FI 3103 Quantum Physics

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1 10/10/014 FI 3103 Quantum Physics Alxandr A. Iskandar Physics of Magntism and Photonics Rsarch Group Institut Tknologi Bandung 1D Potntials Doubl Potntial Wll Alxandr A. Iskandar 1D Potntials 1

2 10/10/014 Doubl Wll Potntial Considr a systm of two idntical wll sparatd by a distanc that is largr than th width of th wlls. Rcall that in th singl wll cas of th lowst nrgy solution (vn parity), th cosin wav function must match th falling xponntial dcay, thus th cosin function dtrmins th rat of th falling xponntials, which in turn dtrmins th binding nrgy. In th doubl wll cas, th xponntial dcay can fall of mor rapidly and still match with th fairly flat wav function in th rgion btwn th wlls. Fast dcay bcaus of larg binding nrgy Alxandr A. Iskandar 1D Potntials 3 Doubl Wll Potntial Fast dcay bcaus of larg binding nrgy Thus, bcaus of th fast xponntial dcay of th tails, th doubl wll potntial can bind particl mor strongly than th singl wll potntial. This is not bcaus that th doubl wll is somwhat strongr than th singl wll, but bcaus w can maintain an almost constant wav function btwn th wlls. And this mans that th bound particl has a high probability of bing anywhr btwn th potntials. Alxandr A. Iskandar 1D Potntials 4

3 10/10/014 Doubl Wll Potntial Whn th wlls ar far apart, thn a particl that is bound in, say, th right wll dos not know that thr is a wll on th lft, and vic vrsa. If th wav function for this right-boundd particl is u R (, and u L ( for th lft-boundd particl, thn a symmtric and antisymmtric wav functions can b formd as follows u vn 1 N 1 uodd( u R u No with som normalization constants. u u R L L Alxandr A. Iskandar 1D Potntials 5 Doubl Wll Potntial Bcaus thr is a small ovrlap of th wav functions u R (, and u L (, th ignvalus of ths wav functions ar not xactly th sam. Thus, u R (, and u L ( do not rprsnt stationary stats. For xampl, th wav function of particl that is localizd on th right will b approximatly givn by If this is th wav function of th systm at t = 0, thn at latr tim th wav function will b with E E o E u C u u u( x, t) C u C vn ie t vn u ie t vn odd u u odd odd ieot iet Alxandr A. Iskandar 1D Potntials 6 3

4 10/10/014 Doubl Wll Potntial iet Aftr a tim t such that 1, i.. t E th wav function bcoms on that is approximatly localizd on th lft sid. Thus th particl oscillats btwn th two wlls with angular frquncy E E o E This phnomna is obsrvd in th ammonia molcul (NH 3 ), whr th thr H nucli forms a triangular bas of a pyramid and th N nuclus oscillats btwn th two apxs of th pyramid. Alxandr A. Iskandar 1D Potntials 7 Considr th potntial barrir givn by V 0 V 0 ( 0 x a x a x a W considr only th cas E < V 0, thn insid th barrir th Schrodingr quation givs d m u dx Whos solution is givn as u( A x B ( 0 x, E V 0 a a E V u 0 for x a m( E V0 ) x Alxandr A. Iskandar 1D Potntials 8 4

5 10/10/014 And th solution at th outsid rgions ar u( ikx R ikx, for x a ikx u( T, for x a me with k Ths solutions ar similar to th potntial wll cas, with th rplacmnt of q i, thus w can rad off th transmission cofficint as ika k T k cosh a i k sinh a Alxandr A. Iskandar 1D Potntials 9 T ika k k cosh a i k sinh a Which implis that th probability flux is proportional to k T k k sinh a Whn a >> 1, sinh a ½ a, and 4k 4a T k i.. thr is transmission vn though th nrgy lis blow th top barrir, this tunnling is a quantum mchanical phnomna. Not that th wav function dos not vanish insid th barrir, hnc thr is som probability of finding th particl with ngativ kintic nrgy. How is this possibl? Alxandr A. Iskandar 1D Potntials 10 5

6 10/10/014 To ovrcom this paradox, w look at th uncrtainty principl. An xprimnt to study th particl insid th potntial barrir must b abl to localizd with accuracy x a. This masurmnt will transfr to th particl momntum, with an uncrtainty p a which corrspond to a transfr of nrgy E 8ma In ordr to obsrv th ngativ kintic nrgy, this uncrtainty must b much lss than E V 0, thus E m 8ma Which implis a >> 1, undr this circumstancs, th quantity to b masurd T is vanishingly small. Alxandr A. Iskandar 1D Potntials 11 Th approximat ratio of transmittd probability flux with rspct to th incidnt probability is xtrmly snsitiv to th width of th barrir (a) and to th amount by which th barrir xcd th incidnt nrgy, sinc ma a ( V 0 E) 1 Alxandr A. Iskandar 1D Potntials 1 6

7 10/10/014 Irrgular In gnral, th potntial barrir is not squar. Th propr way to study an irrgular shap barrir is with Wntzl-Kramrs-Brillouin (WKB) approximation. Howvr, w wantd to know th transmission probability cofficint T through this irrgular shap barrir. Obsrv, that th approximat xprssion 4k 4a T k consist of a product of a pr-factor and a rapidly dcaying xponntial. W can isolat this rapid variation by writing ln T const a i.. th nrgy dpndnt factor multiplis th width a. Alxandr A. Iskandar 1D Potntials 13 Irrgular A smooth curvd barrir, can b considrd as a juxtaposition of th squar potntial barrirs. Th total transmission probability is a product of all transmission probabilitis of ach squar potntials. Thus, th ovrall transmission cofficint is a product of th transmission cofficint of a singl barrir, ln T slics x n is th slic width and is th avrag valu of for that barrir. Alxandr A. Iskandar 1D Potntials 14 ln T slic x n n n 7

8 10/10/014 Irrgular For a continuous summation, th abov bcoms And hnc, T ln T dx barrir C xp barrir dx m V E m V E Alxandr A. Iskandar 1D Potntials 15 8