Quantum Mechanics for Mathematicians
(0, 0 ,1) =∞ (x 1 ,x 2 ,x 3 ) (0, 0 ,0) (0, 0 ,x 3 ) z=x+iy R iR C CP^1 similar triangles: x 1 = x 1 1 −x 3 y 1 = x 2 1 −x 3 Fig ...
takes z= z 1 z 2 → αz+β γz+δ Such transformations are invertible if the determinant of the matrix is non-zero, and one can show ...
S 1 |ψ〉=−^12 |ψ〉 S 2 |ψ〉=−^12 |ψ〉 S 3 |ψ〉=−^12 |ψ〉 S 1 |ψ〉= +^12 |ψ〉 S 2 |ψ〉= +^12 |ψ〉 S 3 |ψ〉= +^12 |ψ〉 Figure 7.4: The Bloch s ...
u+(x) is determined by setting it to be ( 1 0 ) at the North pole, and defin- ing it at other pointsxon the sphere by acting on ...
7.6 For further reading Just about every quantum mechanics textbook works out this example of a spin 1 2 particle in a magnetic ...
Chapter 8 Representations of SU(2) and SO(3) For the case ofG=U(1), in chapter 2 we were able to classify all complex irreducibl ...
This is sometimes called the “vector representation”, and we saw in chap- ter 6 that it is isomorphic to the adjoint representat ...
Proof.Recall that if we diagonalize a unitary matrix, the diagonal entries are the eigenvalues, but their order is undetermined: ...
8.1.2 Lie algebra representations: raising and lowering op- erators To proceed further in characterizing a representation (π,V) ...
witheiθgoing aroundU(1) once asθgoes from 0 to 2π, this means we can choose a basis ofV so that π(ei^2 θS^3 ) = eiθq^10 ...
is called thek’th weight space of the representation. All vectors in it are eigen- vectors ofπ′(S 3 )with eigenvaluek 2. The dim ...
Irreducible representations will be characterized by a highest weight vector, as follows Theorem(Highest weight theorem).Finite ...
Digression.Dropping the requirement of finite dimensionality, the same con- struction starting with a highest weight vector and ...
8.2 Representations ofSU(2): construction The argument of the previous section only tells us what properties possible finite dim ...
Taking the derivative, the Lie algebra representation is given by π′n(X)f= d dt πn(etX)f|t=0= d dt f(etX·z 1 ,etX·z 2 )|t=0 wher ...
We now have an explicit highest weight vector, and an explicit construction of the corresponding irreducible representation. If ...
Forn= 3,s=^32 1 √ 6 z 13 , 1 √ 2 z 12 z 2 , 1 √ 2 z 1 z 22 , 1 √ 6 z 23 8.3 Representations ofSO(3)and spherical har- monics W ...
Taking the derivative, the Lie algebra representation on functions is given by ρ′(X)f= d dt ρ(etX)f|t=0= d dt f(etX·x 1 ,etX·x 2 ...
ρ′(l 1 )f= d dt f e t 0 0 0 0 0 1 0 −1 0 x 1 x 2 x 3 |t=0 = d dt f 1 0 ...
(x 1 ,x 2 ,x 3 ) = (r,θ,φ) x 1 x 2 x 3 r θ φ Figure 8.2: Spherical coordinates. we will have x 1 =rsinθcosφ x 2 =rsinθsinφ x 3 = ...
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