In this course we will develop some of the mathematics needed to describe the physical theory of Quantum Mechanics. Quantum Mechanics was developed in the 1920s by Bohr, Schroedinger, Heisenberg and others in order to describe atomic physics. In order to describe quantum mechanics it is useful, first, to understand the theory it extends namely Classical Mechanics. Classical Mechanics is the physics of Newton. It is the physics of the everyday world. Although Quantum Mechanics is not Classical Mechanics, many ideas are common between the two.
Newtons three laws of motion are the basis of classical mechanics. Taken together the first two tell us that a point-like particle with position and velocity satisfies the second order law of motion
where is the total force acting on the particle, is the mass of the particle and is time. Usually, the mass of a particle is conserved, i.e., constant in time so that . The quantity is called the momentum of the particle and is denoted . So Newton’s law of motion (1) simply states that the rate of change of momentum is the total force acting on . We say total force, because is supposed to be the sum of all forces acting on the particle, where represent the various “fundamental” forces acting on the particle . Newton’s third law — “to every action there is an equal and opposite reaction” — is taken to mean that the particle must exert a force on the origin of the force .
For instance if one considers an ensemble of point-like particles , , with positions , masses and velocities , then we have
where is the momentum of the -th particle and is the force exerted on the -th particle by the -th. Newton’s third law tells us that . In particular we find that
which is the principle of conservation of (the total) momentum.
Conservative forces and Hamiltonian Mechanics
In practice the equations of motions needed to describe physical systems have a bit more structure. It turns out that many forces are conservative, meaning that the force depends only on the positon of the particle and that
where is the potential energy of . Note that the potential energy, when it exists, is only defined up to an additive constant. (For a force field defined on all of a necessary and sufficient condition for to exist is that — this is the PoincarÈ Lemma. If the force field is defined on a domain with more complicated topology then this condition is necessary but may not be sufficient.)
A particle in a conservative force field satisfies the principle of conservation of energy
as one can readily check. Here is the total energy of and is the kinetic energy. More generally, if the potential energy function depends explicitly on time as well as the position of the particle , then
(We say that depends explicitly on time here because the potential energy of the particle always depends on time implicitly through the motion of the particle.)
The Hamiltonian function for is a function of the position and the momentum
along a trajectory satisfying (5). For a system of particles interacting with one another via forces as above we have
where is the force exerted on the -th particle by the -th — the dependence of only on the difference is guaranteed by Newton’s third law. We have introduced here an additional term giving the interaction of the particles with an “external field” or “background potential.”
Phase space and the algebra of observables
The phase space in classical mechanics is the space parameterized by the coordinates and momenta . To simplify notation we will denote by and by so that the Hamiltonian equations of motion (5) are just
For each , let be the evolution map for (8), that is
where satisfy (8) with and . This definition of is quite implicit — and not necessarily easy to compute.
Note that in order to define there must be a unique solution to (8) up to time for any initial condition . A sufficient condition for this would be that the derivative and are Lipschitz functions on phase space. If fails to exist the solution could breakdown in two differnt ways. Here are two examples on the phase space :
- Consider the Hamiltonian . Note that the force pushes the particle away from the origin and increases in magnitude as increases. In fact there are solutions that “escape to infinity in finite time.” For instance, with the initial condition and , the solution is
- Consider the Hamiltonian defined for . In this case the force can draw the particle into the singular point in finite time. In fact, if the initial condition has negative energy — —, then at some finite time beyond which the solution is not defined.
The second example is of physical relevance since it can be generalized to systems of point particles interacting according to Newton’s law of gravitation
which corresponds to the potentials . Gravity can (and does!) cause particles to collide in finite time. The equations of gravitation alone, however, do not describe the physics of collision.
Henceforth we will assume that the Hamiltonian is nice enough that there is a uniqe solution to (8) up to some fixed time for every initial condition. Since has the group property
it follows that there is a unique solution to (8) for all time and for every initial condition. Note that , the identity map. It also makes sense to define for negative times , by integrating the equations (8) “back in time.” In fact the group property (9) for positive and negative times, so is an invertible map (one-to-one and onto) for each and we have
The group property still holds so, in fact, is a one parameter group of transformations of phase space.
An observable in classical mechanics is an infinitely differentiable real valued function on phase space. Think of as some property of the system that can be “observed.” (We take observables to be infinitely differentiable for technical reasons.) Let denote the collection of observables. is an algebra, called the algebra of observables, since under the usual operations
The transformations induce transformations of the algebra as follows:
where satisfy (8) with and . Thus,
Applying the group property we obtain
where we have noted that by conservation of energy. Finally taking we obtain
where the Poisson bracket of two observables is defined to be
where are generalized coordinates and are generalized momenta.