wkb方法介绍PPT优秀课件

1、1 WKB Reading Materials: 曾谨言，量子力学（曾谨言，量子力学（II）第三版，）第三版，2.4 苏铷铿，苏铷铿，量子力学量子力学 第三讲第三讲 2 bound-state e Its principal nergies applications for us will be in calculating and through potentunneling ratetial basrriers. The WKB approximation V (K.E.P.E.) E 1. Classical region ( ), with 2) ) ( ( / ikx xAekm E

2、 EV V 2. Nonclassical region () (tunneli ( ), with 2 () ng regio / n) x xAem EV VE 3. Turning point ()EV In U.S. its WKB (Wentzel, Kramers, Brillouin) In Holland its KWB In France its BWK In England its JWKB (for Jeffreys) constant& constanAt lo:tcal VAk or constant& constanAt lo:tcal VAk or 3 Class

3、ical region 22 2 22 22 2 2 ( ) ( ) 2 , where ( )2( ) ( ): amplitude In general: ( ), where ( ): phas ( The Schrodinger equation Substituting (2) into (1 ) e ( ) ) 2 xi A d V xE m dx dp p xm EV x dx A x xe x p A x iAiAA 2 2 2 2 2 2 ( ) 2 Putting the solutio ()0 ( ) 1 ( ) ( ) ( ) ( )( ) ns to (2) i p

4、x dx A C A A p AA p x dx CC xex p xp x If ( ) is not a constant, but instead varies very slowly on a distance scale of , then it is reasonable to suppose that remains practically sinusoidal, except that twave WKB leng methods: he and thampli V x 2 2 2 change slowly with . Both ( ) tude and x Ap A .

5、(2) . (1) 4 Example for Classical region ( )( ) 12 12 0 2 1 ( ) ( ) 11 ( )sin ( )cos ( ) , with ( )( ) ( ) some specified function, if 0, : ( ) , otherwise. (0), (0)00 ixix x xC eC e p x xCxCxxp x dx p x xa BCV x C 222 2 00 If ( )0 at 0 , putting the solutions to (3) (4) 2 ( )sin ( )0( ) (1,2,3, .)

6、( )2( ) ( ). ( )sin . () . 2 . aa n n n n V xxa pan n aaann np x dxm EV xdx x E xx aa n E ma . (3) . (4) E E 5 Example,potential for harmonic oscillator， how about the energy for this system? 22 1 ( ) 2 V xx The coordinate of Turning points: 22 2 2 1 (42) 2 2/ . . 2/ (43) xE xE i ebaE 1 22 E/=() (44

7、) 2 a b pdxpdxnh 1 E=() (45) 2 n 6 Nonclassical (tunneling) region 1 ( ) Keeping the same definition of ( ), it is now not real but . A similar set of manipulations leads to the solution: ( im ) ( ) aginary p x dx p x C xe p x 00 0 11 ( )( ) 1 ( ) 2 2 2 0 ( ), whth 2/ ( ) ( ) ( )( At 0: At : At tunn

8、eling region (0): . Tran ) 1 , with ( )s Ex missio ample pr no xx a ikxikx ikx p x dxp x dx p x dx a xAeBekmE xFe CD xee p xp x x xa x F e A F Tep x dx A a bability 7 Gamows theory of alpha decay repulsive electr nuclea ice fo r r f c orce e 2 02 12 4r E Ze 2 1 22 11 2 2 2 2 2 0 12 1 21212 2 21 2 12

9、 1 , with ( ), ( )2( ) 1122 21 4 2 sin() , where let sin 2 Typically, 2 2 2 r r rr rr F Tep x dx p xm V xE A rZemE mE drdr rr rmE rr rrrru r mE rr rrr 2 1/2 1 0 121 2 1/2 2 0 2 1.98 MeV 4 , where 4 1.49 fm 4 em K Z KKZr E em K An alpha particle contain Two protons & neutrons unknown 8 1 1 2 1 Averag

10、e velocity of alpha particle: 2 Average time between with the : Frequency of collisions: 2 The probability of escape, per unit time: 2 The lifetime of the pa collisions rent nu wall v r v v r v e r 2 1 2 cleus: r e v 238236 9 6 10 yrs UU 10 note: 1 MeV= 9.7 10 J/molE 9 The connection formulas ( ) 1

11、( ) WKB approximation: ( ) for Classical region ( ) ( ) for Nonclassical region ( ) i p x dx p x dx C xe p x C xe p x 00 00 ( )( ) 11 ( )( ) if 0 1 , ( ) ( ) 1 , ( ) if 0 xx xx ii p x dxp x dx p x dxp x dx AeBe p x x CeDe p x x x ( )2 ( )0(if0) p xm EV xxx 10 Patching wave function . ( . ) (0) strai

12、ght line equ ation V xEVx yabx 2 2 2 1/32 3 22 2 2 (0) 2 2 , where (0 Directly solved by the Schrodinger eqat ) let . Airys equa ion tio S : olution n ( )( s ) p pp p p p p p d EVxE m dx d m xV dx zx d z dz abAi zBi z 11 Airy function: 3/ 23/ 2 3/23/2 1/4 22 ()() 33 1 1 /4 1 22 Solutions 122 sin()co

13、s(), 3434() ( ) 1 2, 2( ( )( ) if 0 ) if p p xx Ai zBiab xx x x z x ee x a bx b a 0 12 00 0 3/ 23/ ( )( ) 1 ( ) 22 ()() 33 3/41/4 1 , ( ) ( ) 1 , ( ) 1 () ( Indirectly solved by WKB approximation: if 0 i0 ) f xx x ii p x dxp x dx p x dx ixix AeBe p x x De p x AeBe x x x x 2 3/ 2 2( ) 3 3/41/4 , 1 if

14、 , 0 if 0 x x D x x 1/3 3/2 2 0 3/2 3/23/2 00 2 ( )2 ()2 (0) ), where (0) 2 ( )(), if 0 3 2 (if ) 3 0(), x xx m p xm EVm EEVxxV p x dxx p x dxxdxx x x 13 2 2 2 2 21 sin( ) For turning poin , if 4( ) ( ) 1 ex t region: p( ), if ( ) x x p x x D p x dxxx p x x D p x dxxx p x Connection formulas ( ) 1 ( ) for Classical WKB approximation: ( ) ( ) ( ) ( region for Nonclassical region ) i p x dx p x dx D xe p x D xe p