Quantum Mechanics for Mathematicians
What is the initial wavefunctionψ(q,0)? Show that at later times|ψ(q,t)|^2 is peaked about a point that moves with velocity~mk. ...
For the three basis elementsljofso(3), show that the momentum map gives functionsμljthat are just the components of the angular ...
B.8 Chapter 17 Problem 1: This is part of a proof of the Groenewold-van Hove theorem. Show that one can writeq^2 p^2 in two way ...
B.9 Chapters 18 and 19 Problem 1: Starting with the Lie algebraso(3), with basisl 1 ,l 2 ,l 3 , consider new basis elements give ...
Show that in the quantized theory the angular momentum operators and theSO(3) Casimir operator satisfy [Lj,H] = 0, [L^2 ,H] = ...
B.11 Chapter 23 Problem 1: For the coherent state|α〉, compute 〈α|Q|α〉 and 〈α|P|α〉 Show that coherent states are not eigenstates ...
The groupSO(3) acts on the system by rotations of the position spaceR^3 , and the corresponding Lie algebra action on the state ...
Problem 2: Prove that, as algebras overC, Cliff(2d,C) is isomorphic toM(2d,C) Cliff(2d+ 1,C) is isomorphic toM(2d,C)⊕M(2d,C) B ...
• Γ = ∏d j=1 (1− 2 a†FjaFj) • Γ =cγ 1 γ 2 ···γ 2 d for some constantc. Computec. γjΓ + Γγj= 0 for allj. Γ^2 = 1 P±= 1 2 ( ...
B.16 Chapter 36 Problem 1: When the single-particle state spaceH 1 is a complex vector space with Her- mitian inner product, one ...
A quantum system corresponding to indistinguishable particles interacting with each other with an interaction energyv(x−y) (wher ...
Problem 4: The Pauli-Lubanski operator is the four-component operator W 0 =−P·L, W=−P 0 L+P×K (same notation as in problem 1) Sh ...
B.20 Chapters 45 and 46 Problem 1: Show that the Yang-Mills equations (46.9 and 46.11) are Hamilton’s equa- tions for the Yang-M ...
Show that, form= 0, the Majorana fermion theory has anSO(2) symmetry given by the action Ψ(x)→cosθΨ(x) + sinθ γ 0 γ 1 γ 2 γ 3 Ψ( ...
Bibliography [1] Orlando Alvarez,Lectures on quantum mechanics and the index theorem, Geometry and quantum field theory (Park Ci ...
[14] Roger Carter, Graeme Segal, and Ian Macdonald,Lectures on Lie groups and Lie algebras, London Mathematical Society Student ...
[31] L. E. Gendenshte ̆ın and I. V. Krive,Supersymmetry in quantum mechan- ics, Uspekhi Fiz. Nauk 146 (1985), no. 4, 553–590. [3 ...
[48] Morris W. Hirsch and Stephen Smale,Differential equations, dynamical systems, and linear algebra, Academic Press, 1974. [49 ...
[63] Dwight E. Neuenschwander,Emmy Noether’s wonderful theorem, Johns Hopkins University Press, 2011. [64] Johnny T. Ottesen,Inf ...
[79] David Shale,Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962), 149–167. [80] David Shale and W. Fo ...
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