41 lines
3.0 KiB
Markdown
41 lines
3.0 KiB
Markdown
# Group Theory
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## Basics
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- 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)
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- Lie Algebra: Vector space + inner product (vector product) that is bilinear, follows the jacobi identity, and has x*x = 0
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## Wigner D/d-matrix (sheet 07) -> Dive into angular momentum maths
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- Generator operators U(R) = exp(-ihJ), J: operator / infinitesimal rotation
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- SO(3) has exactly one irrep of dim 1,3,5,... (=2j+1) -> How does this relate to number of eigenstates?
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- SU(2) allows even dim as well (j = half-integer) because SU(2) is double cover of SO(3) -> research
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- Wigner D-Matrix gives transition probabilities between eigenstates
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- Eigenstates form basis when discrete
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- 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
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- j fixes total angular momentum, m fixes z component
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- angular momentum operator algebra is inherent to so(3) lie algebra
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## Questions / open things
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- Where does the inherent need for SO(3) come from? Why are the possible state spaces the irreps of SO(3)?
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-> 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.
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Enforcing finite-dim and irrep, this fixes dim = 2j+1.
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- 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)
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- Is there an intuitive notion to see how angular momentum emerges from the shapes of the wave functions?
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- How does the emergence of azimuthal dependence in superpositions of (not azimuth-constraining) eigenstates play out mathematically?
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- Why is SU(2) the double cover of SO(3)? What does that mean?
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- What role do Pauli matrices play (generators of SO(2)? idk)
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- 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
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- relation between Lie algebras and Lie groups not exactly clear
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- Whats the algebra of exp of matrices -> https://de.wikipedia.org/wiki/Matrixexponential
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- Is the generator exp(-ihJ) for SO(3) just the most convinient choice or the only choice?
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## Next topics
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- Character tables and relation between conjugacy classes and irreps
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- Further sheets
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- Further lectures, explicitly everything after lecture 09
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- Wigner-Eckart Theorem (in lecture, sheet and indip research)
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- Clebsch Gordan coeff as seperate research?
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