Quantum Mechanics for Mathematicians
Chapter 14 The Poisson Bracket and Symplectic Geometry We have seen that the quantum theory of a free particle corresponds to th ...
is known as “phase space”. Points in phase space can be thought of as uniquely parametrizing possible initial conditions for cla ...
which says that the momentum is the mass times the velocity, and is conserved. For a particle subject to a potentialV(q) one has ...
14.2 The Poisson bracket and the Heisenberg Lie algebra A third fundamental property of the Poisson bracket that can easily be c ...
This isomorphism preserves the Lie bracket relations since [X,Y] =Z↔{q,p}= 1 It is convenient to choose its own notation for the ...
(using the notationcq = (cq 1 ,...,cqd), cp = (cp 1 ,...,cpd)). We will often denote these by (( cq cp ) ,c ) This Lie bracket o ...
ωis antisymmetric:ω(v,v′) =−ω(v′,v) ωis nondegenerate: ifv 6 = 0, thenω(v,·)∈V∗is non-zero. A vector spaceV with a symplectic ...
in terms of basis elements ofV∗, the coordinate functionsvj. There is an anal- ogous theorem in symplectic geometry (for a proof ...
Chapter 15 Hamiltonian Vector Fields and the Moment Map A basic feature of Hamiltonian mechanics is that, for any functionfon ph ...
time evolution), with the functionsμLhaving non-zero Poisson brackets with the Hamiltonian function. 15.1 Vector fields and the ...
Definition(Flow of a vector field and the exponential map).The flow of the vector fieldXonMis the map ΦX: (t,m)∈R×M→ΦX(t,m)∈M sa ...
Definition(Hamiltonian vector field). A vector field onM=R^2 given by ∂f ∂p ∂ ∂q − ∂f ∂q ∂ ∂p =−{f,·} for some functionfonM=R^2 ...
and are given by q(t) =q(0) cost+p(0) sint, p(t) =p(0) cost−q(0) sint The exponential map is given by clockwise rotation through ...
The Jacobi identity implies {f,{f 1 ,f 2 }}={{f,f 1 },f 2 }+{{f 2 ,f},f 1 }={{f,f 1 },f 2 }−{{f,f 2 },f 1 } so X{f 1 ,f 2 }=Xf 2 ...
Digression.For a general symplectic manifoldM, the symplectic two-formω gives us an analog of Hamilton’s equations. This is the ...
are known to physicists as “canonical transformations”, and to mathematicians as “symplectomorphisms”. We will not try and work ...
so we see that it is the map L→π′(L) =−XL that will be a homomorphism. When the vector fieldXLis a Hamiltonian vector field, we ...
Using the formula LX= (d+iX)^2 =diX+iXd for the Lie derivative acting on differential forms (iXis interior product with the vect ...
Takingato be the corresponding element in the Lie algebra ofG=R^3 , the vector field onMcorresponding to this action (by 15.10) ...
15.5 The dual of a Lie algebra and symplectic geometry We have been careful to keep track of the difference between phase space ...
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