Newtonian gravity, at first sight, does not imply new examples for quantum mechanics because of its close similarity with static electric forces. A massive object moving in an approximately constant gravitational field near the surface of Earth is subject to a linear potential https://www.w3.org/1998/Math/MathML" display="inline"> V ( z ) = m g z , while a 2-mass system is subject to Newton's potential https://www.w3.org/1998/Math/MathML" display="inline"> V ( r ) = − G m 1 m 2 / r and therefore resembles the hydrogen model. Moreover, gravitational effects in lab systems are usually weak and would require sensitive measurements to be noticed.
Newtonian gravity, at first sight, does not imply new examples for quantum mechanics because of its close similarity with static electric forces. A massive object moving in an approximately constant gravitational field near the surface of Earth is subject to a linear potential https://www.w3.org/1998/Math/MathML" display="inline"> V ( z ) = m g z , while a 2-mass system is subject to Newton's potential https://www.w3.org/1998/Math/MathML" display="inline"> V ( r ) = − G m 1 m 2 / r and therefore resembles the hydrogen model. Moreover, gravitational effects in lab systems are usually weak and would require sensitive measurements to be noticed.
The unknowns that remain in quantum mechanics are not the mysteries that people often think of. The real mysteries of quantum mechanics are annexed to foundational knowledge of quantum physics that, given all indications, just appeared into existence at our measurement level, or second layer of quantum physics. These miracles of knowledge include Planck's energy equation https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν , Born's rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , elements of the Schrödinger equation, and Dirac's notation. In Quantum Clear we elucidate the physics leading to https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν and https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , but that's it. And it remains certain that those who introduced this wonderful knowledge did not elaborate further except for the explanation of Schrödinger who linked mathematical discovery to ‘like a great gift…’. Ward, on the other hand, described the arrival of knowledge with ‘ no way I can trace the source. ’
Newton’s force law [Formula: see text] is derived from the Schrödinger equation for isolated macroscopic bodies, composite states of e.g. [Formula: see text] atoms and molecules, at finite body temperatures. We first review three aspects of quantum mechanics (QM) in this context: (i) Heisenberg’s uncertainty relations for their center of mass (CM), (ii) the diffusion of the CM wave packet, and (iii) a finite body temperature which implies a metastable (mixed-) state of the body: photon emissions and self-decoherence. They explain the origin of the classical trajectory for a macroscopic body. The ratio between the range [Formula: see text] over which the quantum fluctuations of its CM are effective, and the body’s (linear) size [Formula: see text], [Formula: see text] or [Formula: see text], tells whether the body’s CM behaves classically or quantum mechanically, respectively. In the first case, Newton’s force law for its CM follows from the Ehrenfest theorem. We illustrate this for weak gravitational forces, a harmonic-oscillator potential, and for constant external electromagnetic fields slowly varying in space. The derivation of the canonical Hamilton equations for many-body systems is also discussed. Effects due to the body’s finite size such as the gravitational tidal forces appear in perturbation theory. Our work is consistent with the well-known idea that the emergence of classical physics in QM is due to the environment-induced decoherence, but complements and completes it, by clarifying the conditions under which Newton’s equations follow from QM, and by deriving them explicitly.
People believe that the Schrödinger equation cannot be used to describe macroscopic objects like the Earth, and Newtonian mechanics cannot be used to describe microscopic systems. The old concept of the relationship between the existing laws of quantum mechanics and classical mechanics undoubtedly has a serious impact on people's understanding of the natural world, the development of physics theories, and the application of existing physics theories. The continuous development of physics theory requires constant changes to some incorrect old concepts. The Schrödinger equation that can describe planetary motion was successfully obtained by replacing the potential energy in the Hamiltonian operator from electromagnetic interaction potential energy to gravitational interaction potential energy. If the distance between the sun and the earth is approximated as a constant, the energy eigenvalues obtained by solving the Schrödinger equation for the Earth's revolution are completely consistent with the results obtained directly using classical mechanics. The direct significance of establishing and applying such equations is that they can simultaneously use classical mechanics and wave dynamics to describe all objects (no longer limited by the mass of the objects), simplifying the calculation process of quantum mechanics. It has been proven that classical mechanics and wave dynamics are compatible, and there is no insurmountable gap between them. This result has a huge positive impact on the theoretical updates and applications of quantum mechanics.
The authors review theories and experimental state-of-the-art efforts to study the effects of gravity on massive quantum systems. Classical gravity is the least precisely tested natural force and may be addressed via precision quantum probes. Experiments testing whether the quantum nature of gravity causes decoherence and collapse of matter-wave functions and whether it can mediate entanglement between separate massive particles are underway, and their results will guide the theoretical description of gravity effects on a laboratory scale.
This Chapter introduces the Feynman Path Integral description of quantum transition amplitudes, considering the simplest case of a quantum system described by one quantum variable q and its conjugate momentum p . The transition amplitude from time https://www.w3.org/1998/Math/MathML" display="inline"> t 1 to time https://www.w3.org/1998/Math/MathML" display="inline"> t 2 = t 1 + T is obtained by discretising the time axis in N time steps of amplitude https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , implying T=N https://www.w3.org/1998/Math/MathML" display="inline"> ϵ . For https://www.w3.org/1998/Math/MathML" display="inline"> ϵ → 0 , the Feynman Path Integral formula is derived: https://www.w3.org/1998/Math/MathML" display="inline"> < q N | e - i T H / ℏ | q 0 > = ∫ d [ q ( t ) ] e i S q ( t ) , q ' ( t ) / ℏ , where integration is over all trajectories https://www.w3.org/1998/Math/MathML" display="inline"> q ( t ) which satisfy the boundary conditions at https://www.w3.org/1998/Math/MathML" display="inline"> t 1 and t 2 . https://www.w3.org/1998/Math/MathML" display="inline"> S ( q , q ' ) is the classical Action, a function of https://www.w3.org/1998/Math/MathML" display="inline"> q and of its time derivative. The classical limit, corresponding to https://www.w3.org/1998/Math/MathML" display="inline"> ℏ → 0 , is considered, whereby the solution of the Feynman Path Integral reduces to the classical trajectory, https://www.w3.org/1998/Math/MathML" display="inline"> q c ( t ) , which minimises the classical Action. The lattice approximation , i.e. small, non vanishing https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , with evaluation of the Feynman integral by numerical integration, provides a useful approximation in the case of the fundamental theory of the strong interactions (Quantum Chromodynamics, or QCD), to be considered later; Chapters 17 to19. This chapter is completed by consideration of the extension of gthe formalism to imaginary time, where the Feynman Path Integral becomes the Partition Function of Statistical Mechanics, and by the definition of the Green's functions, ground-state expectation values of products of quantum operators q(t) taken at different times, https://www.w3.org/1998/Math/MathML" display="inline"> t 1 , t 2 … t N .
There is more information in a quantum state than in the classical initial values https://www.w3.org/1998/Math/MathML" display="inline"> ( x 0 , p 0 ) that determine the complete 1-dimensional motion of a point particle for a given force. The initial wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , 0 ) requires an infinite number of parameters, which can be expressed in observational terms as the two basic expectation values https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ x ^ ⟩ and https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ p ^ ⟩ together with the two variances https://www.w3.org/1998/Math/MathML" display="inline"> ( σ x 2 , σ p 2 ) and the covariance 5.1 https://www.w3.org/1998/Math/MathML" display="block"> C x p = 1 2 ⟨ x ^ p ^ + p ^ x ^ ⟩ − ⟨ x ^ ⟩ ⟨ p ^ ⟩ ,