I. Existence of photon

Transcription

1 I. Existee of photo MUX DEMUX 1 ight is a eletromageti wave of a high frequey. Maxwell s equatio H t E 0 E H 0 t E 0 H 0 1 E E E Aos( kzt ) t propagatig eletrial field while osillatig light frequey (Hz) log med short u.short miro. milli. ifra-red visible ultra-violet X-ray There are various pheomea due to wave properties, e.g., iterferee. A B Photoi etworks utilize wave properties of light. Tx (l 1 ) Rx (l 1 ) Tx(l ) Rx (l ) Tx (l ) phase-modulated light Rx (l ) Tx (l 4 ) Rx (l 4 )

2 detetio sigal However,,,,, There are some physial pheomea that aot be explaied by the wave model. Hypothesis: There is a miimum uit of light eergy that aot be divided furthermore. photo Quatum Mehais Advet of lasers, CCD ameras, et. Matter Eletro ight atteuator ultra-high sesitive detetor disrete output time Photo This hapter itrodues pheomea that suggest photo.

3 voltage [Photo-eletri effet] light eletro Pheomeo of eletros jumpig out from a metal surfae elimiated by light metal Millika s experimet V light - Voltage at whih the urret starts to flow is measured. Eergy of a eletro jumpig out from the metal - The flowig urret is measured. The umber of eletros omig out from the metal (results) - The eletro eergy is ot depedet o the light itesity, but depeds o the frequey as E = - P : light frequey h, P: ostat light frequey - The umber of eletros is proportioal to the light itesity These aot be explaied by a wave model. The, Eistei thought,,,, - ight is a assemble of eergy partiles. - The eergy of oe partile is proportioal to the light frequey. light quatum photo His hypothesis a explai the experimetal results.

4 eergy [Blak body radiatio] 4 Heated matters radiate light, whose olor (= frequey = spetrum) depeds o the temperature. ex) molte iro How to theoretially explai this pheomeo?? wavelegth (optial frequey) Blak body radiatio, or Cavity radiatio (radiatio from a thermally equilibrium matter) Rayleigh-Jeas formula A formula based o thermodyamis was proposed to theoretially desribe the eergy spetrum of the radiatio, whih is itrodued i this setio. The disussio starts with osiderig light wave elosed withi a avity. Suh light is a stadig-wave, whih a be regarded as harmoi osillators. (Harmoi osillator: a physial system whose behavior is desribed by a sie wave. ) Stadig-wave i a losed spae Partile oeted with a sprig Swigig partile

5 The eergy of a harmoi osillator system is evaluated as (total eergy) = (the mea eergy of harmoi osillators) (the umber of harmoi osillators), whih will be respetively disussed i the followig. The mea eergy of a harmoi osillator Geerally, the mea value of a stohasti variable y is give by y(x): a variable depedet of x y y( x) P( x) dx P(x): the probability desity of x et us derive the mea eergy of a harmoi osillator, utilizig this formula. As a example of harmoi osillators, we osider a sprig system, whose eergy is give by (kieti eergy) + (potetial eergy). 5 The kieti eergy E m a be expressed as E m = ap Geerally, the probability for a system to have a eergy of E follows the Boltzma distributio. E exp E P( E) exp P( E) k:boltzma ostat kt (ormalized) exp E de T:Temperature p: mometum a: proportioal ostat E m ap exp ap exp ap dp dp The itegral i this eauatio is rewritte as ap ap kt d ap exp dp p exp dp kt dp kt kt ap pexp kt 0 kt ap kt ap exp dp exp dp kt kt E m ( kt / ) exp exp ap dp ap dp O the other had, the potetial eergy of a partile oeted to a sprig is E p = bq q: positio q kq b: proportioal ostat kxdx 0 This expressio is similar to that of the kieti eergy. Thus, kt

6 kt Ep The, the average of the total eergy of a partile fixed with a sprig is obtaied as Em Ep kt kt kt 6 The umber of harmoi osillators Stadig-wave i a avity (losed spae) a be regarded as a harmoi osillator. We will evaluate the umber of stadig-waves i a avity. The wavelegth of a oe-dimesioal stadig-wave is give by l 1 l s I terms of frequey, s x y z : avity legth s: atural umber (l = ) : light veloity As for a stadig-wave i a ube (i.e., three dimesio), its frequey is expressed as s x s y s z s x s y s z l l l 4 4 For a give, the frequey of a stadig-wave is idiated by a set of atural umbers {s x, s y, s z }. The umber of {s x, s y, s z } withi a frequey rage from to + d equals to the umber of stadig-waves i this frequeies rage. The umber of {s x, s y, s z } withi frequey equals to the umber of lattie poits o the surfae of a (1/8)-sphere with a radius of /. The umber of stadig-waves withi a frequey rage from 0 to equals to the umber of lattie poits withi a (1/8)-sphere with a radius of /. s z s x s y

7 radiatio eergy Provided that the lattie poits are quite dese, we a approximate the umber of lattie poit = the volume The, the umber of harmoi osillators g is give by 4 g (degree of freedom of the polarizatio state) (desity per a uit volume) 7 8 dg d 8 g the umber of harmoi osillators per frequey From the above osideratio, the eergy of harmoi osillators per uit frequey (eergy desity) is (eergy deisity) 8 ( average eergy () umber per freq. ) kt 8 kt Rayleigh-Jeas formula exp. low frequey regio: OK high frequey regio: NG frequey Plak s hypothesis The above NG i a high frequey regio omes from the mea eergy of harmoi osillators = kt, whih is derived from E ap bq ( ap bq ap )exp bq kt dqdp itegral alulus assumig the eergy as a otiuous variable reosider Plak thought the eergy is a disrete variable with a miimum uit. E = = 0, 1,,, eergy quata

8 The, E exp( E ) de E exp( E ) de otiuous disrete exp( ) E exp( ) 8 umerator: e 1 e e e (1 e ) kt kt deomiator: e 1 e 1 e E 1 e e 1 Plak s law About I the above, we assume the miimum uit of light eergy. As a matter of fat, is proportioal to the light frequey, whih is suggested by the followig osideratio. Here, we osider a swigig partile as a example of a harmoi osillator. Suppose that we slowly shorte the legth of the strig of a swigig partile. T Equatio of motio: m mgsi mgos mg mgsi with = aos(t + d) 1 g (a, d: ostat) d 1 d g 1 d g d d

9 ሶ ሶ Here, we osider pullig fore T that shortes the legth, whih is expressed as 9 (gravity alog the legth) (etrifugal fore) 1 mg mga { os (t ) si The workload oduted by T (= the eergy iremet of the swigig partile) for the legth to be shorteed by d is T ( d) mg mga 1 { os where, < > deotes the temporal average. Thus, the hage of the swigig eergy is mga de d 4 T = mgosθ + mθሶ mg(1 1 θ ) + mθሶ (t ) si (t )} mga (t )} ( d) mgd 4 iremet of the potetial eergy of the whole system d (iremet of the osillatio eergy) O the other had, the eergy of the swigig partile is E = 1 m θ + 1 mg 1 osθ potetial eergy (kieti eergy) from the lowest positio E de d d d 1 m θ + 1 mgθ = mga a os( t ) ( 1/ ) g / d(e/) = de/ (E/ )d = (E/)(dE/E d/) = 0 d de E d de E E ost The bottom lie is; eergy is proportioal to frequey. = (h:proportioal ost.) h:plak ostat

10 Plak s law, agai radiatio eergy 10 Substitutig = ito the previous Plak s law E e 1 h is experimetally evaluated. (h = erg-seod) The, the eergy desity [= (average eergy) (desity of harmoi osillators)] is give by Eergy desity: 8 e 1 : exp. lie:plak s law (Appedix) Why the lassial model is OK i the low-frequey regio? The miimum uit of eergy is. Whe << kt, the miimum uit is so small that it is regarded to be otiuous. The, Rayleigh-Jeas formula that treats the eergy as otiuous is OK at low-frequeies. I ase of << kt, kt e 1 h 1 1 Rayleigh-Jeas Formula wavelegth

11 Mometum of photo Provided that light has a partile-like property, it a have mometum. Here, we disuss the mometum of a photo. 11 Our disussio here utilizig the avity radiatio i a ube. Situatio here is; Suppose we slowly hage the legth of oe side of a ube. Suppose oe photo propagates i the diretio of the hagig side, whih pushes the surfae whe bumpig agaist it. We move the surfae agaist the pressure from the photo.. I this situatio, the eergy oservatio tells us (hage of the photo eergy) = (work of movig the surfae by the exteral fore) = (pressure by the photo) (movig legth of the surfae). We will evaluate eah term i the above equatio. (hage of the photo eergy) The relatioship betwee ad the frequey of a stadig-wave is s (s: atural umber) Photo eergy is E =, thus d s d d d s d de dv h de hd de d (pressure by the photo) et us deote the mometum of a photo as p, the - the hage of the mometum whe bumped bak at the surfae is p = the pressure that the photo gives the surfae at oe bumpig - the frequey of the bumpig is /. The pressure that the photo gives to the surfae is p p Therefore, (hage of the photo eergy) = (pressure by the photo) (movig legth of the surfae) d p d p Mometum of a photo

12 [Wave property ad partile-like property] ight has partile-like properties, as idiated i the previous setios. O the other had, however, there are pheomeo based o wave properties (e.g., iterferee). How these two properties, that look ompletely differet, are osistet? The aswer is that the eergy of light has a miimum uit whose behavior follows wave properties. 1 (wave) sree A B (photo) photo outig array A B atteuator How the wave ad the partile-like properties are osistet will be theoretially desribed i the ext hapter. Brief summaries i advae are - The state of light is expressed by the probability amplitude of photos: a - The probability amplitude behaves like wave (i.e., it has a phase): a = a e i - Whe we observe light, its eergy has a miimum uit ad thus disrete: - The probability of observig a photo is give by the absolute square of the probability amplitude: a -Whe some probability amplitudes are overlapped, there ours iterferee: a + b = a + b + Re[ab * ] Note: Partile-like property does ot mea that light partiles fly over a spae. It just meas the light eergy has a miimum uit ad othig else.