from the Hamiltonian function üof classical mechanics by the operator substitution h d liri dqiwhere h is Planck's constant.For instance, for a hydrogen atom with fixed nucleus H is the sum of the kinetic energy of the one electron and the potential energy of the electron with respect to the nucleus :The Hamiltonian operator correspondingly involves the Laplacian operator in a term representing the kinetic energy of the electron and a scalar multiplier representing the potential energy:The wave equation is thenIn these equations £ is a number, the energy of the system, and *, the corresponding solution, is the wave function which describes the state of the system when it has that energy, x// is obviously a function of all the spatial coordinates needed to define a configuration of the system.In the case of a simple molecule like Cl 2 , for instance, \p would depend on 108 variables.It is this multiplicity of variables which makes it so hard for physicists to interest mathematicians in some problems they would like very much to have solved.Now not all the solutions of the wave equation are of interest, but only those which can be physically interpreted.This requirement of physical interpretability places certain auxiliary conditions on the functions \//, often merely that they be quadratically integrable or that they remain bounded.Between the physically interprétable wave functions and the stationary states in which the system may be found there is a one-to-one correspondence.Thus in many cases physically satisfactory solutions of the wave equation exist only for certain discrete values of E, and in such cases one finds that the physical system also can possess only these discrete energies.The physically satisfactory solutions \p of such an equation and the 194i]
from the Hamiltonian function üof classical mechanics by the operator substitution h d liri dqiwhere h is Planck's constant.For instance, for a hydrogen atom with fixed nucleus H is the sum of the kinetic energy of the one electron and the potential energy of the electron with respect to the nucleus :The Hamiltonian operator correspondingly involves the Laplacian operator in a term representing the kinetic energy of the electron and a scalar multiplier representing the potential energy:The wave equation is thenIn these equations £ is a number, the energy of the system, and *, the corresponding solution, is the wave function which describes the state of the system when it has that energy, x// is obviously a function of all the spatial coordinates needed to define a configuration of the system.In the case of a simple molecule like Cl 2 , for instance, \p would depend on 108 variables.It is this multiplicity of variables which makes it so hard for physicists to interest mathematicians in some problems they would like very much to have solved.Now not all the solutions of the wave equation are of interest, but only those which can be physically interpreted.This requirement of physical interpretability places certain auxiliary conditions on the functions //, often merely that they be quadratically integrable or that they remain bounded.Between the physically interprétable wave functions and the stationary states in which the system may be found there is a one-to-one correspondence.Thus in many cases physically satisfactory solutions of the wave equation exist only for certain discrete values of E, and in such cases one finds that the physical system also can possess only these discrete energies.The physically satisfactory solutions \p of such an equation and the 194i]
Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation, (−2mℏ2∇2+v)Ψ=EΨ,ℏ=2πh, or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression H=−2mℏ2∇2+V; h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and 2m−ℏ2∇2 the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.
Abstract In a previous paper the theory of perturbations was discussed for system which possess only discrete spectra. The theory is now extended to systems with both discrete and continuous spectra, and in addition a slightly more general method of perturbation is considered. The Hamiltonian H of the perturbed systems is split up into three parts H0, H1, H2 (t), two of which can be chosen arbitrarily. The solutions of the perturbed problem can then be worked out in terms of the characteristic functions corresponding to the Hamiltonian H0 + H2 (T), where T is any arbitrary fixed time. In general, however, the series of perturbations does not converge, and different modes of splitting up the Hamiltonian will lead to different formal results. The perturbation theory does usually give the correct physical results, and it is therefore necessary to give some explanation of how this happens. In (I) the asymptotic nature of certain solutions was emphasised, but this does not extend to the more general systems treated here. It is suggested that the usual boundary conditions do not give an adequate description of any but the simplest atomic problems, and that more detailed restrictions, determined by the experimental conditions, ought to be substituted. As this is impossible in practice, an alternative method is to regard the motion as a perturbation on that Hamiltonian H0 + H2 (T) which most nearly corresponds to the experimental conditions. (For example, we may regard an electron as free or bound according as the experiment determines the number of free or bound electrons respectively, whereas in the usual theory the initial Hamiltonian is taken as fundamental.) This criterion, together with the initial conditions, suffices to fix the division of H into the three parts H0, H1, H2 (t) which at first were chosen arbitrarily. The formal solution can be worked out on this basis, and it is then assumed that this solution is asymptotic to the one which would be obtained if the full boundary conditions were used. This suggestion is in accordance with the fact that a hydrogen atom in an electric field gives rise not only to the usual Stark effect, but also to an ionisation effect. In § 2 the perturbation equation are derived in a slightly more general form than has previously been given. The theories of Born,* Dirac and Oppenheimer are particular cases of that given here, and are obtained by particular choices of H0, H1, H2 (t). To obtain Born’s theory we take H1 = 0, and for Oppenheimer’s we take H2 (t) = H2 (T) and T = 0. In Dirac’s theory we have H1 = 0, H2 (0) = 0 and T = 0. Each of these theories is appropriate for the solution of special problems, but the more general method of this paper is required for the full discussion of the perturbation theory.
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.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
The study of the behaviour of microscopic systems using Schrodinger's wave equation is a branch of quantum mechanics known as wave mechanics. This approach is inadequate to deal with all problems of interest, but a remarkably wide range of processes can successfully be investigated solely with the theory. The physics of a particle which moves in a static potential is of particular interest. Classically, the energy of such a particle is conserved. The wave function is reminiscent of the classical problem of standing waves on a stretched string with fixed endpoints, with the stationary states corresponding to the normal modes of vibration. Just as there are discrete frequencies of vibration on the string, so there are discrete energy levels. The energy, which is a continuous parameter in classical mechanics, is 'quantized' to certain discrete energy levels.