Quantum Mechanics for Mathematicians
It is a general phenomenon that for any Lie algebrag, a Poisson bracket on functions on the dual spaceg∗can be defined. This is ...
Note that in the second example, the standard inner product onR^3 provides an SO(3)invariant identification of the Lie algebra w ...
Chapter 16 Quadratic Polynomials and the Symplectic Group In chapters 14 and 15 we studied in detail the Heisenberg Lie algebra ...
16.1 The symplectic group Recall that the orthogonal group can be defined as the group of linear transfor- mations preserving an ...
or ( 0 αδ−βγ −αδ+βγ 0 ) = ( 0 1 −1 0 ) so det ( α β γ δ ) =αδ−βγ= 1 This says that we can have any linear transformation with un ...
The commutation relations amongst these matrices are [E,F] =G [G,E] = 2E [G,F] =− 2 F which are the same as the Poisson bracket ...
Exponentiating the Lie algebra elementE−Fgives rotations of the plane eθ(E−F)= ( cosθ sinθ −sinθ cosθ ) Note that the Lie alge ...
whereA,B,Caredbydreal matrices, withBandCsymmetric, i.e., B=BT, C=CT Note that, replacing the block antisymmetric matrix by the ...
Another important special case comes from takingA= 0,B= 1 ,C=− 1 in equation 16.12, which by equation 16.13 gives μL= 1 2 (|q|^2 ...
for allg∈Gandh 1 ,h 2 ∈H, the groupGis said to act onHby automorphisms. Each mapΦgis an automorphism ofH. Note that sinceΦgis an ...
adjoint representation operatorsAd(g) discussed in chapter 5. So, in this case the corresponding action by automorphisms on the ...
with Lie bracket [(( cq cp ) ,c ) , (( c′q c′p ) ,c′ )] = (( 0 0 ) ,cqc′p−cpc′q ) = (( 0 0 ) ,Ω (( cq cp ) , ( c′q c′p ))) The l ...
Theorem 16.3.Thesp(2d,R)action onh 2 d+1=M⊕Rby derivations is L·(cq·q+cp·p+c) ={μL,cq·q+cp·p+c}=c′q·q+c′p·p (16.22) where ( c′q ...
The left-hand side of this equation isc′′q·q+c′′p·p, where ( c′′q c′′p ) = (LL′−L′L) ( cq cp ) As a result, the right-hand side ...
Chapter 17 Quantization Given any Hamiltonian classical mechanical system with phase spaceR^2 d, phys- ics textbooks have a stan ...
where the last of these equations is the equation for the time dependence of a Heisenberg picture observableO(t) in quantum mech ...
representation of a larger Lie algebra, that of all quadratic polynomials on phase space, a representation that we will continue ...
angle from 0 to 2π), the phase angle only goes from 0 toπ, demonstrating the same problem that occurs in the case of the spinor ...
Proof.For a detailed proof, see section 5.4 of [8], section 4.4 of [26], or chapter 16 of [37]. In outline, the proof begins by ...
operatorHwill make up a Lie algebra of symmetries of the quantum system, and will take energy eigenstates to energy eigenstates ...
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