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1 = Introduction to mathematical physics, spring 2017
2 Quantum Dynamics =
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4
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6 {{panel}}
7 **Teacher:** [[Jani Lukkarinen>>doc:mathstatHenkilokunta.Lukkarinen, Jani]]
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9 **Scope:** 10 cr
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11 **Type:** Advanced studies
12
13 **Teaching:** //Lectures//: Tue 14-16 and Thu 14-16 in Exactum C123.  //Exercises//: see below.  //"Ratkomo" tutorial time//: Thu 13-14 (Exactum 3rd floor corridor)
14
15 **Topics:** Schrödinger equation with various potentials and boundary conditions, Wigner function, unbounded operators, self-adjointness, basics of Fourier-analysis and of distribution theory, tensor product spaces, multi-particle systems and their creation operator formalism
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17 **Prerequisites: **Basic //measure theory// and //analysis//. Introductory courses to //quantum mechanics// or //functional analysis// are useful, but not necessary.
18 {{/panel}}
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20 {{info title="Course announcements"}}
21 * The course at Aalto by Ville Turunen on applications of Born-Jordan "quantization" in time-frequency analysis begins on 11.4.  More details on the [[course web page>>url:https://mycourses.aalto.fi/course/view.php?id=14479#section-0||shape="rect"]].
22 {{/info}}
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24 {{toc maxLevel="4" minLevel="2" indent="20px"/}}
25
26 == Teaching schedule ==
27
28 Weeks 3-9 and 11-18, Tuesday and Thursday 14-16 in Exactum, room C123.
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30 Easter holiday 13.-19.4.
31
32 == Course material, links and references ==
33
34 Main references:
35
36 * Course lecture notes (see below)
37 * Brian C. Hall, //Quantum Theory for Mathematicians//. (Graduate Texts in Mathematics, Vol. 267) Springer, 2013
38
39 Supplementary material:
40
41 * M. Reed, B. Simon, //Methods of Modern Mathematical Physics I: Functional Analysis// & //II: Fourier Analysis, Self-Adjointness//, Academic Press, 1980 & 1975
42 * G. Teschl, //Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators//, Graduate Studies in Mathematics 99, Amer. Math. Soc., 2009
43 * P. L. Garrido, S. Goldstein, J. Lukkarinen, and R. Tumulka, //Paradoxical Reflection in Quantum Mechanics//, Am. J. Phys. 79 (2011) 1218-1231. The [[published version>>url:http://dx.doi.org/10.1119/1.3636408||shape="rect"]] of the paper contains some misprints in the formulae in the appendices: these are correctly printed in the [[arxiv preprint>>url:http://arxiv.org/abs/0808.0610||shape="rect"]].
44
45 Further reading:
46
47 * Wikipedia entries for the [[Schrödinger equation>>url:http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation||shape="rect"]] and the [[double slit experiment>>url:http://en.wikipedia.org/wiki/Double-slit_experiment||shape="rect"]]
48 * Research paper about a modern [[double slit experiment with electrons>>url:http://dx.doi.org/10.1088/1367-2630/15/3/033018||shape="rect"]] (R. Bach, D. Pope, S.-H. Liou, and H. Batelaan, //Controlled double-slit electron diffraction//, New Journal of Physics 15 (2013) 033018, 7pp.) Includes movies in the supplementary data
49 * [[Lectures notes>>url:http://www.math.rutgers.edu/~~tumulka/teaching/fall11/325/||shape="rect"]] of [[Roderich Tumulka>>url:http://www.math.rutgers.edu/~~tumulka/||shape="rect"]] on foundations and interpretation of quantum mechanics
50 * Repeated (spin-)measurements: Wikipedia entry for the [[Stern–Gerlach experiment>>url:http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment||shape="rect"]]
51 * Research paper about scaling limits of Wigner transforms (P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, //Homogenization limits and Wigner transforms//, Comm. Pure App. Math. 50 (1997) 323-379)
52 * (((
53 M.A. Shubin, //Pseudodifferential Operators and Spectral Theory//, Springer, 1987
54 )))
55 * M. de Gosson and F. Luef, //Preferred quantization rules: Born–Jordan versus Weyl. The pseudo-differential point of view//, J. Pseudo-Differ. Oper. Appl. 2 (2011) 115-(% class="ArticleCitation_Pages" %)139(%%).  (Link to the [[paper>>url:http://dx.doi.org/10.1007/s11868-011-0025-6||shape="rect"]], and to its [[arxiv-version>>url:https://arxiv.org/abs/1102.5132||shape="rect"]].)
56 * O. Bratteli, D. W. Robinson, //Operator Algebras and Quantum Statistical Mechanics 1// & //2//, Springer, 2002 & 2002
57 * J. J. Sakurai, //Modern Quantum Mechanics//, Addison-Wesley, revised edition, 1994
58 * S. J. Gustafson, I. M. Sigal, //Mathematical Concepts of Quantum Mechanics//, Springer, 2nd edition, 2006
59 * E. H. Lieb, R. Seiringer, //The Stability of Matter in Quantum Mechanics//, Cambridge University Press, 2010
60
61
62
63 == Lecture notes ==
64
65 * [[Lectures 17. & 19.1.>>attach:Lect1.pdf]]
66 * [[Lecture 24.1.>>attach:Lect3.pdf]]
67 * [[Lecture 25.1.>>attach:Lect4.pdf]] (tensor products, new version)
68 * [[Lectures 31.1. & 2.2.>>attach:Lect5.pdf]]
69 * [[Lectures 7. & 9.2.>>attach:Lect6.pdf]]
70 * [[Lectures 14. & 16.2.>>attach:Lect7.pdf]]
71 * [[Lectures 21. & 23.2.>>attach:Lect8b.pdf]] ((% style="color: rgb(255,0,0);" %)added page 63b on 20.4.(%%))
72 * [[Lectures 28.2., 2.3. and 14.3.>>attach:Lect9.pdf]]
73 * [[Lecture 16.3.>>attach:Lect10.pdf]]
74 * [[Lecture 21.3.>>attach:Lect11.pdf]] (more details in Shubin's book or the article by de Gosson and Luef above)
75 * [[Lectures 23.3. & 28.3.>>attach:Lect12.pdf]]
76 * [[Lectures 4.4. & 6.4.>>attach:Lect13.pdf]] ([[Link>>url:http://arxiv.org/abs/0808.0610||shape="rect"]] to the paper about step potentials, and to a related [[applet>>url:https://phet.colorado.edu/fi/simulation/quantum-tunneling||shape="rect"]] about quantum tunnelling.)
77 * [[Lectures 11.4. & 20.4.>>attach:Lect14.pdf]]
78 * [[Lectures 25.4. & 27.4.>>attach:Lect15.pdf]] (extra material: [[bosons>>attach:Bosons.pdf]], [[lattice fermions>>attach:LatticeFermions.pdf]])
79 * [[Summary>>attach:Summary.pdf]] (including info for the oral exam)
80
81 == [[Registration>>url:https://weboodi.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57357||shape="rect"]] ==
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83 (% style="color: rgb(96,96,96);" %)Did you forget to register?(%%) (% style="color: rgb(96,96,96);" %) (%%) [[What to do?>>url:https://wiki.helsinki.fi/display/mathstatOpiskelu/Kysymys4||style="text-decoration: underline;" shape="rect"]]
84
85 == Exercises ==
86
87 === Assignments ===
88
89 * [[Set 1>>attach:hw-01.pdf]], due 27.1.
90 * [[Set 2>>attach:hw-02.pdf]], due 3.2.
91 * [[Set 3>>attach:hw-03.pdf]], due 10.2.
92 * [[Set 4>>attach:hw-04.pdf]], due 17.2.
93 * [[Set 5>>attach:hw-05.pdf]], due 24.2.
94 * [[Set 6>>attach:hw-06.pdf]], due 3.3.
95 * [[Set 7>>attach:hw-07.pdf]], due **17.3.**
96 * [[Set 8>>attach:hw-08.pdf]], due 24.3.
97 * [[Set 9>>attach:hw-09.pdf]], due 31.3.
98 * [[Set 10>>attach:hw-10.pdf]], due 7.4.
99 * [[Set 11>>attach:hw-11.pdf]], due Wed 19.4.
100 * [[Set 12>>attach:hw-12.pdf]], due 28.4.
101 * [[Final set 13>>attach:hw-13.pdf]], due 5.5.
102
103 === Exercise classes ===
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124 (% style="color: rgb(255,0,0);" %)C321
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126 [[Brecht Donvil>>url:http://wiki.helsinki.fi/display/mathstatHenkilokunta/Donvil%2C+Brecht||shape="rect"]]
127 )))
128
129 == "Final exam" ~= projects ==
130
131
132 Below are some possible topics for the final exam project (you can also suggest your own topic). Please note that in addition to the project, a minimal amount of points from the exercises //or// an oral exam later in May will be required to get the credits from the course.
133
134 Ideally, the report from the project should be 3-7 pages (certainly not more than 10) and contain at least one theorem with a proof based on results which are either proven in the lecture notes, or for which you can provide a reference, e.g., in a textbook.
135
136 1. Relativistic Hamiltonians (see Section 11.5.b and Exercise 13.1.)
137 1. Quantum particles in external magnetic fields (Section 11.5.a)
138 1. Rotations and spin in quantum mechanics (Section 2.20.)
139 1. Quadratic forms and semibounded operators (Proof of Theorem 10.5.)
140 1. Fock space dynamics of lattice fermions (Section 13)
141 1. Spectrum of the Hydrogen atom (Teschl, Chapter 10)
142 1. Time-dependent Hamiltonians and Dyson series
143 1. Trotter product formula and "Feynman path integrals" (Suggestions and instructions in (% class="confluence-link" %)[[Project8-v2.pdf>>url:http://wiki.helsinki.fi/download/attachments/113259252/Project8-Trotter-v2.pdf?version=1&modificationDate=1387297256410&api=v2||shape="rect"]] (%%).)
144
145 == Course feedback ==
146
147 Course feedback can be given at any point during the course. Click [[here>>url:https://elomake.helsinki.fi/lomakkeet/11954/lomake.html||style="line-height: 1.4285;" shape="rect"]].