Classical Mechanics and Hamiltonian Systems

There are two approaches to model mechanical systems:

I) Lagrangean description

- 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).

II) Hamiltonian description

- 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.

Example: 

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.

Examples of Hamiltonians

1) Harmonic oscillator (one degree of freedom): 

2) Freely falling body: 

3) Pendulum (q is the angle): 

Solving Hamilton equations (H.E.) using first integrals

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)=mx²/2 is the free Lagrangean, then x(p)=p/m, and g(p)=p²/(2m) is the free Hamiltonian corresponding through Legendre transformation.

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 |<y,j>|².

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: R³ 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 .

Quantization

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

Example: “from numbers to generalized functions” (i.e. L² 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).