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Computational Physics - M. Jensen Episode 2 Part 5

Tham khảo tài liệu 'computational physics - m. jensen episode 2 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 13.4. PHYSICS PROJECTS BOUND STATES IN MOMENTUM SPACE 249 First we need to evaluate the integral over p using e.g. gaussian quadrature. This means that we rewrite an integral like r. . . . . . . . _ where we have fixed N lattice points through the corresponding weights . and points. . The integral in Eq. 13.48 is rewritten as I 13.49 . . . - We can then rewrite the SE as e . . 2 Ă . ộ k JcUjPjV k pjyộ j j EýỌĩ . 13.50 m K j l Using the same mesh points for Z as we did for p in the integral evaluation we get pLz X . 2 A TrZ x x _z x h- m TĨ j i with i j 1 2 . N. This is a matrix eigenvalue equation and if we define an N X N matrix H to be rr pL . 2 2 X . . where Sij is the Kronecker delta and an N X 1 vector 13.51 13.52 13.53 we have the eigenvalue problem Ety. 13.54 The algorithm for solving the last equation may take the following form Fix the number of mesh points A . Use the function gauleg in the program library to set up the weights . and the points Pi. Before you go on you need to recall that gauleg uses the Legendre polynomials to fix the mesh points and weights. This means that the integral is for the interval -1 1 . Your integral is for the interval 0 oo . You will need to map the weights from gauleg to your interval. To do this call first gauleg with a 1 b 1. It returns the mesh points and 250 CHAPTER 13. EIGENSYSTEMS weights. You then map these points over to the limits in your integral. You can then use the following mapping . . f71 - . xl Pi const X tan Í - 1 Xi and . . Wị 0 i const ----- ------------T-. . . const is a constant which we discuss below. Construct thereafter the matrixH with -ipiPj UPj -Pi 2 p2 We are now ready to obtain the eigenvalues. We need first to rewrite the matrix H in tri-diagonal form. Do this by calling the library function tred2. This function returns the vector 7 with the diagonal matrix elements of the tri-diagonal matrix while e are the non-diagonal ones. To obtain the eigenvalues we call the function tg 7 On return the .