In a world designed specifically to make easy the learning of quantum physics, the simplest example would be states of position of a free particle or of a particle confined in an atom. Photons allow us to develop the essential formal ideas of quantum physics with an absolute minimum of complexity. A satisfactory theory of photons requires quantum electrodynamics , a branch of advanced quantum theory. The simplest photon polarization state to produce and describe is a state of "linear polarization". It is easy enough to determine whether or not the photons in a beam are in a state of linear polarization. For a linearly polarized beam the state can be exactly specified in terms of the orientation of the polarizer transmission axis for maximum transmission. The behavior of photons in a beam is independent of the intensity of the beam; the photons behave—most conveniently—in a manner independent of one another.
In a world designed specifically to make easy the learning of quantum physics, the simplest example would be states of position of a free particle or of a particle confined in an atom. Photons allow us to develop the essential formal ideas of quantum physics with an absolute minimum of complexity. A satisfactory theory of photons requires quantum electrodynamics , a branch of advanced quantum theory. The simplest photon polarization state to produce and describe is a state of "linear polarization". It is easy enough to determine whether or not the photons in a beam are in a state of linear polarization. For a linearly polarized beam the state can be exactly specified in terms of the orientation of the polarizer transmission axis for maximum transmission. The behavior of photons in a beam is independent of the intensity of the beam; the photons behave—most conveniently—in a manner independent of one another.
The early twentieth century was a revolutionary time in the history of physics. People often think of Einstein's special theory of relativity of 1905, which changed our conceptions of time and space. But among physicists, quantum mechanics is usually regarded as an even more radical change in our thinking about the physical world. Quantum mechanics, which was developed between 1900 and 1926, began as a theory of atoms and light but has now become the framework in terms of which all basic physical theories are expected to be cast. We need quantum ideas not only to understand atoms, molecules, and elementary particles, but also to understand the electronic properties of solids and even certain astronomical phenomena such as the stability of white dwarf stars. The theory was radical in part because it introduced probabilistic behavior as a fundamental aspect of the world, but even more because it seems to allow mutually exclusive situations to exist simultaneously in a “quantum superposition.” We will see later that the possibility of quantum superposition is largely responsible for a quantum computer's distinctive advantage over an ordinary computer. The present chapter is devoted to introducing the basic principles of quantum mechanics. There are essentially four components of the mathematical structure of quantum mechanics. We need to know how to represent (i) states, (ii) measurements, (iii) reversible transformations, and (iv) composite systems. We will develop the first three in stages, starting with a very simple case – linear polarization of photons – and working toward the most general case.
How can we construct a picture of the photon from the wealth of observation material available to us? The photon appears to have a split personality: it is neither a wave nor a particle but something else which, depending on the experimental situation, exhibits a wave- or a particle-like behavior. In other words, in the photon (as in material particles such as the electron) the particle–wave dualism becomes manifest. Whereas classically the wave and the particle pictures are separate, quantum mechanics accomplishes a formal synthesis through a unified mathematical treatment. Let us look first at the wave aspect familiar from classical electrodynamics, which seems to be the most natural description. It makes all the different interference phenomena understandable, such as the “interference of the photon with itself” on the one hand and the appearance of spatial and temporal intensity correlations in a thermal radiation field on the other (which are obviously brought about by superposition of elementary waves emitted independently from different atoms). It might come as a surprise (at least for those having quantum mechanical preconceptions) that the classical theory is valid down to arbitrarily small intensities: the visibility of the interference pattern does not deteriorate even for very small intensities – the zero point fluctuations of the electromagnetic field advocated by quantum mechanics do not have a disturbing effect – and is valid not only for conventional interference experiments but also for interference between independently generated light beams (in the form of laser light).
We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉 * , are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.
In this chapter we build up the basic concepts of quantum mechanics using two simple examples, following a heuristic approach which is more inductive than deductive. We start with a familiar phenomenon, that of the polarization of light, which will allow us to introduce the necessary mathematical formalism. We show that the description of polarization leads naturally to the need for a two-dimensional complex vector space, and we establish the correspondence between a polarization state and a vector in this space, referred to as the space of polarization states. We then move on to the quantum description of photon polarization and illustrate the construction of probability amplitudes as scalar products in this space. The second example will be that of spin 1/2, where the space of states is again two-dimensional. We construct the most general states of spin 1/2 using rotational invariance. Finally, we introduce dynamics, which allows us to follow the time evolution of a state vector. The analogy with the polarization of light will serve as a guide to constructing the quantum theory of photon polarization, but no such classical analog is available for constructing the quantum theory for spin 1/2. In this case the quantum theory will be constructed without reference to any classical theory, using an assumption about the dimension of the space of states and symmetry principles. The polarization of light and photon polarization The polarization of an electromagnetic wave The polarization of light or, more generally, of an electromagnetic wave, is a familiar phenomenon related to the vector nature of the electromagnetic field. Let us consider a plane wave of monochromatic light of frequency ω propagating in the positive z direction.
Physics is based on experimental observables. Some of these observables are functions of other observables. For example, mass is an observable associated with inertia or gravity and velocity is classically measured, but the product mov yields nonrelativistic momentum which may be measured independently through an impulse hit with a target. A further issue with measurement is the case of vectors. For a vector pointing in one direction, one may orient the z axis to lie along this direction. If one always does this, momentum appears as a scalar, but one has to continuously rotate the co-ordinate system to account for different orientations. In this note, we are interested in spin which is linked to a kind of rotation and so we focus on the momentum vector, but only consider a function of momentum which yields a linear E, free particle energy. Thus, for the Lorentz invariant: -EE + p dot p = -momo, we consider the nonrelativistic limit: E = p dot p /2mo. In quantum mechanics, a free particle is represented by the complex probability (wavefunction) exp(ip dot r).This function has physical implications as it manifests itself in n-slit interference experiments so it is physical (not a mathematical convenience), but at the same time is responsible for conservation of momentum in x,y,z because Integral dx exp(ip1x) exp(ip2x) = 0 if p1 is not equal to p2. A physical function (property) through orthogonality creates an important relationship, namely conservation of momentum. We suggest that a similar idea may be at work in a relationship linking a linear E (free particle energy) to a function of momentum. We start with a linear E related to a function of p and only focus on the p vector because if it is treated as a number, the co-ordinate axis must rotate to account for its orientation. We only consider a situation in which p(x), p(y) and p(z) behave in an independent manner in the function of p (i.e. in the same manner as p itself) and suggest that instead of using vector analysis, there should exist a physical co-ordinate axis for each projection p(x), p(y) and p(z) if they are independent. We argue that p dot p = p(x)p(x) + p(y)p(y)+p(z)p(z) does not contain cross terms such as p(x)p(y). This is a kind of orthogonality which we associate with a rotational kind of operator linked with the p(i) operator -id/dx(i). In other words, we seek: p vector = generalized number = p(x)sx+p(y)sy+p(z)sz, where s(i) are generalized numbers or matrices, so that when this multiplied by itself, it yields p dot p. It turns out that the s(i) are the 2x2 Pauli matrices. In the case of a photon, there is no nonrelativistic limit and a linear E equation related to p is: E= |p|c. This equation is not independent in p(x), p(y), p(z) i.e. one cannot write p(x)sx+p(y)sy+p(z)sz an obtain E = {p(x)sx+p(y)sy+p(z)sz} c. Thus we look for another equation containing a linear E and a function of p. In this case, we use energy and momentum densities as given by the continuity equation: d/dt partial energy density + d grad dot ElxB =0 where d is a constant and El and B electric and magnetic 3-vectors. As in the above analysis, one wishes to have a momentum operator linked with a kind of rotational operator which allows one to eliminate vector math and use generalized numbers. In such a case, grad in the continuity equation becomes the momentum vector and eijk (Levi-Civita symbol in the cross product) becomes the spin matrix. Thus in both cases, spin governs the form of the function containing momentum which yields a linear E. In other words, it is not momentum alone which yields the relationship between a linear E (free particle energy) and momentum.