The following equation pair shows the close connection between quantum and classical mechanics. F = m × a = ×( ) ( )average F m average a The first equation is the basis of classical mechanics. It summarizes Newton’s laws, where F is the force, m is the mass, and a is the acceleration. The second equation is the quantum-mechanical generalization. It looks almost the same except for the word average. Averaging is necessary because particles in quantum mechanics are associated with waves, and one cannot precisely determine the position or motion of a wave. Only the averages are known.
The following equation pair shows the close connection between quantum and classical mechanics. F = m × a = ×( ) ( )average F m average a The first equation is the basis of classical mechanics. It summarizes Newton’s laws, where F is the force, m is the mass, and a is the acceleration. The second equation is the quantum-mechanical generalization. It looks almost the same except for the word average. Averaging is necessary because particles in quantum mechanics are associated with waves, and one cannot precisely determine the position or motion of a wave. Only the averages are known.
The aim of this chapter is to present some very basic principles of classical mechanics. They can help us understand the novelty of quantum theory. Classical mechanics was founded by Newton and contemporary 17th-century natural philosophers, and further developed in the 18th and 19th centuries by many physicists and mathematicians, as a result of the search for a rational, causal, and scientific explanation of the universe. In classical mechanics, real-world objects are modeled as a collection of point particles, i.e., objects with negligible size. The motion of a point particle is characterized by a small number of parameters such as its position, mass, velocity, and the forces applied to it. The position of a point particle is defined with respect to an arbitrary fixed reference point in the three- dimensional Euclidean space, which is usually chosen as the origin of some coordinate system. The velocity of a point particle is the rate of change of its position with time. The force applied to a point particle (called the external force) is the cause for the point particle x y z A to undergo a certain change in its velocity, i.e., an acceleration (Newton’s second law of motion). For the sake of simplicity, here we shall refer to a point particle or a collection of point particles simply as an object. Newton’s first law of motion states that if an object experiences no net external force, then its velocity is constant, i.e., the object is either at rest or it moves in a straight line with a constant speed. The first law of motion postulates the existence of a certain set of frames of reference called inertial reference frames, relative to which the motion of an object not subject to external forces is a straight line at a constant speed. For this reason, Newton’s first law is also referred to as the law of inertia.
The branch of physics referred to as classical mechanics describes and predicts with great accuracy the motion of macroscopic objects moving much slower than the speed of light when subject to one or more forces. At the center of classical mechanics is Newton’s famous second law of motion: force = mass × acceleration. Classical mechanics is deterministic in that it exactly describes (within its limitations) the future motion of an object, provided that the object’s mass, initial position, and velocity, as well as the forces acting on the object, are sufficiently well known. The objects in classical mechanics are often approximated as fictional particles of a certain mass, their sizes being negligible or irrelevant. In cases where the physical extent of an object needs to come into play, the object often takes the form of a sphere or some other calculation-friendly shape (recall, for example, our spherical asteroid from the preface to this book).
Classical mechanics and quantum mechanics have long been framed as binary opposites—but this dichotomy has yielded no real benefits, only persistent confusion. The fundamental equations of quantum mechanics cannot logically demarcate a clear boundary between the quantum and classical realms; in fact, the mass m and potential energy V in the Schrödinger equation seamlessly traverse the divide between the microscopic and macroscopic. The electromagnetic potential energy in the Schrödinger equation can be replaced with the potential energy of interactive forces, which can further be converted into force itself using the relation F = V/r. Derivations have now produced a force operator and quantum mechanical equations that explicitly incorporate force—including a quantum formulation of Newton’s second law. These results provide direct evidence that quantum and classical mechanics are not incompatible, but complementary. By resolving the historical absence of “force” as a formal concept in quantum mechanics, we bridge the conceptual gap between macroscopic classical frameworks and microscopic quantum theory, challenging long-held perceptions of their irreconcilability. This unified approach explains why nanoscale particles exhibit quantum properties, simplifies calculations of quantum forces, and lays the groundwork for a generalized wave mechanics that unifies mechanical descriptions across all scales.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dtdP, P=mdtdV, and [X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
It is difficult to understand quantum mechanics. A major reason for this is that people do not realize that classical mechanics can be very well described by quantum mechanics. It would all have been much easier if Newton had not developed classical mechanics from the equation relating force, mass, and acceleration of a particle, but had started from a simple form of the Schrödinger wave equation. The equation F=ma could just as well have waited for its derivation from quantum mechanics by Paul Ehrenfest in 1927.
In reality, micro objects and macro objects coexist harmoniously, and each macro object depends on micro particles to exist (macro is a collection of micro objects, indicating that they have a very extensive relationship and are symbiotic). However, classical mechanics and quantum mechanics have long been regarded as binary oppositions. This is a serious disconnect between theory and phenomena, requiring the establishment of new theories and methods that are compatible between quantum and classical. Prior to this, many reasons and methods for combining quantum theory with classical theory have been found (providing numerous computational examples). The wave mechanics equations of classical mechanics laws have been written. The technical method used is still the conversion rule of the Hermitian operator in the fundamental postulates of quantum mechanics. There are no new assumptions. The key is to use the relationship between force and energy in the planetary model to obtain wave mechanics equations that include force, Newton's second law, and the law of conservation of momentum. Thus, quantum mechanics, which is compatible with classical mechanics, truly becomes a form of 'mechanics'. This kind of mechanics is generalized wave mechanics. Quantum and classical are coordinated and compatible, resulting in no clear boundary between quantum and classical. It will seriously affect the credibility of quantum decoherence and wave function collapse.