Partial differential equation eigenfunctions

In section 7.1.7, eigenfunctions and eigenvalues were obtained numerically. This method is very general and can be used to avoid the use of complicated special function solutions. In section 7.1.8, the separation of variables method which was illustrated earlier for parabolic partial differential equations was extended to elliptic partial differential equations. A total of fourteen examples were presented in this chapter. 

The fourth and final step in the stability analysis is the reduction of the linear system of partial differential equations to a system of ordinary linear differential equations, the solution of which, subject to the appropriate boundary conditions, yields the eigenfunction

associated complex wave velocity c. 

For separable systems the Schrodinger equation, represented by a partial differential equation, is mapped onto uni-dimensional differential equations. The eigenvalues (separation constants) of each of the 3 uni-dimensional (in general n uni-dimensional) differential equations can be used to label the eigenfunctions (r) and hence serve as quantum numbers. Integrability of a n-dimensional Hamiltonian system requires the existence of n commuting observables O/, 1 i in involution ... 

The wave functions rp (r) are called the eigenfunctions of the Schrodinger equation. The term originates from the German adjective eigen (own). The time-independent Schrodinger equation (2.34) does not contain the imaginary unit i. In one-dimensional cases it involves only one independent variable x and is an ordinary differential equation. Three-dimensional version involves more independent variables and is therefore a partial differential equation. 

The time-independent Schrodinger equation in Eq. (2.5) is a second-order partial differential equation. However, it can also be interpreted as an eigenvalue equation. The time-independent wavefunctions (° ( Rif, r) ) are then the eigenfunctions of the Hamiltonian with the energy as eigenvalue. 

The earliest solution of the partial differential equation was given by D Alembert for the case of a vibrating string in 1750 [16]. At the same time, Bernoulli found a solution that was quite different from D Alembert s solution. Bernoulli s solution is based on the eigenfunction and is comparable with the Fourier series. 

The reason total twist is of interest is that the twist distribution w s) at t = 0 resembles a constant plus sinusoid from floor to ceiling. This is the slowest-evolving eigenfunction of the partial differential equation used by Pertsov et al. [46] to model the evolution of twist, and as such it decays exponentially, everywhere by the same fixed fraction per unit time. Thus the total twist also decays by the same fraction per unit time, and it suffices to measure that fraction. (Except in the limits of long time and small twist, which seem not to apply to this experiment, this logic is spoiled if the initial conditions are not a single clean sinusoid or if the approximate dynamical equation is not right, as maintained by Keener [35] and Keener and Tyson [39]... but these are refinements for a later experiment.)... 

This reduces the Schrodinger equation to = 4/. To solve the Schrodinger equation it is necessary to find values of E and functions 4/ such that, when the wavefunction is operated upon by the Hamiltonian, it returns the wavefunction multiplied by the energy. The Schrodinger equation falls into the category of equations known as partial differential eigenvalue equations in which an operator acts on a function (the eigenfunction) and returns the... 

The last stage is to find the eigenfunctions (x) and the corresponding complex wave velocities c. During this procedure the linear system of partial differential derivatives is reduced to a linear system of ordinary differential equations. 

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