what is the physical meaning of diagonalization?

Answer 1

in brief, the diagonalized matrix has the eigen values of the original matrix as its diagonal elements; this must be the result that you should get. To diagonalize a given matrix , follow the given steps
1. find the eigen values and eigen vectors.
2. obtain the matrix that has the eigen vectors as columns( let this be matrix M)
3. multiply the original matrix with the matrix obtained from the above steps.( let A be original matrix then this is AM)
4. find M-1(AM)... i.e M inverse AM.
hope this was useful

Answer 2

physically it means that you are choosing a co-ordinate system(normal co-ordinates)where the interactive term in energy(energy due to coupling between different particles)is eliminated in the equations.

Answer 3

Spectacular increases in computer power now provide opportunities to obtain numerical solutions to the three-body problem. A profound understanding of atomic few-body problems such as interference effects associated with resonances, correlation interactions between the charged particles, rearrangement processes, and others can be obtained by using fully quantal nonperturbative methods.
Among the fully quantum nonperturbative theories, the time-dependent close-coupling (TDCC) method has been successfully employed for calculations of electron-impact ionization [1,2,3]. We have recently [4] studied the dielectronic capture into doubly excited resonances within the time-dependent framework. This study required the development of methods for generating accurate wave functions for doubly excited autoionizing states and time-dependent close-coupling calculations of the capture and the subsequent decay of autoionizing states. Since in future applications we wish to consider resonances in the time-dependent framework and to study processes that involve transitions to and among continuum states, we chose to discretize the wave functions and the action of operators which result from these procedures, working therefore in a numerical lattice, thus solving the problem of an ion inside a box.

Solving real atomic-physical problems with discrete numerical methods is an approximation that relies on strong logical grounds. It becomes a very good approximation to the physical bound states for all orbitals that fit well into the lattice (i.e., for such bound states in which the wave function at the boundaries approaches zero in a practical sense for numerical calculations). Use of discrete lattice functions allows the system to be described by the dynamics of always-square-integrable functions. Even if the states can be represented by fully analytical functions (such as the hypergeometrical functions for the continuum Coulomb waves), a proper calculation with these functions could involve a numerical work, and therefore, finite-lattice discretization. The method presented here has many other important advantages. It gives the exact solutions to the problem of an ion confined in a box. If the discretized allowed energy levels are positive, the eigenfunctions are true continuum solutions of the ion, not necessarily confined inside a box. These solutions form a set selected by a particular boundary condition, i.e., these are the particular true continuum wave functions with zero value at the boundaries of the box. All the solutions obtained by direct diagonalization are naturally orthogonal, and most importantly, the set is complete. Finally, the simplicity of the method is expected to lead to a greater understanding of the close-coupling formalism in general.

The problem of a spatially confined system has been a subject of interest in many branches of physics and chemistry since the early years of quantum mechanics. Nowadays, investigations on confined systems in physics (see, for example, [5]) have focused especially on the study of artificial atoms, also known as quantum dots (essentially a number of electrons confined in a potential well). However, an important theoretical motivation for the study of enclosed systems is to understand in detail the electron correlation effects on the properties of those systems.

In this work, we study how to obtain the exact wave functions in two-electron systems by the direct solution of the Schrödinger equation. Two methods are developed for this purpose. The first method consists of the direct diagonalization of the two-electron Hamiltonian on the radial grid. The second method is a constrained relaxation of the wave functions, until they relax on the successive doubly excited levels of He. The relaxation method was currently used in the TDCC method for calculation of the ground and first excited states of ions, and also for calculation of ground and low-lying excited states in Bose-Einstein condensates [6]. However, a particular treatment is needed for calculation of doubly excited levels.

In order to explain the main features of the different methods, a detailed presentation of the theory is given for the spherically symmetric model [7,8] (also known as the Temkin-Poet model or the S-wave model). The remainder of the paper is the following. Section [3] shows the results for the first doubly excited wave functions obtained in both methods. Section [3b] shows how to use the Feshbach projection-operator formalism in order to compare the results obtained in both methods. In Sec. [3c] we propagate the first doubly excited autoionizing level, showing how the autoionization process evolves in time, and we also calculate the autoionization rate from this level by monitoring the autocorrelation function in time.


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