Quantum Mechanics for Mathematicians

Definition(Hamiltonian vector field). A vector field onM=R^2 given by

∂f

∂p

∂

∂q

−

∂f

∂q

∂

∂p

=−{f,·}

for some functionfonM=R^2 is called a Hamiltonian vector field and will be
denoted byXf. In higher dimensions, Hamiltonian vector fields will be those of
the form

Xf=

∑d

j=1

(

∂f

∂pj

∂

∂qj

−

∂f

∂qj

∂

∂pj

)

=−{f,·} (15.2)

for some functionfonM=R^2 d.

The simplest non-zero Hamiltonian vector fields are those forfa linear
function. Forcq,cpconstants, if

f=cqq+cpp

then

Xf=cp

∂

∂q

−cq

∂

∂p

and the map
f→Xf

is the isomorphism ofMandMof equation 14.6.
For example, takingf=p, we haveXp=∂q∂. The exponential map for this
vector field satisfies

q(exp(tXp)(m)) =q(m) +t, p(exp(tXp)(m)) =p(m) (15.3)

Similarly, forf=qone hasXq=−∂p∂ and

q(exp(tXq)(m)) =q(m), p(exp(tXq)(m)) =p(m)−t (15.4)

Quadratic functionsfgive vector fieldsXfwith components linear in the
coordinates. An important example is the case of the quadratic function

h=

1

2

(q^2 +p^2 )

which is the Hamiltonian function for a harmonic oscillator, a system that will
be treated in much more detail beginning in chapter 22. The Hamiltonian vector
field for this function is

Xh=p

∂

∂q

−q

∂

∂p

The trajectories satisfy
dq
dt

=p,

dp

dt

=−q