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MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1

Tham khảo tài liệu 'mathematical method in science and engineering episode 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 662 GREEN S FUNCTIONS AND PATH INTEGRALS We can write the free particle propagator in terms of the phase space path integral as K x t XQ to N fjpfJxexp I dr J c xo to x t I J to 20.167 After we take the momentum integral and after putting all the new constants coming into D. Equation 20.167 becomes K x t Xo to N i Zhrexpf i dr f mi2 ị . 20.168 Jc x0 t0 x t I Jto 2 J We can convert this into a Wiener path integral by the t it rotation and after evaluating it we return to real time to obtain the propagator as K x t Xo to . .-1. exp ị m x x i2 20.169 v 7 ự 2ivih t-t0 m h 2 t-to v We conclude by giving the following useful rules for path integrals with N 4- 1 segments Eq 20.15 For the pinned Wiener measure I oo 7 00 f J 7V-M dwX T dX1 dXN exp 1 E Xỉ 47rJys J oo J OQ 20.170 For the unpinned Wiener measure 1 oo oc I 1 JV-l-1 L exp t ifc g Ii - 1 rdo . 20.171 J V Ddr 47r e Ar 1 2 7-00 7-00 Also t 1 N l dTX2 r - Y2 i Xi -1 2 20.172 0 zi yi JV l X N l I dTV r e 2 v 20.173 0 i i i i PROBLEMS 663 Problems 20.1 Show that . 1 f x Xo 2 W x t Xo to . exp T v y 4nD t - to I 4Z i - to satisfies the normalization condition ZOO dxW x t Xo to 1. 20.2 By differentiating both sides with respect to t show that the following equation is true 20.3 Show that V rr in Equation 20.64 dW x t xo to d2W x t-x0 t0 . X ------dt--------D-------s------- y w t o to is defined as V x -L-F2 x 4r 2D 2t dx 20.4 Show that the propagator W x t Xo to 1 exp ỵ 4iĩD t to 1 4D t to satisfies the ESKC relation Eq. 20.14 . 20.5 Derive equation WB x t xo to W x t xo to N poo 2 dxjW x t Xj tj V Xj tj W xj tj Xo to NN poo poo 7 2V 22 I dXj dxkW x t Xj tj V xj tj W xj tj Xk tk 2 J-00 xV xk tk W xk tk xo to 4---. 664 GREEN S FUNCTIONS AND PATH INTEGRALS given 20.6 integral in Section 20.3. Using the semiclassical method show that the result of the Wiener W x t XQ to is given as 1 1 2 iv x t.ỉ o. o r _ . L27T Dsinh 2fcv 0 t to . f x2 Xr. cosh 2ky D t to 2 0 X exp k------ _ 7 -------------- ì 2ựỡsinh 2fcựD t - to 20.7 By