LECTURE 5 Guassian Wave Packet

Transcription

1 LECTURE 5 Guassian Wav Pact 1.5 Eampl f a guassian shap fr dscribing a wav pact Elctrn Pact ψ Guassian Assumptin Apprimatin ψ As w hav sn in QM th wav functin is ftn rprsntd as a Furir transfrm r sris. This Furir transfrm is tan with rspct t spac nt tim. Th rsulting transfrm rprsnts th magnitud f th spatial frquncis nd t prduc a particl with a crtain wavfunctin. Ths spatial frquncis ar rfrrd t by thir wav numbr. Thus th transfrmd spac is ftn rfrrd t as spac. is rlatd t th frquncy by th rlatinship. πf & * whru is th phas vlcity, i th vlcity ach frquncy cmpnnt mvs. u Thrfr w hav tw distributins f ψ n in spac and n in th frquncy dmain. Oftn in QM ach spatial frquncy can b attributd with a cmpnnt f th particls mmntum p, and p is dfind as p!. Undr ths cnditins th frquncy dmain is ftn calld th mmntum spac. Th tw distributins f ψ basically rval t us that if w masur th distributin f th lctrns mmntum and psitin in spac w will btain many diffrnt valus but that th distributins f ths valus will b dtrmind by ψ() and ψ(p). Thrfr w nly nw what th prbability f th lctrn bing at a particular plac r vlcity and nt whr it is r hw fast it is ging. As an ampl w can us a Guassian lctrn prbability distributin, w can bviusly btain th transfrm f th Guassian, as it is anthr Guassian. Th Gaussian wav pact prducs a wll-lcalizd particl and mathmatically is asy t dal with. It is a gd idalizd mdl f a QM lctrn. Lctur 4: Gaussian Wav pacts Sptmbr 000 1

2 In th SPACE SPACE: / ( ) ψ ( ) A Cntrd at 0, with a width In th P SPACE (as a Furir transfrm f Guassian is a Guassian) ψ ( ) / ( ) i d ψ ( ) i d W btain: ψ ( ) ( ) / ( )! p 1/ p! / Ψ() Ψ(p) F.T. I.F.T. p p p Nw w as what happns if w nw th psitin f th lctrn vry wll. Thn will b vry small and vry larg. In fact if w us a dlta functin fr th psitin in spac ψ() δ() and ψ(p) is a flat cnstant distributin. ψ() ψ(p) p What ds this man? Wll if w nw th psitin f th lctrn vry wll thn w d nt nw th mmntum f th lctrn vry prcisly. On th thr hand if w nw th mmntum (vlcity pmv) f th lctrn wll thn th psitin f th lctrn is nt Lctur 4: Gaussian Wav pacts Sptmbr 000

3 vry wll dfind. This rlatinship is nwn as th uncrtainty law and using th assumptin f a Guassian w find that w can dfin a prduct. 1/ p! / p h Which indicats th magnitud f this uncrtainty. This is th uncrtainty principl it prdicts a limit n th nwability f psitin and vlcity f QM bjcts. As w said bfr th lctrn can b dfind t hav tw vlcitis a grup and phas vlcity (just li an lctrmagntic puls in a wir). Th phas vlcity is assciatd with a particular frquncy cmpnnt f th lctrn and is th vlcity f a mnchrmatic wav, and in fr spac is f λ ω/ Th grup vlcity is th vlcity f th lctrn i.. th cntr f th distributin functin r pact. Just as in an lctrmagntic puls th grup vlcity is diffrnt frm th phas vlcity. If thr is nt a linar rlatinship btwn ω and it can b fund that th grup vlcity f a wav is (as w dfind it bfr), grup ω Fr a singl mattr wav cmpnnt w hav (S slutin f SE fr a fr lctrn): p m E m E! m E! ω! m ω! ω m ( nnlinar, disprsin) Lctur 4: Gaussian Wav pacts Sptmbr 000 3

4 ! phas ω m! grup m Th phas vlcity f th mattr wav is n half th grup vlcity! Th grup vlcity ( grup ) ds nt qual phas vlcity ( phas ) and th wav pact hibits disprsin (s last plt in lctur 4). Rmmbr hwvr that th grup vlcity quals th lctrn s vlcity. ω! E grup! P p m As a cmparisn, cnsidr an quivalnt EM wav in vacuum (this is th wav quatin intrducd in lctur ). * 1 * * E E, E lctric fild vctr µ ε t * Cnsidr 1-D nly E E,0,0), and E E ( (EM wav mving in dirctin) Th trms in th wav quatin rduc t, j(-ωt ) E E t E ω E s substituting in th abv quatin w hav E ω ε µ E Lctur 4: Gaussian Wav pacts Sptmbr 000 4

5 r ω ω - (linar rlatinship n disprsin) µ ε ω 1 µ ε grup C spd f light ω 1 µ ε phas C fr an EM wav in vacuum, phas grup n disprsin, thus, in th parlanc f QM w can say that a light wav pact (phtn) will nt hibit disprsin. Lctur 4: Gaussian Wav pacts Sptmbr 000 5