- Configuration space (X): “position” of its components
(q);

- Evolution: “trajectories” q(t) in configuration space;

- Equations of motion are derived from a Lagrangean L(q,q’,t) defined on a state space associated to the configuration space (cotangent bundle M=TX).

- State space (M): “position” q and “momentum” p;

- Evolution: “trajectories” (p(t), q(t)) in state space ;

- Equations of motion are derived from a Hamiltonian H(p,q,t):

H.E.

The Hamiltonian is interpreted as the energy of the system.

The Legendre transform is allows to establish a correspondence between the two descriptions. It associates to a Lagrangean a Hamiltonian. In general, it is not invertible.

If the system is “isolated”, the energy is conserved, so the Hamiltonian is time independent.

In a large class of examples the variables q and p of the Hamiltonian are “separated”:

The “free Hamiltonian” represents the kinetic energy, and depends only on the momentum p.

The “interaction Hamiltonian” represents the potential energy and depends only on the position q.

where n is the number of “particles” of the system
(components without structure).

__Proposition: __

For the above Hamiltonian, the H. E. of motion are equivalent to Newton’s equations:

where Ñ_{i}
is the gradient of V with respect to q_{i}.

1) Harmonic oscillator (one degree of freedom):

2) Freely falling body:

3) Pendulum (q is the angle):

The function *f(p,q)* is a first integral if
it is constant on all solutions to H.E.:

which is the Poisson bracket of f and *H.*

__Proposition__: *f* is a first integral iff
*{f,H}=0*.

__Example:__* H *is a first integral, since
*{,}* is skew-symmetric: *{H,H}=0*.

__Example:__ “N particles with central 2-body interaction”
*V _{ij}(|q_{i}-q_{j}|).*

First integrals:

1) Momentum: (vector),

2) Angular momentum: *J=(*tensor
of order 2),

3) Center of gravity: *G=tP-MR*
(*M *total mass, *R* “average position vector”),

4) Energy: *H*.

__Example:__Charged
particle in a potential (“total
momentum” = particle’s momentum (*p*) + field’s momentum (the vector
potential *–eA*)).

If *B*=const, *A=1/2B x q*, and a possible
gauge condition is *div A=0*.

__The Legendre Transformation (V.I. Arnold)__

__Definition:__

Let *y=f(x)* convex function (*f’’(x)>0*).
The Legendre transform *T**(f)*
of *f(x)* is the function:

*g(p)=Maximum of F _{p}(x), where F_{p}(x)=px-f(x)*

is the “vertical distance” between the line of slope
*p* and the graph of *f(x)*:

__Example__:

If *f(x)=mx ^{2}/2* is the free Lagrangean,
then

__Properties of T__

1) Its is involutive (if *f* is convex, then
*g* is convex and *T**
(g)=f*),

2) Young’s inequality:* px£f(x)+g(p)*,

3) If *f(x)* is quadratic, then *f(x)=g(p(x))*.

__Application__: If ,
then Lagrange equations are equivalent to H.E.

__Example__: ,
with T quadratic, then *H=T(L)=T+U.*

__Postulates
of Quantum Mechanics__

__Descriptions of classical mechanics:__

1) N-particle system “moves” in state space c:R->M, observables are functions f:M->R (e.g. q, p, H etc.); p(c(t)),q(c(t)) change according to H.E..

2) Observables for a system change with time: f:R->R according to Poisson’s eq., “states don’t change”.

__Quantum mechanics:__

__PI__. Pure states of a system: unit vectors in
a Hilbert space;

__PII__.States
evolve according to Schrödinger’s equation:;
i.e. –i/h H is the infinitesimal time-flow., where H is a selfadjoint operator
representing the energy of the system. The formal solution is : .

__P III__. An observable *a* is represented
by a self-adjoint operator.
An outcome of a measurement of *a* is an eigenvalue *A* of (range
of *a* = point spectrum of ).

So far Q.M. is deterministic and the description is based on trajectories, in the same way as classical mechanics.

__Example:__ Wave functions, q, p, H as operators,
energy levels.

__P IV__ (Interpreting measurements). Ifthe
system is in the state y,
then a measurement of *a* will yield the outcome *A* corresponding
to the eigenvector j* _{A
}*with the probability

__P V__ (Projection postulate) Immediately after
a measurement which yields the value A, the state of the system is j* _{A}*.

__P VI__ (Average value) The expected value of
an observable a for a system in state y
is:

.

__Motivation__: If the state is a __superposition__
of pure states ,
which are eigenvalues of the observable a, then the weightsare
the probabilities of outcomes *A _{k}*, and the average value
of the random variable “measuring the quantity a of the system” is .

__Statistical mixtures__: (or
a trace class operator); the expectation value:

__Note__. A superposition of pure states with weights
which are amplitudes of probability, is a “mixture” at “the quantum level”,
in contrast to a density matrix, i.e. a sum of projection operators with
weights probabilities, representing a “mixture” at a “macroscopical level”.

__P VII__ (“Exp. S’s Eq.”). The Hamiltonian operator is
the infinitesimal one parameter unitary group of time translations .

If Schrodinger’s equation is interpreted as “*d/dt*=infinitez.gen.
of *R* (time)=*H* (Hamiltonian)”, then through exponentiation
one “obtaines” that the dynamical group *U(t)* is *exp(iH/h).*

__Example:__ R^{3 }as the group of translations,
the unitary transformations on the L2 space, p=d/dx the infinitesimal generator
for the unitary group.

“Taylor’s Theorem”: *exp(H)f(x,y,z)=f(x+a,y+b,z+c),*
where *H=a ¶ _{x}
+ b ¶_{y}
+c ¶_{z}
.*

__P VIII__ (Statistics) A quantum state is symmetric
with respect to permutation of bosons, and antisymmetric w.r.t. permutations
of fermions.

If particles have spin (e.g. , S a representation of SU(2,R)), then the state of a system of N particles belongs to the symmetrized (bosons) or antisymmetrized (fermions) tensor product of the Hilbert space .

Phase space-> Hilbert space, functions -> operators,
s.t. Poisson bracket -> commutator.

Example: “from numbers to generalized functions” (i.e.
L^{2} space & ops.).

Note: H.E. = trajectories = grad H -> Schrodinger’s equation (not E-> d/dt!).

The two classical descriptions -> Scrodinger’s & Heisenberg pictures.

__Schrodinger picture:__

The evolution of the system is modeled by trajectories in the Hilbert space governed by Schrodinger’s equation. Observables are operators, expectation values change <y(t),A y(t)>.

__Heisenberg picture:__

States are fixed, the observables are time-dependent (compatible with the duality H x B(H)), satisfying Heisenberg’s equation:

.

A formal solution of Hg’s Eq.:, i.e. .

__The correspondence:__

The expectations values should correspond:.

Equivalently: It satisfies Heisenberg’s equation.

The two descriptions are analogous to the fluid dynamics alternative: label “the points” and the particles move (S’s picture) or label “the particles” and the points, or other observables, change (H’s picture).

__Remarks:__

1) The results of experiments (transition amplitudes <y(t), c(t)>) are time independent.

2) Besides the continuous symmetries of the Hilbert space (U(t)) , there are discrete symmetries (e.g. time inversion).