Quantum Mechanics for Mathematicians

same Lie bracket operation on pairs of vectors. This operation is familiar in yet another context, that of the cross-product of ...

For the adjoint representation on antisymmetric matrices Ad(g)   0 −v 3 v 2 v 3 0 −v 1 −v 2 v 1 0  =g   0 −v 3 v 2 v 3 0 − ...

Given such a construction ofSpin(n), we also need to explicitly construct the homomorphism Φ, and show that its derivative Φ′is ...

6.2.2 Rotations and spin groups in four dimensions Pairs (u,v) of unit quaternions give the product groupSp(1)×Sp(1). An element ...

1 1 − 1 Φ ~v7→u~vu−^1 u −u inSO(3) inSp(1) =Spin(3) Figure 6.1: Double coverSp(1)→SO(3). Bothuand−uact in the same way on~v, so ...

space of all pure imaginary quaternions, which can be identified withR^3 by w=   w 1 w 2 w 3  ∈R^3 ↔~w=w 1 i+w 2 j+w 3 k∈H U ...

This is the orthogonal transformation ofR^3 given by v=   v 1 v 2 v 3  →   cos 2θ −sin 2θ 0 sin 2θ cos 2θ 0 0 0 1     ...

between quaternionsHand a space of 2 by 2 complex matrices, and work with matrix multiplication and complex numbers. The Pauli m ...

This isomorphism identifies basis vectors by i 2 ↔−i σ 1 2 ↔l 1 etc. The first of these identifications comes from the way we ch ...

Recall that anySU(2) matrix can be written in the form ( α β −β α ) α=q 0 −iq 3 , β=−q 2 −iq 1 withα,β∈Carbitrary complex number ...

6.4 For further reading For another discussion of the relationship ofSO(3) andSU(2) as well as a construction of the map Φ, see ...

Chapter 7 Rotations and the Spin 1 2 Particle in a Magnetic Field The existence of a non-trivial double coverSpin(3) of the thre ...

Definition(Spinor representation). The spinor representation ofSpin(3) = SU(2)is the representation onC^2 given by g∈SU(2)→πspin ...

vector representation is a real representation ofSO(3) orSpin(3), the spinor representation is a complex representation. 7.2 The ...

~in the definition of the momentum operator. Our definitions ofSj and of rotations using (see equation 6.3) Ω(θ,w) =e−iθw·S=eθw· ...

In the case of a spin^12 particle, the groupSpin(3) =SU(2) acts on states by the spinor representation with the element Ω(θ,w)∈S ...

so is a rotation aboutwtaking place with angular velocityge 2 mc|B|. The amount of non-trivial physics that is described by this ...

where the “H” subscripts indicate the Heisenberg picture choice for the treat- ment of time-dependence. It can easily be seen th ...

For some insight into this construction, consider first the analog for real numbers, where (R^2 − 0 )/R∗can be thought of as the ...

on the plane corresponding to ( 1 0 ) does not have a well-defined value: as one approaches this point one moves off to infinity ...