text: "use strict";(self.webpackChunkdocusaurus_tecuntecs=self.webpackChunkdocusaurus_tecuntecs||[]).push([[595],{5661:function(e,a,t){t.r(a),t.d(a,{frontMatter:function(){return m},contentTitle:function(){return p},metadata:function(){return l},toc:function(){return c},default:function(){return d}});var n=t(7462),s=t(3366),i=(t(7294),t(3905)),r=t(8331),o=["components"],m={sidebar_position:1},p="Pre-requisite Knowledge",l={unversionedId:"quantum-mechanics/prerequisite-knowledge",id:"quantum-mechanics/prerequisite-knowledge",isDocsHomePage:!1,title:"Pre-requisite Knowledge",description:'Grant Sanderson on "not being gifted enough"',source:"@site/docs-advanced/quantum-mechanics/prerequisite-knowledge.mdx",sourceDirName:"quantum-mechanics",slug:"/quantum-mechanics/prerequisite-knowledge",permalink:"/docs-advanced/quantum-mechanics/prerequisite-knowledge",tags:[],version:"current",sidebarPosition:1,frontMatter:{sidebar_position:1},sidebar:"tutorialSidebar",previous:{title:"TecunTecs Intro",permalink:"/docs-advanced/intro"},next:{title:"Schrodinger Equation",permalink:"/docs-advanced/quantum-mechanics/schrodinger-equation"}},c=[{value:"Grant Sanderson on &quot;not being gifted enough&quot;",id:"grant-sanderson-on-not-being-gifted-enough",children:[],level:4},{value:"The case for taking pre-requisite knowledge seriously (but fearlessly)",id:"the-case-for-taking-pre-requisite-knowledge-seriously-but-fearlessly",children:[],level:4},{value:"Physics background",id:"physics-background",children:[],level:2},{value:"Mathematical background",id:"mathematical-background",children:[{value:"Column spaces (i.e., the vector space generated by the columns of a matrix)",id:"column-spaces-ie-the-vector-space-generated-by-the-columns-of-a-matrix",children:[],level:4},{value:"Independence, basis, and dimension",id:"independence-basis-and-dimension",children:[],level:4},{value:"Eigenvalues and Eigenvectors",id:"eigenvalues-and-eigenvectors",children:[],level:4}],level:2},{value:"References",id:"references",children:[],level:2}],h={toc:c};function d(e){var a=e.components,t=(0,s.Z)(e,o);return(0,i.kt)("wrapper",(0,n.Z)({},h,t,{components:a,mdxType:"MDXLayout"}),(0,i.kt)("h1",{id:"pre-requisite-knowledge"},"Pre-requisite Knowledge"),(0,i.kt)("h4",{id:"grant-sanderson-on-not-being-gifted-enough"},'Grant Sanderson on "not being gifted enough"'),(0,i.kt)("div",{class:"video-container"},(0,i.kt)("iframe",{width:"560",height:"315",src:"https://www.youtube.com/embed/-bc9EWhmDZg?start=412&end=532&controls=0&rel=0&disablekb=1",title:"YouTube video player",frameborder:"0",allow:"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture"})),(0,i.kt)("h4",{id:"the-case-for-taking-pre-requisite-knowledge-seriously-but-fearlessly"},"The case for taking pre-requisite knowledge seriously (but fearlessly)"),(0,i.kt)("p",null,'When it comes to mathematics (and quantum mechanics has a lot of it),\nit is common currency to hold the belief that some gifted people simply "get it"\nfaster than others. But does this belief actually hold true to reality?\n',(0,i.kt)("strong",{parentName:"p"},"For the most part, at least, it seems that background is the real dealbraker.")),(0,i.kt)("div",{className:"admonition admonition-tip alert alert--success"},(0,i.kt)("div",{parentName:"div",className:"admonition-heading"},(0,i.kt)("h5",{parentName:"div"},(0,i.kt)("span",{parentName:"h5",className:"admonition-icon"},(0,i.kt)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},(0,i.kt)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"}))),"Understand the pre-requisites")),(0,i.kt)("div",{parentName:"div",className:"admonition-content"},(0,i.kt)("p",{parentName:"div"},"Quantum mechanics is not really that hard; ",(0,i.kt)("strong",{parentName:"p"},"it just has a\nlot of pre-requisite knowledge.")," If the pre-requisite knowledge is practiced\nand learned at the right pace at the right\ntime, anyone can understand, master, and apply the theory."))),(0,i.kt)("div",{className:"admonition admonition-warning alert alert--danger"},(0,i.kt)("div",{parentName:"div",className:"admonition-heading"},(0,i.kt)("h5",{parentName:"div"},(0,i.kt)("span",{parentName:"h5",className:"admonition-icon"},(0,i.kt)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"},(0,i.kt)("path",{parentName:"svg",fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"}))),"Do not overlook the pre-requisites")),(0,i.kt)("div",{parentName:"div",className:"admonition-content"},(0,i.kt)("p",{parentName:"div"},"The same is true for the opposite scenario: ",(0,i.kt)("strong",{parentName:"p"},"without the right pre-requisite\nknowledge, the most seemingly straightforward lecture on quantum mechanics will\nbe as strange as an unfamiliar alphabet and as hard as trying to read it"),".\nBut to blame is not the lack of talent, but the pedagogy\n(whether self-taught or not)."))),(0,i.kt)("h2",{id:"physics-background"},"Physics background"),(0,i.kt)("p",null,"Contrary to the way that quantum mechanics is usually taught, it is possible\nto understand the theory without most of the math.\nThe mathematical formulation of the theory is not the same as the theory,\nbut a representation of it.\nWith enough words and good prose, it should be possible explain quantum mechanics\nwith verbal logic."),(0,i.kt)("p",null,"It is also a good idea explain the history of the\nsubject.\nBy understanding how it came about, we gain a lot of insight into the theory that\nwe don't get if we start with the modern mathematical formulation."),(0,i.kt)("p",null,"wave particle duality khan academy"),(0,i.kt)("p",null,(0,i.kt)("a",{parentName:"p",href:"https://youtu.be/iZ96TuP_veY"},"https://youtu.be/iZ96TuP_veY")," (heisenberg?)"),(0,i.kt)("p",null,"This below is still the mathematical formulation! (but at least still focused\non the concepts and not on the mathematical jargon)"),(0,i.kt)("p",null,(0,i.kt)("a",{parentName:"p",href:"https://youtu.be/payp7simhBM"},"https://youtu.be/payp7simhBM")," (just a language)"),(0,i.kt)("p",null,(0,i.kt)("a",{parentName:"p",href:"https://youtu.be/BFTxP03H13k"},"https://youtu.be/BFTxP03H13k")),(0,i.kt)("p",null,'"It is easy to find a system in which the SE cannot be solved analytically" (\nThis is where a solution like TecunTecs comes in)\nEven best computers cannot handle systems that are larger than a few dozens\natoms. Example: Helium atom.'),(0,i.kt)("p",null,'"The Schrodinger Equation tells you how the quantum state changes with respect\nto time based on its energy"'),(0,i.kt)("p",null,"What is a quantum system?"),(0,i.kt)("div",{className:"admonition admonition-info alert alert--info"},(0,i.kt)("div",{parentName:"div",className:"admonition-heading"},(0,i.kt)("h5",{parentName:"div"},(0,i.kt)("span",{parentName:"h5",className:"admonition-icon"},(0,i.kt)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},(0,i.kt)("path",{parentName:"svg",fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"}))),"What are examples of quantum systems?")),(0,i.kt)("div",{parentName:"div",className:"admonition-content"},(0,i.kt)("ul",{parentName:"div"},(0,i.kt)("li",{parentName:"ul"},"If you are interested in how a hydrogen atom behaves, then your quantum\nsystem is that hydrogen atom (in addition to any extra necessary information\nsuch as the position and what's happening in its surroundings)."),(0,i.kt)("li",{parentName:"ul"},"If you place the hydrogen atom in a magnetic field, then your quantum system\nis that hydrogen atom in the magnetic field."),(0,i.kt)("li",{parentName:"ul"},"If you look deeper at the hydrogen atom, the subatomic particles such as the\nelectron, neutron, proton, ane even photon may constitute your quantum system.")))),(0,i.kt)("div",{className:"admonition admonition-info alert alert--info"},(0,i.kt)("div",{parentName:"div",className:"admonition-heading"},(0,i.kt)("h5",{parentName:"div"},(0,i.kt)("span",{parentName:"h5",className:"admonition-icon"},(0,i.kt)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},(0,i.kt)("path",{parentName:"svg",fillRule:"evenodd",d:"M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"}))),"What is a quantum state? ")),(0,i.kt)("div",{parentName:"div",className:"admonition-content"},(0,i.kt)("p",{parentName:"div"},'A quantum state is the mathematical description of our quantum system, i.e.,\nit\'s an expression that "stores" as much information as we know about the\nquantum system.'),(0,i.kt)("p",{parentName:"div"},(0,i.kt)("strong",{parentName:"p"},"Indeed, the quantum state object holds everything about the state of\nthe quantum system!")))),(0,i.kt)("p",null,(0,i.kt)("a",{parentName:"p",href:"https://youtu.be/-WSWz1H3mJg"},"https://youtu.be/-WSWz1H3mJg")),(0,i.kt)("p",null,(0,i.kt)("a",{parentName:"p",href:"https://youtu.be/so1szjHu7jY"},"https://youtu.be/so1szjHu7jY")),(0,i.kt)("p",null,'"The wave function is all the mathematical information we know about our system."'),(0,i.kt)("p",null,'"Operators are mathematical entities that correspond to making a measurement in real life"'),(0,i.kt)("h2",{id:"mathematical-background"},"Mathematical background"),(0,i.kt)("div",{className:"admonition admonition-note alert alert--secondary"},(0,i.kt)("div",{parentName:"div",className:"admonition-heading"},(0,i.kt)("h5",{parentName:"div"},(0,i.kt)("span",{parentName:"h5",className:"admonition-icon"},(0,i.kt)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},(0,i.kt)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.3 5.69a.942.942 0 0 1-.28-.7c0-.28.09-.52.28-.7.19-.18.42-.28.7-.28.28 0 .52.09.7.28.18.19.28.42.28.7 0 .28-.09.52-.28.7a1 1 0 0 1-.7.3c-.28 0-.52-.11-.7-.3zM8 7.99c-.02-.25-.11-.48-.31-.69-.2-.19-.42-.3-.69-.31H6c-.27.02-.48.13-.69.31-.2.2-.3.44-.31.69h1v3c.02.27.11.5.31.69.2.2.42.31.69.31h1c.27 0 .48-.11.69-.31.2-.19.3-.42.31-.69H8V7.98v.01zM7 2.3c-3.14 0-5.7 2.54-5.7 5.68 0 3.14 2.56 5.7 5.7 5.7s5.7-2.55 5.7-5.7c0-3.15-2.56-5.69-5.7-5.69v.01zM7 .98c3.86 0 7 3.14 7 7s-3.14 7-7 7-7-3.12-7-7 3.14-7 7-7z"}))),"First Postulate of Quantum Mechanics")),(0,i.kt)("div",{parentName:"div",className:"admonition-content"},(0,i.kt)("p",{parentName:"div"},"The state of a physical system is characterized by a ",(0,i.kt)("strong",{parentName:"p"},"state vector")," that belongs\nto a complex vector space ",(0,i.kt)("span",{parentName:"p",className:"math math-inline"},(0,i.kt)("span",{parentName:"span",className:"katex"},(0,i.kt)("span",{parentName:"span",className:"katex-mathml"},(0,i.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,i.kt)("semantics",{parentName:"math"},(0,i.kt)("mrow",{parentName:"semantics"},(0,i.kt)("mi",{parentName:"mrow",mathvariant:"bold"},"V")),(0,i.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\mathbf{V}")))),(0,i.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,i.kt)("span",{parentName:"span",className:"base"},(0,i.kt)("span",{parentName:"span",className:"strut",style:{height:"0.68611em",verticalAlign:"0em"}}),(0,i.kt)("span",{parentName:"span",className:"mord"},(0,i.kt)("span",{parentName:"span",className:"mord mathbf",style:{marginRight:"0.01597em"}},"V")))))),", called the ",(0,i.kt)("strong",{parentName:"p"},"state space")," of the system."))),(0,i.kt)("p",null,"The first postulate from quantum mechanics encourages you to get a refresher on\nthe basic properties of vector spaces from linear algebra."),(0,i.kt)("div",{class:"video-container"},(0,i.kt)("iframe",{width:"560",height:"315",src:"https://www.youtube.com/embed/hJoWM9jf0gU?start=82&end=321&rel=0&controls=0&disablekb=1",title:"YouTube video player",frameborder:"0",allow:"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture",allowfullscreen:!0})),(0,i.kt)("div",{className:"math math-display"},(0,i.kt)("span",{parentName:"div",className:"katex-display"},(0,i.kt)("span",{parentName:"span",className:"katex"},(0,i.kt)("span",{parentName:"span",className:"katex-mathml"},(0,i.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},(0,i.kt)("semantics",{parentName:"math"},(0,i.kt)("mrow",{parentName:"semantics"}),(0,i.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"})))),(0,i.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"})))),(0,i.kt)("p",null,"For a more comprehensive study on vector spaces and the operations we can perform\non them, watch the following lectures by Prof. Gilbert Strang:"),(0,i.kt)("h4",{id:"column-spaces-ie-the-vector-space-generated-by-the-columns-of-a-matrix"},"Column spaces (i.e., the vector space generated by the columns of a matrix)"),(0,i.kt)("div",{class:"video-container"},(0,i.kt)("iframe",{width:"560",height:"315",src:"https://www.youtube.com/embed/n98ilenWoak?controls=0&disablekb=0&rel=0",title:"YouTube video player",frameborder:"0",allow:"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture",allowfullscreen:!0})),(0,i.kt)("div",{className:"math math-display"},(0,i.kt)("span",{parentName:"div",className:"katex-display"},(0,i.kt)("span",{parentName:"span",className:"katex"},(0,i.kt)("span",{parentName:"span",className:"katex-mathml"},(0,i.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},(0,i.kt)("semantics",{parentName:"math"},(0,i.kt)("mrow",{parentName:"semantics"}),(0,i.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"})))),(0,i.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"})))),(0,i.kt)("h4",{id:"independence-basis-and-dimension"},"Independence, basis, and dimension"),(0,i.kt)("blockquote",null,(0,i.kt)("p",{parentName:"blockquote"},'"The basis is a bunch of vectors"')),(0,i.kt)("blockquote",null,(0,i.kt)("p",{parentName:"blockquote"},'"The dimension is how many vectors"')),(0,i.kt)("div",{class:"video-container"},(0,i.kt)("iframe",{width:"560",height:"315",src:"https://www.youtube.com/embed/eeMJg4uI7o0?controls=0&disablekb=1&rel=0",title:"YouTube video player",frameborder:"0",allow:"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture",allowfullscreen:!0})),(0,i.kt)("h4",{id:"eigenvalues-and-eigenvectors"},"Eigenvalues and Eigenvectors"),(0,i.kt)("div",{class:"video-container"},(0,i.kt)("iframe",{width:"560",height:"315",src:"https://www.youtube.com/embed/DzqE7tj7eIM?controls=0&disablekb=1&rel=0",title:"YouTube video player",frameborder:"0",allow:"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture",allowfullscreen:!0})),(0,i.kt)("div",{class:"video-container"},(0,i.kt)("iframe",{width:"560",height:"315",src:"https://www.youtube.com/embed/cdZnhQjJu4I",title:"YouTube video player",frameborder:"0",allow:"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture",allowfullscreen:!0})),(0,i.kt)("div",{class:"video-container"},(0,i.kt)("iframe",{width:"560",height:"315",src:"https://www.youtube.com/embed/PFDu9oVAE-g",title:"YouTube video player",frameborder:"0",allow:"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture",allowfullscreen:!0})),(0,i.kt)("div",{className:"math math-display"},(0,i.kt)("span",{parentName:"div",className:"katex-display"},(0,i.kt)("span",{parentName:"span",className:"katex"},(0,i.kt)("span",{parentName:"span",className:"katex-mathml"},(0,i.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},(0,i.kt)("semantics",{parentName:"math"},(0,i.kt)("mrow",{parentName:"semantics"}),(0,i.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"})))),(0,i.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"})))),(0,i.kt)("p",null,'Other important operations we can perform on vector spaces are the scalar or "dot" product.'),(0,i.kt)("div",{className:"admonition admonition-note alert alert--secondary"},(0,i.kt)("div",{parentName:"div",className:"admonition-heading"},(0,i.kt)("h5",{parentName:"div"},(0,i.kt)("span",{parentName:"h5",className:"admonition-icon"},(0,i.kt)("svg",{parentName:"span",xmlns:"http://www.w3.org/2000/svg",width:"14",height:"16",viewBox:"0 0 14 16"},(0,i.kt)("path",{parentName:"svg",fillRule:"evenodd",d:"M6.3 5.69a.942.942 0 0 1-.28-.7c0-.28.09-.52.28-.7.19-.18.42-.28.7-.28.28 0 .52.09.7.28.18.19.28.42.28.7 0 .28-.09.52-.28.7a1 1 0 0 1-.7.3c-.28 0-.52-.11-.7-.3zM8 7.99c-.02-.25-.11-.48-.31-.69-.2-.19-.42-.3-.69-.31H6c-.27.02-.48.13-.69.31-.2.2-.3.44-.31.69h1v3c.02.27.11.5.31.69.2.2.42.31.69.31h1c.27 0 .48-.11.69-.31.2-.19.3-.42.31-.69H8V7.98v.01zM7 2.3c-3.14 0-5.7 2.54-5.7 5.68 0 3.14 2.56 5.7 5.7 5.7s5.7-2.55 5.7-5.7c0-3.15-2.56-5.69-5.7-5.69v.01zM7 .98c3.86 0 7 3.14 7 7s-3.14 7-7 7-7-3.12-7-7 3.14-7 7-7z"}))),"Second Postulate of Quantum Mechanics")),(0,i.kt)("div",{parentName:"div",className:"admonition-content"},(0,i.kt)("p",{parentName:"div"},"A physical quantity ",(0,i.kt)("span",{parentName:"p",className:"math 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