Discretizing the 3-D Schrödinger equation for a Central Potential

Transcription

1 Discetizing the 3-D Schödinge equation fo a Centa Potentia By now, you ae faiia with the Discete Schodinge Equation fo one Catesian diension. We wi now conside odifying it to hande poa diensions fo a centa potentia. The ethod wi be the sae, with one ati pobe pe diension. Fo a centa potentia one that ony depends on distance, but not diection, it is convenient to phase the Schödinge equation in tes of spheica coodinates. Giffiths section 4. waks us though that pocess. The thee esuting equations ae d d adia: d d d d Angua: d d d Aziutha: d E 0 a a 3 The thid eation is tivia, and thee s no point in esoting to coputationa eans fo soving it. So we focus on the othe two. The one petinent esut fo anayzing equation 3 is that its soution coes and es dictates that be an intege so the soutions ae ge vaued fo a ge diection in space that is, to ensue that and etun the sae vaue. The Angua Equation The second equation can be ewitten as d cos d d d 0 Then we can ewite the deivatives as finite diffeences to give cos. 0 The appoiation fo the fist-ode deivative beas a itte epanation. In the iit, this is of couse equivaent to the taditiona way of witing the deivative,

2 d d 0 i ; howeve, the way I ve witten it,, has the advantage it gives the aveage sope stadding the point of inteest at ange athe than the aveage sope eading up to the point of inteest at ange / which is what the coon epession gives. A itte eaanging and egouping gives b O phased in tes of subscipted anges,. c Modifying you Poga You ay ecognize that this has the sae basic fo as the finite-diffeence appoiation you d used fo the -D Schodinge equation, specificay, E. So it takes vey itte wok to odify you eisting DiscetePIB.py poga to find the angua coponent of the wave function. I suggest saving a copy of the poga as DisceteAnge.py, and then aking the foowing odifications. Constants and Finite Steps In the oigina vesion, was neve epicity defined, but now you wi want to define a. So N steps take you fo = 0 to, you want dq = *pi/n

3 You aso want to define, which Giffiths notes ust be an intege positive o negative. Haitonian Mati Tes Whee you define the tes in the Haitonian ati, H, H[,-] becoes - + dq/*tan+0.5*dq H[,] becoes + *dq/+0.5*dq** And H[,+] becoes - - dq/*tan+0.5*dq Adding 0.5 to offsets the ange of anges you evauate by / and, in the pocess, ensues that you won t ask the copute to evauate /tan0, /tan, o /tan which bow up. Potting Eigenvaues In you od vesion of the poga, you d potted the eigen vaues against the inde nube to see if they vaied ike fo the patice in the bo o ike fo the sipe haonic osciato. Coputationay soving the angua equation, 4b, gives eigen vaues fo + the facto on the ight of equation 4b. eaanging that, 4eigenvaue / / The anaytica wok that Giffiths does tes us that ust be a positive intege and. So, you shoud pot this epession fo against the inde,. To see if the vaues it finds ae indeed vey appoiatey integes, at east fo the owe vaues of eebe, the siuation beaks down fo when unduations in the eigen functions get as sa as the step size,. The foowing ines wi ceate the ist of vaues that you siuation etuns. Ls =zeosn fo in aange,n+: Ls[-] = sqt+4*vaues[-]/dq**-/ As Giffiths points out on the botto of page 37, thee ae actuay two independent eigen functions fo each eigen vaue though ony one is physicay eaistic, so you shoud epect to see two instances of each vaue. Potting Wave functions

4 athe than potting wave functions against, it wi ake oe sense to pot against the anges. If you define a ist of the anges as Qs = 0.5+aange0,N-+0.*dQ Then you can use Qs as the -vaues when you pot. Again, keep in ind that ony eveyothe soution to the equation shoud be physicay eaistic. Eecise : Afte aking the above odifications, satisfy yousef that the siuation is woking popey: the L vaues shoud be integes fo at east the owe quate of states, and thee shoud be two states fo each L vaue note: the poga wi not find the tivia, =0 soution; each soution s vaue shoud coespond to the nube of waveengths wapping fo = 0 to. The adia Equation The adia equation, a, can be ewitten as E d d d d. Appying the finite-diffeence appoiation, it becoes E. Gouping ike tes gives E b The soutions to this eation shoud we appoiate the anaytica soutions as ong as we espect its fundaenta appoiation: that D is infinitesia. So, we shoud avoid <D and we shoud keep. Modifying you Poga eation b has the sae genea fo as the -D equation and as the adia equation, c. So odifying DisceteAnge.py to becoe Disceteadia.py shoud be staight fowad.

5 Constants and Finite Steps You want to define constants hba and now, epesenting ass athe than the quantu nube, though you ae fee to choose units in which both ae. To ensue that you keep the step size sa enough, I d suggest defining it to be d = 0.0*hba/sqt* With N = 00, you get sooth wavefunctions and easonabe enegies. Equation b assues you aeady have an vaue in hand. So you want to define a constant fo that, and when it coes to unning the poga, you can see how vaying keeping it an intege affects the soutions. Fo a stat, et =0. Haitonian Mati Tes Based on Equation b, odify the eations fo H[,-], H[,], and H[,+]. I ecoend defining = d+*d, whee the d= ensues you don t evauate fo <d o divide by 0 ce equation b has a few tes divided by. Potentia We stat off consideing the sipest potentia, the infinite spheica we of Giffiths Eape 4.. That essentiay has = 0. Potting Wave functions athe than potting wave functions against, it wi ake oe sense to pot against the. So you want to define a ist of adii to be used as you coodinate in you pots: s = d+aange0,n-+0.*d Potting Enegies functions You pobaby aeady have in you poga soe ines fo potting the eigen vaues, though you ay have coented it out o odified it fo the vesion of the poga that handed angua soutions. Get that code woking again.

6 Eecise : Fo = 0, un the poga and convince yousef that it s woking popey. Accoding to Giffiths Eape 4., fo = 0, the enegies scae ike n, ust ike a patice in a -D bo, so you shoud check that that is the case fo the ow-enegy eigen vaues. It aso cais that the soutions have the fo of n/a/ Eq n 4.44; so check that you poga etuns soutions of this fo note: you poga s eigen functions ae not noaized, so they agee with Eq n 4.44 ony to within a utipicative facto. Toube with / potentias At the heat of the finite-diffeence appoiation is the assuption that the fist and secondode deivatives ae faiy constant on the ode of. Unfotunatey, a potentia o even an effective potentia that bows up at 0 esses that up. So this poga wi have toube with > 0 o the Couob, / potentia. You can eediate a itte by inching away fo appying Equation b too cose to = 0, but things won t be pefect. Fo eape, with the infinite spheica potentia with =, you can get quaitativey coect esuts by oving d out to 3.