The goal is to imagine how we could select quantum state with an experimental apparatus

--------------------------------SPIN & ANGULAT MOMENTUM ------------------------

What is the most basic and straightforward property that one can assign for an object?

One answer might be location, where the object is located.

Consider this as a fundamental characteristic. In a 3-D world this will require three pieces of information (x,y,z). Along with every coordinate, the motion of an object generates a conjugate momentum in this case we would use p_x, p_y, p_z. Alternatively one might choose to represent the location with spherical coordinates (r, θ,φ). Here the angular momentum would emerge when looking for the conjugate momenta. So momentum and angular momentum are two similar ways to characterize the motion of an object. This object should be referred to as a point particle.

point particle

Has two observable features connected with space: Position and Momentum. It may have other properties such as charge and mass.

and

One can also add the additional properties such as mass, charge, strangeness, isospin and others. These typically represent an intrinsic property unrelated to the particles spatial description. Mass through general and special relativity is related to space and time but for nonrelativistic QM it is an intrinsic characteristic but it is not completely separated from space because mass relates how difficult it is to change the particle’s motion (force, inertia). So the real meaning of mass requires relativity where

Newton’s laws or some equivalent formulation of physics will be chosen to understand the particle’s motion if there are interactions that must be included. For quantum we add a potential to the energy to account for the interactions.

This point particle is considered to be something with no spatial dimension and naturally appeals to physicists as a starting point or building block. It contains in some sense a fundamental character that has been isolated or idealized.

The notion that an object has no dimensions is a bit troublesome but as an abstract limit it has appeal. Zero size even classically leads to certain dilemmas. Clearly the energy cost to build the particles may be infinite. The force between particles at close distances tends toward infinity. If you want to assemble any amount of charge in building the point particle it would cost an infinite amount of energy. It is perhaps not unreasonable to assume that if these entities exist then perhaps the notion of building them doesn’t apply. Nature has these basics structures and so we don’t include these energies in any real calculation (renormalization).

Interestingly, the real behavior of an elementary particle as the distance gets ever smaller requires further development. Indeed one outcome of QM is that nothing exists in isolation. The void or empty space is really a dynamic entity. Particles, at some level, may be more equivalent to bubbles in a glass of water than small BBs or point particles in an empty space. The vacuum percolates with particles and antiparticles and this sea of temporary charge, mass, energy etc. responds to the presents of a particle. The vacuum can polarize, for example, to screen charge. At present physicists model the elementary nature of matter by assigning properties to bare particles which are idealized characteristics of a particle separate from the vacuum. The complete picture emerges when one places bare particle in a dynamic vacuum. The combination establishes the reality. This separation may not be optimal. The quest for what constitutes the best basic elements both theoretically and experimentally is ongoing. The LHC promises to reveal new quirks about nature. We may be poised to peel back one more layer of the onion to reveal a new view of substructure of matter.

We are about to consider angular momentum. It has a natural analog in classical physics. The manifestation of angular momentum in QM offers some new bizarre behavior (can’t simultaneously know all of the components simultaneously) but this is not really new. It is derive from the application of quantum rules already derived and the construction of the angular momentum from . The more interesting aspect of angular momentum is how it emerges as a fundamental property that is part of the basic structure that we need to impose on the building blocks or elements of out theory.

Considering this further we return to the notion of a point particle. One could assemble two of these objects thereby needing 6 positions and 6 momenta. However if the particles are connected in some fashion it becomes more convenient to represent their COM motion (X,Y,Z) and their relative motion (Δx, Δy, Δz). Here the notion of rotating an object about the COM becomes important. If we view the compound object as a new object then it must possess some aspects of the 6 degrees of freedom that were built in by choosing to construct it from 2 point particles. If we constrain the system by fixing the distance between the objects then the motion of the COM and the rotation about the COM adequately describe the system. This constructed system possesses a location and an intrinsic angular momentum. This is the classical view of angular momentum. It is viewed as a property that is based on construction.

However we have already had an inkling that this is not a sufficient way to view elementary particles. If we assume that E&M field needs to be quantized and that the basic quantum of this field is a fundamental particle then we need to be able to find a photon that only possess location as a property (point particle) and then construct the field (vector character) using some construction mechanism in an analogous fashion to the combining of two particles to get angular momentum. Or we can imbue the photon with and additional structure (polarization) that carries the information about the field directions as a natural element. The second approach is the one taken and the photon is a vector or spin-1 particle.

We dealt with the special case of light. Light consisted of the solutions to Maxwell’s equations with no charge or current sources. A wave equation emerges with fields transverse to propagation. For transverse fields and the wave direction are all perpendicular. Also the free photon has zero mass. There are basic quanta for the Coulomb field and all fields generated when charges and currents are present. These are virtual quanta with polarizations along the direction of propagation. Thus both free fields and static fields are present and the polarization is a vector which can be used to define the direction of the fields.

We usually write the field as a 4-vector in terms of the potentials

Just like typical vectors the photons can be represented in terms of their components or their magnitude and direction. So now rather than representing the photon as a point particle at some point in space {or by a wavefunction spread out over space} we need to provide a richer structure with components or a direction. A E&M field has vector nature at each point in space.

We eventually link this structure with an intrinsic character. The fields are spin 1 particle fields.

SO(3) ROTATIONS

Representation: orthogonal 3X3 matrices that act on 3-d vectors (x,y,z)

Important relationships

Rotation about z

This shows that any function of φ will be translated to a new function of φ

Next we work out the commutator relations and we find

It is easy to see that we get terms containing second order derivatives and terms with first order derivatives only. Because of the symmetry of the second order derivatives, all these terms cancel, and we are left with first order derivatives only. These terms are due to the fact that the derivative on the left may act on a spatial coordinate of the second angular momentum factor. We are left with