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Handbook of mathematics for engineers and scienteists part 188. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | T8.1. Parabolic Equations 1277 2 . In the cases where the eigenfunctions Gn x form an orthonormal basis in L2 R the solution of the Cauchy problem for Schrodinger s equation with the initial condition w f x at t 0 T8.1.10.2 is given by w x t G x t f d G x t V n x n exp f-- - . J-oc h J n 0 Various potentials U x are considered below and particular solutions of the boundary value problem T8.1.10.1 or the Cauchy problem for Schrodinger s equation are presented. T8.1.10-2. Free particle U x 0. The solution of the Cauchy problem with the initial condition T8.1.10.2 is given by w x t - i exp 2 VinT j_œ x - G 2 4iT f G dG ht 2m f e G if a 0 Via I e G E if a 0. T8.1.10-3. Linear potential motion in a uniform external field U x ax. Solution of the Cauchy problem with the initial condition T8.1.10.2 w x t - exp -ibTx-3ib2T3 i exp 2v inT J- x x bT2 - G 21 -------t------- f G dG 4iT ht 2am 2m h 1 T8.1.10-4. Linear harmonic oscillator U x mw2x2. Eigenvalues En hw n 2 n 0 1 . Normalized eigenfunctions 2n x 1 4 L eM- 2 2 Hn G x0 n1 2nn x0 2 x0 V mw where Hn G are the Hermite polynomials. The functions v . x form an orthonormal basis in L2 R . T8.1.10-5. Isotropic free particle U x a x2. Here the variable x 0 plays the role of the radial coordinate and a 0. The equation with U x a x2 results from Schriidinger s equation for a free particle with n space coordinates if one passes to spherical cylindrical coordinates and separates the angular variables. The solution of Schrodinger s equation satisfying the initial condition T8.1.10.2 has the form . exp-2in 1 sign t w x t ---1 2 t f vxy exp J f Ç f y dy 2 T ht 2m v 2m 1 -1 where J G is the Bessel function. 1278 Linear Equations and Problems of Mathematical Physics T8.1.10-6. Morse potential U x U0 e 2x a - 2e x a . Eigenvalues En -U0 1 - 1 n 1 p 0 n p - 2. P 2 ft Eigenfunctions n x .e 2 -n 2s 1 2e s V-2mEn ft where a b is the degenerate hypergeometric function. In this case the number of eigenvalues energy levels En and .
