In his extremely important paper [1] E. Schro¨dinger proposed a wave equation which is the basic equation of the “undulatorical” mechanics and can be considered as a substitution to the Hamilton-Jacobi partial differential equations (HJ) of the ordinary (classical) mechanics. The wave equation was established on condition that the Lagrange function contains no terms linear in velocities. Schro¨dinger writes (footnote on p. 514 l.c.): “In relativistic mechanics and in the case of a magnetic field the expression of the (HJ) is more complicated. In the case of a single electron this equation means the constancy of the four-dimensional gradient diminished by a given vector (the four-dimensional potential). The wave mechanical translation of this Ansatz meets some difficulties.”
In his extremely important paper [1] E. Schro¨dinger proposed a wave equation which is the basic equation of the “undulatorical” mechanics and can be considered as a substitution to the Hamilton-Jacobi partial differential equations (HJ) of the ordinary (classical) mechanics. The wave equation was established on condition that the Lagrange function contains no terms linear in velocities. Schro¨dinger writes (footnote on p. 514 l.c.): “In relativistic mechanics and in the case of a magnetic field the expression of the (HJ) is more complicated. In the case of a single electron this equation means the constancy of the four-dimensional gradient diminished by a given vector (the four-dimensional potential). The wave mechanical translation of this Ansatz meets some difficulties.”
In the previous chapter we have seen that electrons and other subatomic particles sometimes exhibit properties similar to those commonly associated with waves: for example, electrons of the appropriate energy are diffracted by crystals in a manner similar to that originally observed in the case of x-rays. We have also seen that the energy and momentum of a free particle can be expressed in terms of the angular frequency and wave vector of the associated plane wave by the de Broglie relations (4.8).
According to non-relativistic quantummechanics, the evolution of the wave function of a free particle of mass m and momentum p is described by the Schro¨dinger equation: i ∂ψ ∂t = Hψ , (6.1) with the Hamiltonian operator H = −∇2/2m obtained from the expression for energy E = p2 2m , (6.2) using the substitution E → i ∂ ∂t , p → −i∇ . (6.3) Schro¨dinger himself [6] first suggested a generalisation of equation (6.1) based on the use of the relativistic energy equation: E2 = p2 +m2 . (6.4) (+m2)ψ = 0 , (6.5) Multiplying by ψ∗ and subtracting the product of ψ and the complex conjugate of (6.5) from the result, the continuity equation is obtained ∂ρ ∂t = ∇j , (6.6) with ρ = ψ∗ ∂ψ ∂t − ψ∂ψ ∂t (6.7) and j = −∇ (ψ∗∇ψ − ψ∇ψ∗) . (6.8) However, the ρ which appears in equation (6.6) cannot be identified with the probability density, in analogy with the similar quantity obtained from the Schro¨dinger equation, because it does not have the required property of being always positive-definite. To be convinced of this it is sufficient to substitute i∂/∂t→ E in (6.7). The result, ρ = E|ψ|2 , shows that ρ can be either positive or negative, following from the fact that the Klein-Gordon equation has solutions for the energy with both signs E = ± √ p2 +m2 , Note that in the non-relativistic limit E ≈ m > 0, and the familiar result ρ ∝ |ψ|2 is recovered.
In this chapter, the authors introduce the Hamilton-Jacobi equation showing that classical mechanics can be described with a wave equation for the classical action. In order to find the equation showing that classical mechanics already contains a wave behavior in matter particles, the people need to go back to the Lagrange/Hamilton formalism. Davisson-Germer experiment has shown without any doubt that electrons behave as massive particles whose dynamics is governed by some sort of wave. A free particle is apparently the conceptually simplest system to study because the TISE will reduce to its simplest form. The simple (linear) harmonic oscillator is of paramount importance in classical as well as quantum mechanics. A proper solution to the Harmonic oscillator problem can be found by solving Schrodinger equation analytically. Charles Hermite was a French mathematician who gave important contributions in many fields of mathematics used in physics like, for example, Hermitian operators and Hermite polynomials.
Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density ρ=R2 coupled to Hamilton's principal function S. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair (R,S) into a single complex field. Starting from a completely general complex ansatz ψ=f(R,S)eig(R,S), and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose ∣κ∣→0 limit reproduces the HJ formulation exactly. Remarkably, when Re(κ)=0, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.
Introduction Under this name at present two theories are being carried on, which are indeed closely related but not identical. The first, which follows on directly from the famous doctoral thesis by L. de Broglie, concerns waves in three-dimensional space. Because of the strictly relativistic treatment that is adopted in this version from the outset, we shall refer to it as the four-dimensional wave mechanics. The other theory is more remote from Mr de Broglie's original ideas, insofar as it is based on a wave-like process in the space of position coordinates ( q -space) of an arbitrary mechanical system. We shall therefore call it the multi-dimensional wave mechanics. Of course this use of the q -space is to be seen only as a mathematical tool, as it is often applied also in the old mechanics; ultimately, in this version also, the process to be described is one in space and time. In truth, however, a complete unification of the two conceptions has not yet been achieved. Anything over and above the motion of a single electron could be treated so far only in the multi -dimensional version; also, this is the one that provides the mathematical solution to the problems posed by the Heisenberg–Born matrix mechanics. For these reasons I shall place it first, hoping in this way also to illustrate better the characteristic difficulties of the as such more beautiful four-dimensional version.
Implementation of the equivalence principle, stating that all physical systems can be connected by a coordinate transformation (trivializing map) to the free system with vanishing energy, uniquely leads to the quantum analogue of the Hamilton–Jacobi (HJ) equation. A property of the formalism is manifest p–q duality, a consequence of the involutive nature of the Legendre transformation and of the recently observed relation with second–order differential equations. This makes clear the intrinsic ψD –ψ duality between linearly independent solutions of the Schrödinger equation. Being a third–order non–linear differential equation, the Quantum Stationary HJ Equation (QSHJE) gives rise to initial conditions which cannot be seen in the Schrödinger equation. Solutions of the QSHJE provide a trajectory representation of quantum mechanics, depending on the Planck length. Unlike Bohm’s theory, there is a non– trivial action even for bound states and no wave guide is present. A fundamental aspect of the formulation is that no use of any axiomatic interpretation of the wave function is made. The QSHJE is defined only if the ratio ψD /ψ is a local homeomorphism of the extended real line into itself. In this respect the quantization of the energy is strictly related to the S duality of the Schwarzian derivative. As a result, the equivalence principle, according to which all the potentials are equivalent under coordinate transformations, by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization. Finally, we derive the quantum HJ equation also in the higher dimensional case with time dependent potentials and in the relativistic case. Contents