27.1 The Heisenberg uncertainty principles

Transcription

1 7.1 Te Heisenberg uncerainy rinciles Naure is bilaeral: aricles are waves and waves are aricles.te aricle asec carries wi i e radiional conces of osiion and momenum; Te wave asec carries wi i e conces of waveleng and frequency. Naure laces naural limis on e recision of our measuremens; some knowledge and informaion forever is srouded from our rying eyes. 1

2 7.1 Te Heisenberg uncerainy rinciles 1. Te osiion-momenum uncerainy rincile Single sli diffracion y of elecrons: r..... θ 1 z a sinθ 1 λ aθ 1 z a θ 1 z 7.1 Te Heisenberg uncerainy rinciles Te osiion uncerainy: Te momenum y uncerainy in y direcion: y y θ 1 y z z θ 1 y a r θ 1 y y If we accoun for e secondary maima y y z z z

3 7.1 Te Heisenberg uncerainy rinciles Te Heisenberg uncerainy rincile saes a ere eiss a fundamenal limi o e een o wic we can simulaneously deermine e osiion and corresonding momenum comonen of a wave-aricle in any given direcion.. Te energy-ime uncerainy rincile Recall maemaical form of a monocromaic wave Acos k ω π Acos πν λ 7.1 Te Heisenberg uncerainy rinciles Subsiuion for λ and ν λ E ν π π Acos E Te role of E and in e equaion are maemaically idenical o e role of and. E 3.Te imlicaions of e uncerainy rinciles 1 e uncerainy rincile indicae a i is no eac describing e microscoic aricle by classical eory. 3

4 7.1 Te Heisenberg uncerainy rinciles e uncerainy rincile gives an limiaion using classical model. Imlicaions of e osiion-momenum uncerainy rincile In classical ysics e osiion and momenum comonen can be measured wi arbirary recision in rincile bu in quanum mecanics is is no sricly rue because of e naure of aricles. Eamle 1:Wy aoms do no collase? Te minimum momenum of an elecron is r 7.1 Te Heisenberg uncerainy rinciles Te oal energy of e elecron is E KE + PE e e m 4πε r mr 4πε r Te value of r minimizes e oal energy saisfied de e dr m r 4πε 0 r r 4πε 0 me I is same order of magniude as e ficiious of e Bor model

5 7.1 Te Heisenberg uncerainy rinciles Eamle : You measure e diameer of a sogun elle of mass 1.0 g o be 1.00±0.01 mm wi a micromeer. Wa is e minimum uncerainy of is momenum if e magniude of is momenum is measured simulaneously? Soluion: kg m/s mv kg m s I is eceedingly small. As a resul for suc a macroscoic aricle e uncerainy rincile as no effec. 7.1 Te Heisenberg uncerainy rinciles Eamle 3: Te acceleraing oenial difference of TV kinescoe is 9kV e diameer of ei of elecron gun is 0.1 mm is i reasonable o describe e elecrons as classical aricles? Soluion: Te uncerainy of e seed m v v 7.7m/s 31 3 m Te seed of e elecrons v 19 3 KE e V m/s 31 m m

6 7.1 Te Heisenberg uncerainy rinciles Eamle 4: Minimum energy of a aricle in a bo zero oin energy Soluion: For a aricle in a bo of leng L av E 0 m ml av Te microscoic aricle canno be res! 7.1 Te Heisenberg uncerainy rinciles Eamle 5: A roon is known o be in e nucleus of an aom. Te size of e nucleus is abou m. Find a. min?; b. v min? For nonrelaiveisic siuaion. c. is i reasonable o eress is momenum in classical eression? Soluion: a. According o e uncerainy rincile kg m/s b. Te minimum seed of e roon vmin 4.0 m/s 7 m

7 7.1 Te Heisenberg uncerainy rinciles c. Te fracion of e seed of lig 7 v c γ v / c I is reasonable o eress is momenum in classical eression. 7.1 Te Heisenberg uncerainy rinciles Imlicaions of e energy-ime uncerainy rincile a. Te mass of fundamenal aricles According o secial relaiviy Eres mc Eres c m According o e energy-ime uncerainy rincile c m Te mean lifeime of a free neuron is 888 s en m c kg 7

8 7.1 Te Heisenberg uncerainy rinciles Te mass of e neuron can be deermined quie recisely because i is a relaively longlived aricle. Te values of e masses of very sor-lived aricles will ave inrinsic sread because of e Heisenberg uncerainy rincile. b. Te naure of a vacuum vacuum is no emy Te idea of a vacuum in quanum ysics is quie differen from classical idea of a vacuum as simly a volume wi noing in i. 7.1 Te Heisenberg uncerainy rinciles Classical mecanics KE E PE 0 m Relaiviy quanum mecanics 4 4 E c + m c E ± c + m c Dirac sea: osiive energy saes are emy negaive energy saes are filled fully. E + mc E0 E + mc E0 E - -mc E - -mc 4 E E+ E c + m c + c + m c 4 8

9 7.1 Te Heisenberg uncerainy rinciles Te quanum mecanical vacuum is a seeing sea of aricle-aniaricle airs called virual aricles since eir eisence is eemeral. Te airs of virual aricles well u ou of noing live for very sor ime and en disaearanniilae eac oer. According o energy-ime uncerainy rincile E For a virual elecron-osiron air E mc s Te Heisenberg uncerainy rinciles c. Te verificaion of eerimen: H. B. G. Casimir effec: Lamb sif: Eamle: Lig of waveleng 63.8 nm is inciden on an eremely fas suer a cos e beam ino ulses. Te suer says oen for only s. Wa is e aroimaely minimum range of wavelengs λ in e lig ulses a ass e suer? 9

10 7.1 Te Heisenberg uncerainy rinciles E E.66 E since erefore Eλ λ c c c E ν E λ λ λ J m 7. Paricle-waves and e wavefuncion 1. Te srange beavior of microscoic aricles

11 7. Paricle-waves and e wavefuncion We will ge one aern on e disan screen if e aricles go roug a single sli bu a comleely differen aern if a e aricles go roug wo or more slis. Tus we canno ink of e aricles as going roug eier on e sli or e oer in e double slis arrangemen. Te idea of a aricle as a discree billiard ball eniy mus be abandoned in favor of some absrac aricle-wave wic can diffrac and inerfere wi i self. 7. Paricle-waves and e wavefuncion Dislay e roery of aricle resrain e roery of wave. Dislay e roery of wave resrain e roery of aricle. 11

12 7. Paricle-waves and e wavefuncion. Te rincile of comlemenariy Microscoic aricles roagae as if ey were waves and ecange energy as if ey were aricles a s e wave-aricle dualiy.wen we measure e arrival of one of em a a deecor we always measure e energy i delivers o a single oin even oug a oin may be ar of a aern a could only be creaed by a wave. In eerimenal even suc eniies of wave do one or e oer ey canno simulaneously manifes e roeries of wave and aricle. 7. Paricle-waves and e wavefuncion Classical aricles ave recise a of moion. Quanum aricle ave no a of moion. 1

13 7. Paricle-waves and e wavefuncion 3. Wavefuncion How are e aricle-waves o be reresened in a maemaical cone by a wavefuncion? Wa equaion governs e roagaion of e aricle-waves? Te wavefuncion of an individual aricle raveling in one dimension in free sace. Acos k ω wen 0 A cos k 7. Paricle-waves and e wavefuncion Te robabiliy of finding a aricle in sace for a lane monocromaic wave sould no deend on e value of. i k ω Ae A[cos k ω + i sin k ω] i k ω i k ω * A* e Ae A* A Te wavefuncion waever maemaical form i akes in a aricular cone yically is a comle-valued funcion. 13

14 7. Paricle-waves and e wavefuncion 4. Te roeries of wavefuncions 1Te sandard condiion of wavefuncions: Single valued coninuous and finie. Normalized condiion: y z * y z dv 1 For one-dimension * d 1 Quaniy * d is usefully defined as e robabiliy of finding e aricle in a region of sace beween and +d a a aricular ime. is known as robabiliy amliude. 7. Paricle-waves and e wavefuncion Noice: Since e wavefuncion is a comlevalualed funcion iself is no a ysical quaniy a can be observed direcly in any eerimen; measuremens in eerimens always yield real numbers. In e quanum domain we lose e abiliy o redic wi cerainy wa aens o individual aricles suc as elecrons. 14

15 7.3 Oeraors and e Scrödinger equaion 1. Oeraors A maemaical oeraor is a symbolic way of reresening a maemaical oeraion. Noice: Some oeraors do no commue and bu cerain oeraors do commue. and 7.3 Oeraors and e Scrödinger equaion 1Te momenum oeraor i k since [ Ae π π k λ en i i i i ω ] ik --momenum oeraor 15

16 7.3 Oeraors and e Scrödinger equaion Te energy oeraor since en i [ Ae E ν ω k ω ] iω E i i E i i --- energy oeraor 7.3 Oeraors and e Scrödinger equaion Various maemaical oeraors corresond o ysical observable roeries suc as momenum and energy. Te oeraors ac on e wavefuncion and e resuling eigenvalues are e aroriae values of a ysical observable. Wi e oeraor formalism we say a e wavefuncion conains all e informaion abou e ysical sysem. Tis elains wy suc a remium is laced in quanum mecanics on discovering e form for e wavefuncion in a eculiar ysical cone; e wavefuncion says i all. 16

17 Oeraors and e Scrödinger equaion. Te scrödinger equaion Te rocedure for finding suc an equaion was by no means obvious en nor is i really clear even in indsig. Te bes a can be done is o surmise an equaion for e aricle-waves based on wa we know of em and en es e equaion and is redicions in siuaions a can be comared wi eerimen. Te oal energy of a aricle for nonrelaivisic siuaion V m E Oeraors and e Scrödinger equaion Mulily is eression by e wavefuncion ] [ E V m + i E and i By using e oeraors of and E i V i i m + ] 1 [ i V m +

18 Oeraors and e Scrödinger equaion Tis is called one-dimensional Scrödinger equaion. i V m + H V m + Is called Hamilonian of e sysem. i H