The transformation theory of quantum mechanics is constructed on the mathematical basis of the theory of the linear functionals and the physical notion of the wave-packet formalism of the theory. The wave-packet function, which is described by an indefinitely differentiable and rapidly decreasing function, specifies the regions of the values of some physical quantities corresponding to the physical situation. The wave functions or the transformation functions are considered to be the linear continuous functionals which have the wave-packet functions as their functional arguments and transform the wave-packet functions to the other ones. The dynamics of quantum mechanics is constructed on the former (the wave functions as functionals) but not on the latter (the wave-packet functions). The theory never contains improper functions and integrals. The present formalism is physically interpreted as a sort of the wave-packet theory of quantum mechanics. The formal theory of scattering is formulated in the framework of this theory without resort to any ε-manipulation and the adiabatic assumption, in order to investigate the conditions for potentials and the behaviors of the evolution operator at infinite past and future. Finally the definition is given for the higher-order functional derivatives. They may suggest a method of treating the overlapping singularities.
The transformation theory of quantum mechanics is constructed on the mathematical basis of the theory of the linear functionals and the physical notion of the wave-packet formalism of the theory. The wave-packet function, which is described by an indefinitely differentiable and rapidly decreasing function, specifies the regions of the values of some physical quantities corresponding to the physical situation. The wave functions or the transformation functions are considered to be the linear continuous functionals which have the wave-packet functions as their functional arguments and transform the wave-packet functions to the other ones. The dynamics of quantum mechanics is constructed on the former (the wave functions as functionals) but not on the latter (the wave-packet functions). The theory never contains improper functions and integrals. The present formalism is physically interpreted as a sort of the wave-packet theory of quantum mechanics. The formal theory of scattering is formulated in the framework of this theory without resort to any ε-manipulation and the adiabatic assumption, in order to investigate the conditions for potentials and the behaviors of the evolution operator at infinite past and future. Finally the definition is given for the higher-order functional derivatives. They may suggest a method of treating the overlapping singularities.
The time-dependent theory of scattering is reformulated. In §1, it is developed on the basis of a new limiting process which is self-consistent, and the equivalence is shown of Heisenberg's S-matrix and Dyson's one, when the total Hamiltonian permits the existence of bound states. In §2, a theory of scattering of wave packet is proposed in conformity with the physical picture. The damping factor e−ε|t| is derived from the amplitude of the wave packet. In §3, the rearrangement scattering is treated on the basis of the wave packet formalism.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol, as defined in Eq. (3.6.6). The following steps are the local and global conservation laws associated to the Schrödinger equation, the probabilistic interpretation of the wave function, the spreading of wave packets and transformation properties of wave functions. The solution of the Schrödinger equation for a given initial condition is then studied. For this purpose, one first expands the initial condition in terms of eigenfunctions of the Hamiltonian operator. This is then ‘propagated’ to determine the evolution of the wave function. Thus, one is led to consider the Green kernel of the Schrödinger equation, and the boundary conditions on the wave function that lead to complete knowledge of stationary states. A dynamics involving an isometric non-unitary operator is also studied. The chapter ends with an elementary introduction to harmonic oscillators and to the JWKB method in wave mechanics, plus examples on the Bohr–Sommerfeld quantization. From classical to wave mechanics The correspondence between differential operators and polynomials in k relies on the notion of the symbol defined in Eq. (3.6.6). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is non-trivial because the map from symbols to operators contains some ambiguity. Such a transition is always possible if polynomials in p have constant coefficients. When this is not the case, i.e. if the coefficients are functions on the configuration space, there is an obvious ambiguity in the association of an operator with xp or px , for instance.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
The study of wave packet is of great significance in quantum mechanics, optics and fluid mechanics. However, in order to solve the strict evolution behavior of wave packet, it is necessary not only to determine the parameters of various physical quantities such as mass, but also to carry out the complex integration process in configuration space and momentum space. In the final analysis, the evolution behavior of wave packet is the evolution behavior of distribution function parameters with time. Using the method of grey system theory, the time-varying parameters are replaced by time-varying response series, and then the formal structure is constructed according to the physical meaning of time evolution to realize the approximate simulation and approximate calculation of wave function expression. The advantage of this method is that it can simulate the evolution of wave packet under the uncertainty of each physical quantity. Based on the grey system theory, this paper uses GM (1,1) model to replace the time response function of wave packet distribution, obtains the simulated evolution behavior of wave packet wave function by means of mathematical techniques such as variational method, and makes error analysis and correction. In practical application, the approximate evolution behavior of wave function can be obtained only by determining the parameters of expansion coefficient of wave function. In this paper, the Gauss wave packet is taken as an example to discuss the method. The grey system theory of mathematical modeling is introduced into physical calculation for the first time, and an approximate calculation method which can be widely used in wave packet evolution calculation is given.