Abstract This Chapter gives an in-depth discussion of four simple examples, and of how they illustrate the Schema: (i) Position-momentum duality in elementary quantum mechanics amounts to a Fourier transformation. The position and momentum models each represent the Hilbert space and its algebra of operators. (ii) Wave-particle duality, though often called a ‘duality’, is in general not a duality in the Schema’s sense. Thus the Chapter disambiguates different uses of ‘wave-particle duality’ and clarifies the relation of these uses to the conception used in the Schema. It also discusses ‘complementarity’. (iii) Electric-magnetic duality in Maxwell’s theory. Here, the interpretation of the common core is discussed in analogy with the interpretation of the Lorentz transformations. Dirac monopoles are also discussed. (iv) Kramers-Wannier duality for the Ising model. This is paradigmatic for many physical behaviours associated with dualities, and is the book’s first example of a duality for a probabilistic theory.
Abstract This Chapter gives an in-depth discussion of four simple examples, and of how they illustrate the Schema: (i) Position-momentum duality in elementary quantum mechanics amounts to a Fourier transformation. The position and momentum models each represent the Hilbert space and its algebra of operators. (ii) Wave-particle duality, though often called a ‘duality’, is in general not a duality in the Schema’s sense. Thus the Chapter disambiguates different uses of ‘wave-particle duality’ and clarifies the relation of these uses to the conception used in the Schema. It also discusses ‘complementarity’. (iii) Electric-magnetic duality in Maxwell’s theory. Here, the interpretation of the common core is discussed in analogy with the interpretation of the Lorentz transformations. Dirac monopoles are also discussed. (iv) Kramers-Wannier duality for the Ising model. This is paradigmatic for many physical behaviours associated with dualities, and is the book’s first example of a duality for a probabilistic theory.
Complementarity, e.g., the wave–particle duality of nature, lies at the heart of quantum mechanics. It distinguishes the world of quantum phenomena from the reality of classical physics. In the 1920s, quantum theory as we know it today was still new, and examples used to illustrate complementarity emphasized the position (particle-like) and momentum (wave-like) attributes of a quantum mechanical object, be it a photon or a massive particle. This is the historical reason why complementarity is often superficially identified with the so-called wave-particle duality of matter . Complementarity, however, is a more general concept. We say that two observables are complementary if precise knowledge of one of them implies that all possible outcomes of measuring the other one are equally probable. We may illustrate this by two extreme examples. The first example consists of the position and momentum (along one direction) of a particle: if, say, the position is predetermined then the result of a momentum measurement cannot be predicted, all momentum values are equally probable (in a large range). The second extreme involves two orthogonal spin components of a spin- 1/2 particle: if, say, the vertical spin component has a definite value ( up or down ) then upon measuring a horizontal component both values ( left or right , for instance) are found, each with a probability of 50%. Thus, in the microcosmos complete knowledge in the sense of classical physics is not available. The classic example of the merger of wave and particle behavior is provided by Young's double-slit experiment.
Abstract Momentum and position operators are essential in many areas of quantum mechanics. This chapter introduces these operators and discusses their mathematical and conceptual foundations. These operators are then applied to the widely used simple harmonic oscillator model. Simple methods lead to the formal description of both the coordinate-space and momentum-space wave functions for the oscillator model. The Heisenberg uncertainty relation for the oscillator model embodies the important concept of wave–particle duality in quantum mechanics.
Assuming a background in basic classical physics, multivariable calculus, and differential equations, A Concise Introduction to Quantum Mechanics provides a self-contained presentation of the mathematics and physics of quantum mechanics. The relevant aspects of classical mechanics and electrodynamics are reviewed, and the basic concepts of wave-particle duality are developed as a logical outgrowth of experiments involving blackbody radiation, the photoelectric effect, and electron diffraction. The Copenhagen interpretation of the wave function and its relation to the particle probability density is presented in conjunction with Fourier analysis and its generalization to function spaces. These concepts are combined to analyze the system consisting of a particle confined to a box, developing the probabilistic interpretation of observations and their associated expectation values. The Schrödinger equation is then derived by using these results and demanding both Galilean invariance of the probability density and Newtonian energy-momentum relations. The general properties of the Schrödinger equation and its solutions are analyzed, and the theory of observables is developed along with the associated Heisenberg uncertainty principle. Basic applications of wave mechanics are made to free wave packet spreading, barrier penetration, the simple harmonic oscillator, the Hydrogen atom, and an electric charge in a uniform magnetic field. In addition, Dirac notation, elements of Hilbert space theory, operator techniques, and matrix algebra are presented and used to analyze coherent states, the linear potential, two state oscillations, and electron diffraction. Applications are made to photon and electron spin and the addition of angular momentum, and direct product multiparticle states are used to formulate both the Pauli exclusion principle and quantum decoherence. The book concludes with an introduction to the rotation group and the general properties of angular momentum.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation: There are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be the subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. This latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
In this chapter, the physical content of previous chapters is summarized, and the reader is suided towards some key areas of modern physics: the Feynman formulation, the unificaton of quantum theory with special relativity, and dualities in quantum mechanics. This very short chapter is intended to be a bridge aimed at motivating readers to attend first course in quantum field theory. Towards an overall view At this stage, the student or the general reader might want to know what picture of the physical world emerges from the material covered so far and from the areas related to it but not covered in our book, so as to be introduced to what is in sight not only for the beginner but also for the scientific community. For this purpose, we summarize very briefly the previous pictures and we arrive at three key concepts: the Lagrangian in quantum mechanics via the Feynman sum over histories; the unification of quantum mechanics and special relativity; and new duality symmetries in quantum mechanics. From Schrödinger to Feynman Quantum mechanics provides a probabilistic description of the world on atomic or sub-atomic scale. It tells us that, on such scales, the world can be described by a Hilbert space structure, or suitable generalizations. Even in the relatively simple case of the hydrogen atom, the appropriate Hilbert space is infinite-dimensional, but finite-dimensional Hilbert spaces play a role as well.