It is shown that the relation between ray optics and wave optics is analogous to the relation between the classical mechanics of a point particle and quantum mechanics. The time coordinate of quantum mechanics has to be replaced with a length coordinate and the light wavelength λ takes the place of Planck’s constant h. A well-known result of quantum mechanics is that it coincides with classical mechanics in the limit h → 0. Correspondingly, wave and ray optics must coincide in the limit λ → 0.
It is shown that the relation between ray optics and wave optics is analogous to the relation between the classical mechanics of a point particle and quantum mechanics. The time coordinate of quantum mechanics has to be replaced with a length coordinate and the light wavelength λ takes the place of Planck’s constant h. A well-known result of quantum mechanics is that it coincides with classical mechanics in the limit h → 0. Correspondingly, wave and ray optics must coincide in the limit λ → 0.
It is argued that from a formal point of view, the classical limit of light quanta or photons is not that of a point-like particle but that of a geometric ray. According to this view, standard particle-wave dualism, which is often used in schools to describe the quantum behavior of massive objects, could be replaced by a ray-wave dualism (or even a particle-ray-wave trialism), which seems to be more appropriate for massless quantum objects such as photons. We compare the limits leading from quantum electrodynamics to a classical (Hamiltonian) theory of particles for electrons with those leading from photons via Maxwell’s equations to geometric ray optics. We also discuss the question to which extent Maxwell’s theory for electromagnetic waves should be considered as being on the same formal level as Schrödinger’s or Dirac’s theory.
Abstract Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.
Wave mechanics In the classical dynamics the state of the system whose coordinates are q 1 … q s and momenta p 1 … p s is denoted by a point in 2s-dimensional space (the q-space); changes in the system are indicated by the passage of the point along some curve in this space, a ‘ray,’ it may be called. Schrödinger's new concept is that changes in the system should be represented in the q-space not by ‘rays’ but by ‘waves.’ He regards the classical mechanics as only an approximation to actual mechanical events, which he says can only be described by wave mechanics; the former in his view bears to the latter just the same relation as geometrical (or ‘ray’) optics does to physical (or ‘wave’) optics. A large scale (macroscopic) mechanical process corresponds to a wave signal in the g-space, which can be regarded as a point in comparison with the geometrical structure of its path. But for small scale (microscopic) processes, such as atomic phenomena are, the rigorous wave theory must be used. The wave equation must replace the usual dynamical equations. The latter are just as unfruitful for the purposes of atom mechanics as geometrical optics alone would be to clear up the fine points of diffraction phenomena.
According to Schroedinger's ideas, classical dynamics of point particles should correspond to the " geometrical optics " limit of a linear wave equation, in the same way as ray optics is the limit of wave optics. It is shown that, using notions of modern wave theory, the " geometrical optics " analogy leads to the correspondence between a classical Hamiltonian H and a " quantum " wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. In the line of Schroedinger's ideas, it is also attempted to justify the occurrence, in QM, of eigenvalues problems, not merely for energy, but also for momentum. It is shown that the wave functions of pure momentum states can be defined in a physically more satisfying way than by assuming plane waves. In the case of a spatially uniform force field, such momentum states have a singularity and move undeformed according to Newton's second law. The mentioned unambiguous correspondence allows to uniquely extend the Klein-Gordon relativistic wave equation to the case where a constant gravitational field is present. It is argued that Schroedinger's wave mechanics can be extended to the case with a variable gravitational field only if one accepts that the wave equation is a preferred-frame one. From this viewpoint, generally-covariant extensions of the wave equations of QM seem rather formal. Finally, it is conjectured that there is no need for a quantum gravity.
Quantum mechanics is one of basics theories of modern physics, and it is also a theory established with some debates of assumptions. However, in the teaching process, if the assumption of wave-particle duality is directly given, it will be particularly difficult for students to understand. Therefore, based on the teaching practice and previous research, we assume that light corresponds to a kind of real particle starting from the classical electromagnetic theory, according to the description methods of wave optics and geometric optics, and the form of functional equations. Subsequently, employing Planck's quantum conditions, we discuss the connection between the wave nature and particle nature of light. And then the Planck-Einstein relation and the de Broglie relationship are derived, which reflect the wave-particle duality of light and real particle.
According to Schrödinger's ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrödinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton's second law in the given force field.