FI 3103 Quantum Physics

Transcription

1 7//7 FI 33 Quantum Physics Axan A. Iskana Physics of Magntism an Photonics sach oup Institut Tknoogi Banung Schoing Equation in 3D Th Cnta Potntia Hyognic Atom

2 7//7 Schöing quation in 3D Fo a 3D pobm, w xtn th psntation of th wav function an potntia function of th D cas into 3D as x x, y, z,, V x V V x y z An th 3D ina momntum opato bcoms p x p px py pz m m m m x m m x y z So that th Tim-inpnnt Schöing quation in 3D is givn as m V E Ĥ E Axan A. Iskana Schoing Equation in 3D 3 Schöing quation in 3D Intptation of wavfunction:, t) 3, t) pobabiity nsity at pobabiity p unit voum) pobabiity of fining patic in a voum mnt cnt on Axan A. Iskana Schoing Equation in 3D

3 7//7 Hamitonian fo a hyognic atom H-atom is ou fist xamp of th 3D Schöing quation. In a hyognic atom o ion with nuca chag +Z th is th Couomb attaction btwn cton an nucus. This has sphica symmty potntia ony pns on. This is known as a CENTAL POTENTIAL. - Z V) +Z Th Hamitonian opato is ˆ Z H m Axan A. Iskana Schoing Equation in 3D 5 Hamitonian fo a hyognic atom Th natua cooinat systm is sphica poas. In this cas th Lapacian opato bcoms Lˆ Thn th Hamitonian bcoms ˆ ˆ Z L Z H m m m An TISE fo H-ik atom is Ĥ E o m = m fom now on) ) Lˆ ) Z ) E ) m m Axan A. Iskana Schoing Equation in 3D 7 3

4 7//7 Th angua wavfunction In th sphica cooinat systm, this suggsts w ook fo spaat soutions of th fom ) ) ) Y m ) Th angua pat a th ignfunctions of th tota angua momntum opato ˆL. Ths a th sphica hamonics, so w aay know th cosponing ignvaus an ignfunctions Lˆ Y, m Y, z m m ˆ m, m, L Y Y = obita quantum numb. m = magntic quantum numb + possib vaus). Not: this agumnt woks fo any sphicay-symmtic potntia V), not just th Couomb potntia. Axan A. Iskana Schoing Equation in 3D 8 Th aia quation Substitut spaat soution into th tim-inpnnt Schöing quation Lˆ Z ) E ) m m Using spaation of vaiabs mtho, w obtain th quation fo th aia pat as o,,, ) ) Y m, ) ) Z E m m m Z ) ) E m Axan A. Iskana Schoing Equation in 3D 9

5 7//7 5 Th aia quation Intouc th paamt Th abov quation cou b cast as wh, Th soution of th abov quation is obtain ik pvious xamp, i.. by fist fining th asymptotic bhaviou ag as w as sma ). Axan A. Iskana Schoing Equation in 3D m E 8 ) ) E m Z Th aia quation Th bhavio of ) if >> : Thus, w wit th soution fo ) as Axan A. Iskana Schoing Equation in 3D o ) )

6 7//7 6 Th aia quation With, yis th quation fo ) as Th bhavio as << is govn by th quation With soutions Axan A. Iskana Schoing Equation in 3D ) ) o Th aia quation Thus th gna soution fo ) is wittn as Substituting, yis th govning quation fo H) as Whos soution can b obtain as a sis soution. Futh, to hav a convgnt soution, = n + + ) has to b an intg, an th soution is known as th Laug poynomias. Axan A. Iskana Schoing Equation in 3D 3 H H H H

7 7//7 Th aia soution ) pns on n an but not on m ) ) ) H ) Th associat Laug poynomias n ) L ) n ) Axan A. Iskana Schoing Equation in 3D Th aia soution n ) pns on n an but not on m 3/ Z ) xp Z / a) a 3/ Z Z Z ) xp 3 a a a 3/ Z Z Z ) xp a a a 3/ Z Z Z 3 ) xp 7 3a a 3a 3/ Z 3) 9 3a Fo atomic units st a = Z Z Z xp 6a a 3a 3/ Z Z Z Z 3 ) xp 3a 3a 7a 3a Axan A. Iskana Schoing Equation in 3D 5 7

8 7//7 Hyognic Soution Th wav function soution is obtain as with, An th ngy associat with this wav function is Z m Z) E mc ) E wh,,, ) ) Y, ) n ) L ) n ) is ca th fin stuctu constant. n m c 37 Axan A. Iskana Schoing Equation in 3D 6 Engy Spctum an Dgnacy ca that w foun fo a Hyognic atom, th ngy is givn by mc Z E ) n n wh w intouc th pincip quantum numb n as n n with,,, n n,, n k,, Thus, th goun stat, n = has ony on possibiity n Whi, th fist xcit stat, n = has two possibiitis n, an n, Axan A. Iskana Schoing Equation in 3D 7 8

9 7//7 Engy Spctum an Dgnacy Axan A. Iskana Schoing Equation in 3D 8 Engy Spctum an Dgnacy n = 3 n = n = n = Axan A. Iskana Schoing Equation in 3D 9 9