Quantum Mechanics Lecture Notes 10 April 2007 Meg Noah

Transcription

1 The -Patice syste: ˆ H V This is difficut to sove. y V 1 ˆ H V ˆ = ad with 1 1 Hˆ Cete of Mass ˆ fo Patice i fee space He Reative Haitoia eative coodiate of the tota oetu Pˆ the tota oetu tota p R M ˆ ˆ P e H V( ) M Hˆ Hˆ Hˆ eigefuctio ad eigeeegy e e We eaed peviousy that x, p x i Fo the tasfoed syste: [ ˆ, ] i ad [ Rˆ, Pˆ ] i So the Haitoia is sepaated ito two idepedet copoets. 1 x Whe the Haitoia ca be sepaated ito idepedet copoets: The Schodige equatio has poduct eigefuctios: R ( ) ad suatio eigevaues: whee Hˆ Hˆ Hˆ Hˆ e e e Hˆ e e e e e

2 Fist Sove the Fee Patice () Pat: Hˆ Ae M i R Now Sove the Reative Haitoia with Reduced Mass: ˆ e H V ( ) e this is the educed ass The Schodige equatio is: e ( ) V ( ) ( ) e sig spheica coodiates, we ca wite it this way: e V ( ) ( ) ( ) e V ( ) ( ) ( ) e Lˆ ˆ L Hee is the adia pat of the oetu ad is the otatioa pat. The otatioa pat is idepedet of the othe two tes which ae fuctios of ad ot,. The soutio is spheica haoics: Y,, ( ) R( ) Y,,, e Lˆ V ( ) R( ) Y, R( ) Y, i ( 1) Y, R( ) R( ) Y, V ( ) R( ) Y, R( ) Y, The adia copoet: ( 1) effective potetia V ( ) R( ) R( ) So, we have a poduct of eigefuctios: a adia pat ad a agua pat. The agua pat is sovabe without owig the potetia which depeds oy o. The agua pat ae spheica haoics defied i Tabe 9.1 o page 373.

3 The Reative Haitoia fo a Hydogeic Ato: Hee: V () Ze the Couob potetia. Hydogeic Atos Z H 1 He + Li ++ 3 Oy coside the boud state (owe tha fee patice eegy): 1 ( 1) Ze ( ) R Chage the depedet vaiabe to: ( ) R( ) 1 ( 1) Ze ( ) 1 ( ) ad ( 1) ( ) Ze ( ) ( ) ( 1) Ze a whee Z a ad Z a e ( 1) 1 ( ) ( ) is Rydbeg costat. Sipify the adia equatio: e M e V

4 Loo at the soutio to the adia eative Haitoia equatio i the iits, fist at ± : ( 1) 1 ( 1) ( ) (fiite) 1 Ae B e Ae B has to be zeo because i the iit this te is ifiite Now oo at the soutio to the adia eative Haitoia equatio i the iit at : ( 1) 1 ( 1) 1 ( ) (fiite ube) ( 1) ( 1) Substitute tia soutio qq ( 1) q ( q) ( 1) q( q 1) ( 1) q 1 o q A 1 B So that the soutio vaishes at the oigi B ad Ae q 1

5 Fo these esuts at the iits, we ca buid the eigefuctio of (ρ). ( ) e 1 F( ) whee F( ) c Substitute this ito the Schodige equatio ad sove. ( ) ( 1 ) ( ) F The soutio is: c c c ( 1) 1 ( 1)( ) I the iit that c What satisfies this c eatio? F( ) e 1 1 Coside tota adia soutio fo Cooub potetia eative Haitoia: 1 1! 1 ( ) e F( ) = e e e e c c! 1 The pobe is that as ad, the ( ). Theefoe, ad has to have a cut-off o axiu vaue. The axiu is at the poit whee. ax 1 We defie a ube accodig to this. The picipe quatu ube. 1 1,,3, ax ax ax, i( ) i( ) i( ) 1 ax 1,1,,..., 1 < Z ad 13.6 ev ( ) ( ) e F A e c c c ad ad Z 1 a Puttig togethe the adia ad agua copoets of the eative Haitoia: A,, R ( ) Y, whee R,, Ze Z e ad 13.6 ev (Rydbeg Costat) ax

6 Let s Cacuate the Degeeacy of : 1 ax 1,1,,..., 1 Thee ae vaues of., 1,...,,..., 1, Thee ae ( 1) vaues of Degeeacy of 1 1 ( 1) If you coside spi, the degeeacy is doube: D( ). =+ =+-1 =-1 =-+3 =-+ =-+1 =- D P S = =1 = = =1 = =-1 =- =1 = =-1 = Whe you excite the ato to aothe state, the it ca give ight, the ight wi coe out. Next cass we wi ea about the seectio ues fobidde/aowed, etc. =3 = S P D = =1 = F =3 =1