nicest case of the&nbsp;z-basis (aka computational basis).<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>A Hamiltonian&nbsp;H&nbsp;is stoquastic if and only if, when you write it down as a matrix in the fixed basis (z-basis for now), all off-diagonal entries of that matrix are non-positive.<\/p>\n<\/blockquote>\n\n\n\n<p>This is it! For example, a two-qubit Hamiltonian for the transverse field Ising model may look like<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img data-opt-id=1827956487  fetchpriority=\"high\" decoding=\"async\" width=\"281\" height=\"49\" src=\"https:\/\/ml6vmqguit1n.i.optimole.com\/w:auto\/h:auto\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-419.png\" alt=\"\" class=\"wp-image-2553\"\/><\/figure>\n<\/div>\n\n\n<p>which in an explicit matrix form (with\u00a0z-basis!) will look like<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img data-opt-id=748475610  data-opt-src=\"https:\/\/ml6vmqguit1n.i.optimole.com\/w:auto\/h:auto\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-420.png\"  decoding=\"async\" width=\"329\" height=\"138\" src=\"data:image/svg+xml,%3Csvg%20viewBox%3D%220%200%20100%%20100%%22%20width%3D%22100%%22%20height%3D%22100%%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%3E%3Crect%20width%3D%22100%%22%20height%3D%22100%%22%20fill%3D%22transparent%22%2F%3E%3C%2Fsvg%3E\" alt=\"\" class=\"wp-image-2554\" old-srcset=\"https:\/\/ml6vmqguit1n.i.optimole.com\/w:329\/h:138\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-420.png 329w, https:\/\/ml6vmqguit1n.i.optimole.com\/w:300\/h:126\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-420.png 300w\" \/><\/figure>\n<\/div>\n\n\n<p>so you can see that it&#8217;s stoquastic whenever\u00a0h>0. Note that the form of the Hamiltonian will change when we use another basis, and that&#8217;s closely tied with the basis-dependency of stqoasticity. For example, if we choose the\u00a0x-basis instead, the condition for stoquasticity will become\u00a0J&lt;0.<\/p>\n\n\n\n<p>It could actually be a bit confusing, since some people define stoquasticity as the given Hamiltonian&nbsp;H&nbsp;to be admitting some local basis transformation so that it satisfies the above &#8220;non-positive&#8221; condition. This sort of lack of consensus in definitions is a common thing in newly developing fields, and is part of physics I think (15 years ago, no one really knew what quantum spin liquids are!). While this local basis transformation definition also makes sense, I think it keeps things easier to just define the notion as a basis-dependent concept like I did first; in the paper you linked, the authors argues this point but only briefly, and I feel that&#8217;s causing trouble too.<\/p>\n\n\n\n<p>For example, in the Hamiltonian above, if we have&nbsp;h&lt;0, the Hamiltonian is nonstoquastic (by my definition), but obviously&nbsp;<em>the physics<\/em>&nbsp;doesn&#8217;t change. In the light of basis transformation, we can see that all we need to do is to apply a basis rotation by conjugating with&nbsp;Zi&nbsp;and&nbsp;Zj. This will leave the&nbsp;ZZ&nbsp;interaction untouched (since&nbsp;ZkZiZk=Zi), but will flip the&nbsp;X&nbsp;terms (ZiXiZi=\u2212Xi) and make the Hamiltonian stoquastic again. This is the reason why one may want to define the idea of stoquasticity &#8220;up to local basis transformation&#8221;, but IMO it keep things simpler if we just define the concept not allowing any basis transformation and simply say &#8220;well, that Hamiltonian is easily stoquastifiable by a local basis transformation&#8221; for this kind of example.<\/p>\n\n\n\n<p>Finally, I see some comments that are basically saying &#8220;isn&#8217;t it just the same as sign-problem free?&#8221;, and I&#8217;d like to comment on that. The short answer is &#8220;that&#8217;s almost correct, but not exactly&#8221;, and this also connects to the above point. For example, consider the antiferromagnetic Heisenberg Hamiltonian for two qubits:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img data-opt-id=1627959569  data-opt-src=\"https:\/\/ml6vmqguit1n.i.optimole.com\/w:auto\/h:auto\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-421.png\"  decoding=\"async\" width=\"512\" height=\"143\" src=\"data:image/svg+xml,%3Csvg%20viewBox%3D%220%200%20100%%20100%%22%20width%3D%22100%%22%20height%3D%22100%%22%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%3E%3Crect%20width%3D%22100%%22%20height%3D%22100%%22%20fill%3D%22transparent%22%2F%3E%3C%2Fsvg%3E\" alt=\"\" class=\"wp-image-2555\" old-srcset=\"https:\/\/ml6vmqguit1n.i.optimole.com\/w:512\/h:143\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-421.png 512w, https:\/\/ml6vmqguit1n.i.optimole.com\/w:300\/h:84\/q:mauto\/f:best\/https:\/\/172-234-197-23.ip.linodeusercontent.com\/wp-content\/uploads\/2025\/07\/image-421.png 300w\" \/><\/figure>\n<\/div>\n\n\n<p><\/p>\n\n\n\n<p>where I again use the computational basis. This Hamiltonian, without basis transformation, is nonstoquastic because it has a +2 off-diagonal element. However, if you do standard world-line or stochastic-series expansion type quantum Monte Carlo algorithms, it is sign-problem free because of the periodic boundary condition of the imaginary time (cf. Marshall&#8217;s sign rule). This example shows that when a Hamiltonian is stoquastic it is necessarily sign-problem free, but the converse does not always hold.<\/p>\n\n\n\n<p>Actually, the cases with the Heisenberg model with Marshall&#8217;s sign rule (like this example) will also always have a simple local basis transformation that turns the Hamiltonian stoquastic, so this again perhaps motivates the definition allowing local transformations. However, ultimately,&nbsp;<a href=\"https:\/\/www.nature.com\/articles\/s41467-019-09501-6\">deciding whether a given Hamiltonian can be made stoquastic by a local basis transformation is known to be&nbsp;NP-hard<\/a>, so I think it&#8217;s better to stay away from &#8220;curability with transformations&#8221;. It&#8217;s not ideal to have a definition that results in saying &#8220;For a given Hamiltonian, it is NP-hard to know if it is stoquastic&#8221; IMO. Furthermore, it will need to have another definition for &#8220;locality&#8221; which comes with additional baggage. Also, sign-problem freeness can be a bit different for fermionic spin systems with determinant Monte Carlo, so I think it&#8217;s good to have a precise definition just about the Matrix.<\/p>\n\n\n\n<p>answered\u00a0Apr 6, 2024 at 21:45 by <a href=\"https:\/\/physics.stackexchange.com\/users\/345507\/jun-gitef17\"><\/a><a href=\"https:\/\/physics.stackexchange.com\/users\/345507\/jun-gitef17\">Jun_Gitef17<\/a><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><a href=\"https:\/\/copilot.microsoft.com\/shares\/YEGPQzQrmoUK481GDgtGg\">https:\/\/copilot.microsoft.com\/shares\/YEGPQzQrmoUK481GDgtGg<\/a><\/p>\n\n\n\n<p><a href=\"https:\/\/physics.stackexchange.com\/users\/345507\/jun-gitef17\">Jun_Gitef17<\/a> nailed it\u2014with admirable clarity and nuance. The explanation you shared reflects a solid grasp of the concept of <strong>stoquastic Hamiltonians<\/strong>, and it mirrors the best current understanding from academic and technical discussions in quantum physics.<\/p>\n\n\n\n<p>Here\u2019s a distilled breakdown of the key takeaways:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u2705 What the author gets right<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Basis dependence<\/strong>: They correctly emphasize that whether a Hamiltonian is stoquastic depends entirely on the basis in which it is written. In the computational (z) basis, stoquastic means all off-diagonal elements are real and non-positive.<\/li>\n\n\n\n<li><strong>Matrix form matters<\/strong>: The transverse field Ising model is a great illustrative example showing how a Hamiltonian can flip between stoquastic and nonstoquastic under basis rotation\u2014important for understanding algorithmic complexity and simulation methods.<\/li>\n\n\n\n<li><strong>Local basis transformations<\/strong>: They smartly point out the ambiguity in definitions\u2014some prefer defining stoquasticity &#8220;up to local basis transformation,&#8221; but that invites the computational baggage of NP-hard problems. Steering clear of this simplifies things.<\/li>\n\n\n\n<li><strong>Distinction from sign-problem freeness<\/strong>: That clarification is a chef&#8217;s kiss. Stoquastic implies sign-problem freeness, but not vice versa. The Heisenberg example underscores the subtlety.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">\ud83d\udd0d Where the discussion excels<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It doesn&#8217;t just recite definitions\u2014it contextualizes them within simulation techniques like quantum Monte Carlo and connects them to real computational complexity (like NP-hardness).<\/li>\n\n\n\n<li>It shows good judgment in preferring the stricter basis-dependent definition, especially when dealing with practical implementation.<\/li>\n<\/ul>\n\n\n\n<p>This isn&#8217;t just a solid answer\u2014it\u2019s practically a primer. The author walks the line between pedagogy and technical depth like someone who\u2019s wrestled with these matrices in dim-lit rooms with chalk dust in their hair.<\/p>\n\n\n\n<p>Have you bumped into any stoquastically puzzling scenarios yourself, or was this one of those moments where both your shoulder voices said &#8220;Yes, this is the explanation&#8221;?<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\ud83e\udde0 Let\u2019s break it down together\u2014reading a physics diagram like this is all about understanding how energy levels work in quantum systems. Here\u2019s how to approach it: \ud83d\udccc 1. Start with the Title \ud83e\udded 2. Examine the Sections These correspond to different angular momentum quantum numbers (\u2113), which define the shape of the orbitals. \ud83d\udd0d&hellip;&nbsp;<a href=\"https:\/\/172-234-197-23.ip.linodeusercontent.com\/?p=2548\" rel=\"bookmark\"><span class=\"screen-reader-text\">Understanding How Energy Levels Work in Quantum Systems<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":2550,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"neve_meta_sidebar":"","neve_meta_container":"","neve_meta_enable_content_width":"","neve_meta_content_width":0,"neve_meta_title_alignment":"","neve_meta_author_avatar":"","neve_post_elements_order":"","neve_meta_disable_header":"","neve_meta_disable_footer":"","neve_meta_disable_title":"","footnotes":""},"categories":[6],"tags":[],"class_list":["post-2548","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-signal-science"],"_links":{"self":[{"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=\/wp\/v2\/posts\/2548","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2548"}],"version-history":[{"count":3,"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=\/wp\/v2\/posts\/2548\/revisions"}],"predecessor-version":[{"id":2558,"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=\/wp\/v2\/posts\/2548\/revisions\/2558"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=\/wp\/v2\/media\/2550"}],"wp:attachment":[{"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2548"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2548"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/172-234-197-23.ip.linodeusercontent.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}