105 lines
6.2 KiB
Markdown
105 lines
6.2 KiB
Markdown
# Group Theory
|
|
|
|
## Points
|
|
|
|
Sheet 01: 16.5 / 20
|
|
Sheet 02: 17.0 / 20
|
|
Sheet 03: 16.5 / 20
|
|
Sheet 04: 15.5 / 20
|
|
Sheet 05: 12.0 / 20
|
|
Sheet 06: 17.5 / 20
|
|
Sheet 07: 17.5 / 20
|
|
Sheet 08: 14.5 / 20
|
|
Sheet 09: 16.5 / 20
|
|
Sheet 10: 14.0 / 20
|
|
Sheet 11: 00.0 / 20
|
|
Sheet 12: 00.0 / 20
|
|
Sheet 13: 00.0 / 20
|
|
|
|
|
|
16.5 + 17.0 + 16.5 + 15.5 + 12.0 + 17.5 + 17.5 + 14.5 + 16.5 + 14.0
|
|
|
|
## Basics
|
|
|
|
- Representation: p: G -> GL(V), a map from the group the set of linear maps from V to V (ambigous if Representation means the map p or the resulting set of linear maps)
|
|
- Lie Algebra: Vector space + inner product (vector product) that is bilinear, follows the jacobi identity, and has x*x = 0
|
|
|
|
## Wigner D/d-matrix (sheet 07) -> Dive into angular momentum maths
|
|
|
|
- Generator operators U(R) = exp(-ihJ), J: operator / infinitesimal rotation
|
|
- SO(3) has exactly one irrep of dim 1,3,5,... (=2j+1) -> How does this relate to number of eigenstates?
|
|
- SU(2) allows even dim as well (j = half-integer) because SU(2) is double cover of SO(3) -> research
|
|
- Wigner D-Matrix gives transition probabilities between eigenstates
|
|
- Eigenstates form basis when discrete
|
|
- When j= 3, m=-3,-2,-1,0,1,2,3 -> states |3m> are allowed -> real states are superpositions of these basis states / eigenstates -> 7-dim space needed to describe them (even though angular momentum needs only 3d classically) -> remember in general that degrees of freedom explode as 2^n because superpositions are allowed
|
|
- j fixes total angular momentum, m fixes z component
|
|
- angular momentum operator algebra is inherent to so(3) lie algebra
|
|
|
|
## Questions / open things
|
|
|
|
- Where does the inherent need for SO(3) come from? Why are the possible state spaces the irreps of SO(3)?
|
|
-> The operators that rotate states (in physical space) can be called U(R(theta)), where R in SO(3). There is a homomorphism between {U(R)} and SO(3), since U(R1R2)=U(R1)U(R2). Thus U(R) is a representation.
|
|
Enforcing finite-dim and irrep, this fixes dim = 2j+1.
|
|
- What about that L/J algebra? See "Lie Algebra" german wikipedia article -> Beispiele -> Aus der Physik -> Seems as though the relation does not only hold for those three "basis matrices", but all matrices, as long as your in the basis of the L_i (wtf even is a basis of matrices)
|
|
- Is there an intuitive notion to see how angular momentum emerges from the shapes of the wave functions?
|
|
- How does the emergence of azimuthal dependence in superpositions of (not azimuth-constraining) eigenstates play out mathematically?
|
|
- Why is SU(2) the double cover of SO(3)? What does that mean?
|
|
- What role do Pauli matrices play (generators of SO(2)? idk)
|
|
- How does one come up with the idea that operators and eigenvalues/vectors are the appropriate mathematical tools to describe QM? How does it arise? BC this alone imposes a lot of the quantization maths, since eigenvalues are discrete
|
|
- relation between Lie algebras and Lie groups not exactly clear
|
|
- Whats the algebra of exp of matrices -> https://de.wikipedia.org/wiki/Matrixexponential
|
|
- Is the generator exp(-ihJ) for SO(3) just the most convinient choice or the only choice?
|
|
|
|
## Next topics
|
|
|
|
- Character tables and relation between conjugacy classes and irreps
|
|
- Further sheets
|
|
- Further lectures, explicitly everything after lecture 09
|
|
- Wigner-Eckart Theorem (in lecture, sheet and indip research)
|
|
- Clebsch Gordan coeff as seperate research?
|
|
|
|
# Ex1-3 Übersichtsprüfung
|
|
|
|
## VL-Inhalte Ex1
|
|
|
|
Grundlagen (Größen, Einheiten; Skalare, Vektoren, trigonometrische Funktionen, differenzieren,
|
|
partielle und totale Ableitungen, integrieren, komplexe Zahlen, Gradient, Divergenz, Rotation);
|
|
Mechanik des Massenpunktes (Kinematik, Dynamik, Relativbewegung; *beschleunigte Bezugssysteme*,
|
|
Impuls, Drehimpuls, Arbeit, Energie, Massenmittelpunkt);
|
|
Relativistische Kinematik (Lorentz-Transformationen, Längenkontraktion, Zeitdilatation).
|
|
Gravitation und *Keplerbewegung*
|
|
Mechanik des Starren Körpers (Kraft, Drehmoment, Statik, Dynamik, Starrer Rotator, freie Achsen,
|
|
Trägheitsmoment, *Kreisel*, Schwingungen, *Festkörperwellen*);
|
|
Mechanik deformierbarer Medien (Aggregatzustände, Verformungseigenschaften fester Körper, *ruhende
|
|
Medien*, statischer Auftrieb, Oberflächenspannung, *bewegte Medien*, Wellen und Akustik, *dynamischer
|
|
Auftrieb)*;
|
|
*Mechanik der Vielteilchensysteme* (Gaskinetik, Temperatur, Zustandsgrößen, Hauptsätze der
|
|
Wärmelehre, Wärmekraftmaschinen, Entropie und Wahrscheinlichkeit, Diffusion,
|
|
Transportphänomene)
|
|
|
|
## VL-Inhalte Ex2
|
|
|
|
Elektromagnetismus, Vergleich mit Gravitation. Elektrostatik (Ladung, Coulomb-Gesetz, Feld, Dipol,
|
|
elektrische Struktur der Materie, Fluss, Gauß-Gesetz, *Poisson-Gleichung*, Ladungsverteilung,
|
|
Kapazität). Elektrische Leitung (Stromdichte, Ladungserhaltung, Ohmsches Gesetz, Rotation des
|
|
Vektorfeldes, Stokes-Satz, Stromkreise, *Kirchhoff-Gesetze*, Leitungsmechanismen). Magnetische
|
|
Wechselwirkung, (Magnetismus als relativistischer Effekt, Magnetfeld, stationäre Maxwell-Gleichungen,
|
|
Lorentz-Kraft, Hall-Effekt, Magnetdipol, Vektorpotential, Biot-Savart-Gesetz). Materie in stationären
|
|
Feldern (induzierte und permanente Dipole, Dielektrikum, Verschiebungsfeld, elektrische Polarisation,
|
|
magnetische Dipole, magnetisiertes Feld H, Magnetisierungsfeld, Verhalten an Grenzflächen).
|
|
Zeitabhängige Felder (Induktion, Maxwellscher Verschiebungsstrom, technischer Wechselstrom,
|
|
Schwingkreise, Hochfrequenz-Phänomene, Abstrahlung, freie EM-Wellen, Hertz-Dipol, Polarisation,
|
|
Reflexion). Vollständige Maxwell-Gleichungen, Symmetrie zwischen elektrischen und magnetischen
|
|
Feldern.
|
|
|
|
## VL-Inhalte Ex3
|
|
|
|
Optik: Strahlenoptik und Matrizenoptik; Abbildungen und Abbildungsfehler; Mikroskop und Teleskop;
|
|
Wellenoptik; Wellentypen; Gaußstrahlen; Kirchhoffsche Theorie der Beugung; Fraunhofer-Beugung;
|
|
Fourier-Optik; Brechung und Dispersion; Polarisation und Doppelbrechung; Kohärenz und
|
|
Zweistrahl-Interferometer; Vielstrahl-Interferometer; Michelson-Interferometer; Holographie, Laser-Speckel;
|
|
Wellenmechanik: Wellen- und Teilchenphänomene mit Licht,Wellenpakete, Tunnel-Effekt; Eingesperrte
|
|
Teilchen, Kastenpotential, Harmonischer Oszillator, Paul-Falle; Meßgrößen in der Quantenphysik;
|
|
Photo-, Compton-Effekt, Franck-Hertz-Versuch; Rutherford-Experiment; elementares Wasserstoff-Atom;
|
|
Stern-Gerlach-Experimente; Manipulation einzelner Teilchen
|