matrix1.m

求解量子力学的薛定谔方程

%> The program <matrix1.m> solves for approximate eigenvalues and eigenfunctions 
%> using the matrix diagonalization methods built into MATLAB.
%> Hamiltonian = kinetic part + potential defined in [0,1] +
%> BC: u(0) = u(1) = 0.
%> The potential is chosen from a set defined in <pot1.m> using <choice.m>, 
%> and the strength of it is defined by a multiplier. The multiplier must be 
%> chosen large enough in order to get interesting effects. 
%> Recall the scale of the spectrum of a particle confined to [0,1] is 
%>  E = (n*pi)^2/2,  approx = 5, 20, 44, 79, 123, 178..
%> The basis set are harmonic functions in [0,1] adapted to the BCs, you
%> choose the number of them used, say from 50 up to 150 depending on your PC. 
%> The matrix elements of the kinetic energy are defined analytically, 
%> those of the potential are calculated using the FFT built into MATLAB.
%> Then the eigenvalues and eigenfunctions of the resulting finite matrix are
%> found using the MATLAB <eig> algorithm and displayed graphically.
%> POSSIBLE ERRORS: Too few  basis functions will result in easily observed   
%> errors in the order of the eigenfunctions: the well-known relation between 
%> the number of nodes and the order of the eigenvalue can then fail.
%> The so calculated eigenfunctions can then be used to find the evolution of
%> an initially Gaussian wave packet.
%> The resulting evolution will be interesting
%> if the potential and velocity are suitably chosen.
%> ================================
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