[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-466236-105":59,"doc-detail-466236-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","quantum-physics-i-fall-2015-final-exam","Quantum Physics I - Fall 2015 - Final Exam","","Final exam materials for Quantum Physics I (Fall 2015) covering core principles and solved reference formulas across relativistic relations, photons, operator commutators, Schrödinger dynamics, Fourier transforms, and quantum uncertainties. The document includes a formula sheet with stationarity, infinite and finite square wells, delta-function potentials, harmonic oscillator ladder operators and spectra, scattering in one dimension, orbital angular momentum and spherical harmonics, central potential reduction, and hydrogen atom bound states. It also lists six exam problems totaling 105 points, with clear constraints on allowed resources and expectations for showing work.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/quantum-physics-i-fall-2015-final-exam/466236/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/quantum-physics-i-fall-2015-final-exam/466236.png","ImageObject",300,407,{"name":92,"@type":93},"Jordan Avery","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are the exam time and point totals for Quantum Physics I (Fall 2015)?","Question",{"text":112,"@type":113},"The exam lasts 3 hours (180 minutes) and contains six questions totaling 105 points.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What resources are allowed during the exam?",{"text":117,"@type":113},"No books, notes, or calculators are allowed.",{"name":119,"@type":110,"acceptedAnswer":120},"What should students do regarding showing work and documentation?",{"text":121,"@type":113},"Answer all problems in the provided white books, write the name and section on each white book, and show work clearly.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},466236,1792173965,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},1099523882367,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","8.04, Quantum Physics I, Fall 2015  \nFINAL EXAM  \nFriday December 18, 1:30-4:30 pm  \nYou have 3 hours = 180 minutes.  \nAnswer all problems in the white books provided. Write YOUR NAME and YOUR SECTION on your white book(s) .  \nThere are six questions, totalling 105 points.  \nThe 􀀌rst three questions are shorter, the last three questions are longer.  \nNo books, notes, or calculators allowed.  \nShow your work CLEARLY!  \nFormula Sheet  \n• ~c ≃ 197.3 MeV · fm , me c2 ≃ 0.511 MeV, mpc2 = 938 MeV , ~~ ~~2c ≃ 1~~1~~37  \n• Relativity: p = γ mv , E = γmc2 , E 2 = p2 c2 + m2 c4 , γ = √1β~~ ~~2 , β = ~~v~~c  \n• Photons: E = hν , p = ~~h~~λ , or E = ~ω , p = ~k  \n• Wavelengths  \nh  h  de Broglie: λ = , Compton: λC =  \np mc .  \n• Momentum and position operators  \n~ ∂  ~  ~ ~  ~ ∂f  \np = [x, p ] = i , p = ∇ , [xi , pj ] = i δij , [pi , f (x)] =  \ni ∂x , i i ∂xi  \n• Schro¨dinger equation  \ni~􀀉t (x, t) = 􀀐 − ~~ ~~~22m∇2 + V(x, t)􀀑 􀀉(x, t) ,  \nρ (x, t) + ∇ · J(x, t) = 0  \nρ (x, t) = |􀀉(x, t)| 2 ; J (x, t) = ~mIm[􀀉 ∗ ∇􀀉]  \n• Fourier transforms:  \n􀀉(x) = √12π Z dk 􀀈(k)eikx , 􀀈(k) = √12π Z dx 􀀉(x)e−ikx , Z dx |􀀉(x)| 2 = Z dk |􀀈(k)| 2  \n􀀉(x) = (21π)~~ ~~32~~ ~~ Zd3 k 􀀈(k)eik·x , 􀀈(k) = (21π)~~ ~~32~~ ~~ Zd3 x 􀀉(x)e−ik·x , Zd3 x |􀀉(x)| 2 = Z d3 k |􀀈(k)| 2  \n21π Z− eikx dx = δ(k) , ~~ ~~(21π)3 Z− eik·x d3 x = δ(3) (k)  \nZ− dx exp 􀀀−ax2 + bx 􀀁= r πa exp 􀀐 b24a􀀑 , when Re(a) > 0 .  \n• Wavepackets  \ndω  \nvgroup = , 􀀁k 􀀁x ≃ 1 , shape preserving : t 􀀁v ≤ 􀀁x  \ndk  \n• Hermitian conjugation:  \nZ dx (K􀀉(x, t))∗ 􀀉(x, t) = Z dx 􀀉∗ (x, t)(K†􀀉(x, t))  \nIf K† = K, then K is Hermitian.  \n• Expectation values  \nhQi(t) = Z dx 􀀉∗ (x, t)(Q􀀉(x, t))  \n• Time evolution of expectation value. For Q Hermitian  \n d  \ni~ hQi = 􀀊 [Q, H]􀀋  \ndt  \n• Commutator identity  \n[A, BC] = [A, B]C + B[A, C]  \n• Uncertainty 􀀁Q of a Hermitian operator Q  \n(􀀁Q)2 = hQ2 i − hQi2 = 􀀊 (Q − hQi)2 􀀋  \n• Uncertainty principle: 􀀁x 􀀁p ≥ ~2  \n􀀁x =  and 􀀁p = √ for ψ ∼ exp 􀀐 − ~~ ~~12 x􀀁22􀀑  \n• Stationary state:  \n􀀉(x, t) = ψ(x)e−iEt/~ , − ~~ ~~~2~~ ~~ ~~ ~~d22m dx2 ψ (x) + V(x)ψ (x) = E ψ(x)  \n• In􀀌nite square well  \nV (x) = ( , for 0 \u003C x \u003C a,  \notherwise  \nψn (x) = r ~~ ~~2a  sin nπax , En = ~22π2~~ ~~nma22 , n = 1 , 2 , . . .  \n• Finite square well bound states: E ≤ 0  \nV (x) = (  \n−V0 , for 0 for  \n|x|  \n|x|  \n\u003C a, V0 > 0> a  \nη 2 ≡ 2m(E~~ ~~V0~~ ~~)a2 , ξ 2 ≡ 2m~|~~ ~~E2|~~ ~~a2 , z20 ≡ 2m0~~ ~~a2  \n→ ~~ ~~|~~ ~~EV0|~~ ~~ = ξz220 , ξ 2 + η 2 = z20 Even solutions: ξ = η tanη Odd solutions: ξ = −η cotη  \n• Delta function potential:  \nV = −α δ(x), α > 0 , Bound state: E = − m22  \n• Harmonic Oscillator  \nˆH = 21mpˆ2 + 12mω 2 xˆ2 = ~ω (ˆN + ~~1~~2) , ˆN = aˆ†aˆ  \naˆ = r m2 􀀒xˆ + ipmˆω􀀓 , aˆ† = r m2 􀀒xˆ − ipmˆω􀀓 ,  \nxˆ = r ~~ ~~2~mω ( + aˆ†) , pˆ = ir mω2~ (aˆ† − ) ,[xˆ , pˆ] = i~ , [aˆ, aˆ†] = 1 , [ˆN ,  ] = − aˆ , [ˆN , aˆ† ] = aˆ† .  \naˆφ0 = 0 , φ0 (x) = 􀀐 mπ􀀑 1/4 exp 􀀐 − m2x2 􀀑 .  \nφn = ! (a†)n φ0  \nˆH φn = En φn = ~ω 􀀀n + ~~1~~2􀀁 φn , ˆN φn = n φn , (φm , φn ) = δmn  \naˆ†φn =√n + 1 φn+1 , aˆ φn = √n φn−1 .  \n• Positive energy states  \nψ (x) = Aeikx + Be−ikx , J = ~km 􀀀|A|2 − |B|2 􀀁 , E = ~22k2m  \n• Scattering in 1D. V (x) = ∞ for x ≤ 0. Solution φ (x) = sin kx when V = 0 .  \nψ (x) = eiδ sin(kx + δ) , x > R (R is the range)  \nScattered wave: ψ = φ + ψs  \nψs = As eikx , As = eiδ sinδ  \nTime delay: 􀀁t = 2~ ddδE~~ ~~ → ~~ ~~1~~ ~~ dR dδk = free~~ ~~tr􀀁antsit~~ ~~time 1  \nNbound = π 􀀀δ(0) − δ(∞ )􀀁 (Levinson’s theorem)  \nResonances: Rapid growth in δ, large time delay, large amplitude in the inner region.  \n• Orbital angular momentum  \nˆLx = yˆpˆz − zˆpˆy , ˆLy = zˆpˆx − xˆpˆz , ˆLz = xˆpˆy − yˆpˆx .[ ˆLx , ˆLy ] = i~ ˆLz , [ ˆLy , ˆLz ] = i~ ˆLx , [ ˆLz , ˆLx ] = i~ ˆLy .  \nˆL2 ≡ ˆLx ˆLx + ˆLy ˆLy + ˆLz ˆLz , [ ˆL2 , ˆLi ] = 0  \n∇2 = 1r 2 r + 1r2 􀀒 2 + cotθ + si1n2~~ ~~θ 2~~ ~~ 􀀓  \nˆL2 = −~2 􀀒 2 + cotθ + si1n2~~ ~~θ 2~~ ~~ 􀀓  \nˆLz = ~i  ; ˆL± = ~e±iφ 􀀒 ±  + i cotθ􀀓  \n• Spherical Harmonics  \nYℓ,m (θ, φ) ≡ Nℓ,m Pℓm (cosθ)eimφ  \nˆLz Yℓm = ~mYℓm  \nˆL2 Yℓm = ~2 ℓ (ℓ + 1)Yℓm  \nZ d􀀊Yℓm′ (θ, φ) Yℓm (θ, φ) = δℓ′,","cbCaicbCFfiacJvU","https://ap.wps.com/l/cbCaicbCFfiacJvU","pdf",214708,13,"English","# FINAL EXAM\n## Instructions\n## Formula Sheet Topics\n### One-dimensional operators and dynamics\n### Bound states and exactly solvable potentials\n### Scattering and time delay\n### Angular momentum and spherical harmonics\n### Central potentials and hydrogen atom\n## Problems","[{\"question\":\"What are the exam time and point totals for Quantum Physics I (Fall 2015)?\",\"answer\":\"The exam lasts 3 hours (180 minutes) and contains six questions totaling 105 points.\"},{\"question\":\"What resources are allowed during the exam?\",\"answer\":\"No books, notes, or calculators are allowed.\"},{\"question\":\"What should students do regarding showing work and documentation?\",\"answer\":\"Answer all problems in the provided white books, write the name and section on each white book, and show work clearly.\"}]","Quantum Physics I - Fall 2015 - Final Exam | PDF",1790770843,33]