The electromagnetic scattering of a spin-0 charged particle off a fixed center is calculated in first-order quantum perturbation theory. This implies evaluating the square of a `Dirac delta-function,' an operation that is not defined in Schwartz distribution theory, and which in elementary text-books is dealt with according to `Fermi's golden rule.' In this paper these conventional calculations are carefully reviewed, and their crucial parts reformulated in a Colombeau algebra -- in which the product of distributions is mathematically well defined. The conclusions are: (1) The Dirac delta-function insuring energy conservation in first order perturbation theory belongs to a particular subset of representatives of the Schwartz distribution defined by the Dirac measure. These particular representatives have a well-defined square, and lead to a physically meaningful result in agreement with the data. (2) A truly consistent mathematical interpretation of these representatives is provided by their redefinition as Colombeau generalized functions. This implies that their square, and therefore the quantum mechanical rule leading from amplitudes to probabilities, is rigorously defined.
The electromagnetic scattering of a spin-0 charged particle off a fixed center is calculated in first-order quantum perturbation theory. This implies evaluating the square of a Dirac delta-function,' an operation that is not defined in Schwartz distribution theory, and which in elementary text-books is dealt with according to Fermi's golden rule.' In this paper these conventional calculations are carefully reviewed, and their crucial parts reformulated in a Colombeau algebra -- in which the product of distributions is mathematically well defined. The conclusions are: (1) The Dirac delta-function insuring energy conservation in first order perturbation theory belongs to a particular subset of representatives of the Schwartz distribution defined by the Dirac measure. These particular representatives have a well-defined square, and lead to a physically meaningful result in agreement with the data. (2) A truly consistent mathematical interpretation of these representatives is provided by their redefinition as Colombeau generalized functions. This implies that their square, and therefore the quantum mechanical rule leading from amplitudes to probabilities, is rigorously defined.
We calculate the scattering rate of a quantum particle, based on Fermi’s golden rule for a quantum particle as a distribution of matter, where the density of states is a function of the particle Lagrangian, not of the Hamiltonian, as in the conventional theory based on the Schrödinger equation. Thus, the density of states does not refer to the energies as in the conventional form, but to the Lagrangian functions, which, besides the energies, include the momentum-velocity products, that are also constants of motion for the initial and the final states. We obtain the relativistic dependence on the momentum, and the two scattering rates for the two possible cases, with the spin conservation, or the spin inversion.
It is well-accepted that the Dirac delta function can be defined by the limitation of the square function covering a unit area with decrease of the width of the square to zero, and the eigenvectors of the position operator take the form of the delta function. When discussing the position measurement in quantum mechanics, one is prompted by the mathematical convention that uses the square function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large. The continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the delta function to represent the physical states of position.
It is explicitly shown the weak points (being not rigorous) in the reasoning of Dirac, of Landau, of Von Neumann, of Weyl, of Fadeev and of others, deriving many fundamental relations, which later became classical and are being commonly used in Quantum Theory up to now. It is because of their derivations were made when functional analysis was not yet sufficiently developed. Those weak points are of two following reasons:1. Unbounded property of the physical operators is not yet paid by attention.2. The “Dirac delta function” defined by Dirac is not always distinguished clearly with the Dirac distribution defined by Schwartz and with the generalized functions defined by Gel'fand.Particularly, the formulation of the closure relation by using equality of the functionals is not rigorous. Consequently, it is necessary to improve Quantum Theory.
We propose an improved scheme of perturbation theory based on our exact solution [An Min Wang, quant-ph/0611216] in general quantum systems independent of time. Our elementary start-point is to introduce the perturbing parameter as late as possible. Our main skills are Hamiltonian redivision so as to overcome a flaw of the usual perturbation theory, and the perturbing Hamiltonian matrix product decomposition in order to separate the contraction and anti-contraction terms. Our calculational technology is the limit process for eliminating apparent divergences. Our central idea is ``dynamical rearrangement and summation" for the sake of the partial contributions from the high order even all order approximations absorbed in our perturbed solution. Consequently, we obtain the improved forms of the zeroth, first, second and third order perturbed solutions absorbing the partial contributions from the high order even all order approximations of perturbation. Then we deduce the improved transition probability. In special, we propose the revised Fermi's golden rule. Moreover, we apply our scheme to obtain the improved forms of perturbed energy and perturbed state. In addition, we study an easy understanding example of two-state system to illustrate our scheme and show its advantages. All of this implies the physical reasons and evidences why our improved scheme of perturbation theory are actually calculable, operationally efficient, conclusively more accurate. Our improved scheme is the further development and interesting application of our exact solution, and it has been successfully used to study on open system dynamics [An Min Wang, quant-ph/0601051].
Abstract Through a series of postulates, we define a function γ(x) whose square acts as Dirac's δ(x) and exhibits several unusual properties. Though the square root of δ cannot be defined among distributions, it appears in quantum theory if one wants to associate a wave function to a (quasi)classical particle having charge distribution δ(x). The newly defined function γ(x) serves to describe quasi‐classical particles using part of the quantum formalism (eg, wave functions, operators, expectation values) but exhibiting classical properties. The function γ(x) appears to be useful to define model wave functions for simple (quasi)quantum systems. In a spherical coordinate system, γ(r − r0) leads to a quasi‐classical “bubble” model of the hydrogen atom, where the electron is distributed on the surface of a sphere with radius r0, and it provides exact quantum mechanical energies of its total symmetric levels. For other simple quantum systems, it provides approximate but meaningful energies. In particular, exact energy differences for harmonic oscillator levels are obtained, with the zero‐point energy missing.
In the framework of distributionally generalized quantum theory, the object Hψ is defined as a distribution. The mathematical significance is a mild generalization for the theory of para- and pseudo-differential operators (as well as a generalization of the weak eigenvalue problem), where the ψ-do symbol (which is not a proper linear operator in this generalized case) can have its coefficient functions take on singular distributional values. Here, a distribution is said to be singular if it is not Lp(Rd) for any p≥1. Physically, the significance is a mathematically rigorous method, which does not rely upon renormalization or regularization of any kind, while producing bound state energy results in agreement with the literature. In addition, another benefit is that the method does not rely upon self-adjoint extensions of the Laplace operator. This is important when the theory is applied to non-Schrodinger systems, as is the case for the Dirac equation and a necessary property of any finite rigorous version of quantum field theory. The distributional interpretation resolves the need to evaluate a wave function at a point where it fails to be defined. For d=2, this occurs as Ko(a∣x∣)δ(x), where Ko is the zeroth order MacDonald function. Finally, there is also the identification of a missing anomalous length scale, owing to the scale invariance of the formal symbol(ic) Hamiltonian, as well as the common identity for the logarithmic function, with a,b∈R+, log(ab)=log(a)+log(b), which loses unitlessness in its arguments. Consequently, the energy or point spectrum is generalized as a family (set indexed by the continuum) of would-be spectral values, called the C-spectrum.