29.4: Photon Momentum

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Malengo ya kujifunza

Mwishoni mwa sehemu hii, utaweza:

Kuhusiana kasi linear ya photon kwa nishati yake au wavelength, na kuomba linear kasi uhifadhi kwa michakato rahisi kuwashirikisha chafu, ngozi, au tafakari ya photons.
Akaunti kwa ubora kwa ongezeko la wavelength ya photon inayozingatiwa, na kuelezea umuhimu wa wavelength ya Compton.

Kupima kasi ya Photon

Kiwango cha mionzi ya EM tunayoiita fotoni ina mali zinazofanana na zile za chembe tunazoweza kuona, kama vile nafaka za mchanga. Photon inaingiliana kama kitengo katika migongano au inapokwisha kufyonzwa, badala ya kama wimbi kubwa. Quanta kubwa, kama elektroni, pia hufanya kama chembe macroscopic - kitu tunachotarajia, kwa sababu ni vitengo vidogo vya jambo. Chembe hubeba kasi pamoja na nishati. Licha ya photoni zisizo na wingi, kwa muda mrefu kumekuwa na ushahidi kwamba mionzi ya EM hubeba kasi. (Maxwell na wengine waliosoma mawimbi ya EM walitabiri kwamba wangeweza kubeba kasi.) Sasa ni ukweli ulioanzishwa vizuri kwamba photons zina kasi. Kwa kweli, kasi ya photon inapendekezwa na athari ya photoelectric, ambapo photons kubisha elektroni nje ya dutu. Kielelezo\(\PageIndex{1}\) inaonyesha ushahidi macroscopic ya photon kasi.

Trajectory ya comet yenye kiini na mkia inapopita kwa Jua inaonyeshwa kama njia ya sehemu ya parabolic na Jua karibu na kipeo cha njia ya parabolic. — Kielelezo\(\PageIndex{1}\): Mkia wa comet ya Hale-Bopp huelekea mbali na Jua, ushahidi kwamba mwanga una kasi. Vumbi vinavyotokana na mwili wa comet huunda mkia huu. Vipande vya vumbi vinasukumwa mbali na Jua kwa mwanga unaoonyesha kutoka kwao. Mkia wa gesi ionized bluu pia huzalishwa na photoni zinazoingiliana na atomi katika nyenzo za kimondo. (mikopo: Geoff Chester, Marekani Navy, kupitia Wikimedia Commons)

Picha ya kimondo cha Hale Bopp kilichosonga katika nafasi kinaonyeshwa kama kitu kilichoangazwa. — Kielelezo\(\PageIndex{1}\): Mkia wa comet ya Hale-Bopp huelekea mbali na Jua, ushahidi kwamba mwanga una kasi. Vumbi vinavyotokana na mwili wa comet huunda mkia huu. Vipande vya vumbi vinasukumwa mbali na Jua kwa mwanga unaoonyesha kutoka kwao. Mkia wa gesi ionized bluu pia huzalishwa na photoni zinazoingiliana na atomi katika nyenzo za kimondo. (mikopo: Geoff Chester, Marekani Navy, kupitia Wikimedia Commons)

Kielelezo\(\PageIndex{1}\) kinaonyesha comet na mikia miwili maarufu. Kile ambacho watu wengi hawajui kuhusu mikia ni kwamba daima huelekeza mbali na Jua badala ya kufuatilia nyuma ya kimondo (kama mkia wa kondoo wa Bo Peep). Mkia wa kimondo hujumuisha gesi na vumbi vinavyotokana na mwili wa comet na gesi ionized. Chembe za vumbi zinarudi mbali na Jua wakati photoni zinapoenea kutoka kwao. Kwa dhahiri, photons hubeba kasi katika mwelekeo wa mwendo wao (mbali na Jua), na baadhi ya kasi hii huhamishiwa kwenye chembe za vumbi katika migongano. Atomi za gesi na molekuli katika mkia wa buluu huathiriwa zaidi na chembe nyingine za mionzi, kama vile protoni na elektroni zinazotoka Jua, badala ya kasi ya fotoni.

Connections: Uhifadhi wa kasi

Si tu ni kasi iliyohifadhiwa katika ulimwengu wote wa fizikia, lakini kila aina ya chembe hupatikana kuwa na kasi. Tunatarajia chembe na wingi kuwa na kasi, lakini sasa tunaona kwamba chembe massless ikiwa ni pamoja na photons pia kubeba kasi.

Momentum ni kuhifadhiwa katika quantum mechanics kama ilivyo katika relativity na fizikia classical. Baadhi ya ushahidi wa kwanza wa moja kwa moja wa majaribio ya hili ulitokana na kutawanyika kwa fotoni za eksirei na elektroni katika vitu, jina lake Compton kutawanya baada ya mwanafizikia wa Marekani Arthur H. Compton (1892—1962). Karibu mwaka wa 1923, Compton aliona kwamba mionzi ya x iliyotawanyika kutoka vifaa ilikuwa na nishati iliyopungua na kuchambua kwa usahihi hili kama kuwa kutokana na kueneza kwa photoni kutoka kwa elektroni. Jambo hili linaweza kushughulikiwa kama mgongano kati ya chembe mbili-photon na elektroni wakati wa kupumzika katika nyenzo. Nishati na kasi huhifadhiwa katika mgongano (Kielelezo\(\PageIndex{2}\)) Alishinda Tuzo ya Nobel mwaka wa 1929 kwa ugunduzi wa kueneza hii, sasa inaitwa athari ya Compton, kwa sababu imesaidia kuthibitisha kwamba kasi ya photon inatolewa na

\[p = \dfrac{h}{\lambda},\]

wapi\(h\) Planck ya mara kwa mara na\(\lambda\) ni wavelength photon. (Kumbuka kuwa kasi ya relativistic iliyotolewa kama\(p = \gamma mu\) halali tu kwa chembe zilizo na wingi.)

Mgongano wa elektroni na photon ya nishati E sawa na h f inavyoonyeshwa. Electroni inawakilishwa kama mpira wa spherical na photon kama duaradufu inayozunguka wimbi. Baada ya mgongano nishati ya photon inakuwa E mkuu sawa na h f mkuu na nishati ya mwisho ya elektroni K E ndogo e ni sawa na E minus E mkuu. Mwelekeo wa elektroni na photon kabla na baada ya mgongano unawakilishwa na mishale. — Kielelezo\(\PageIndex{2}\): Athari ya Compton ni jina lililopewa kueneza kwa photon na elektroni. Nishati na kasi huhifadhiwa, na kusababisha kupungua kwa wote kwa photon iliyotawanyika. Kujifunza athari hii, Compton alithibitisha kwamba photons zina kasi.

Tunaweza kuona kwamba kasi ya photon ni ndogo, tangu\(p = h/\lambda\) na\(h\) ni ndogo sana. Kwa sababu hii kwamba hatuwezi kuchunguza kasi ya photon. Vioo vyetu hazirudi wakati mwanga unaonyesha kutoka kwao (isipokuwa labda katika katuni). Compton aliona madhara ya kasi ya fotoni kwa sababu alikuwa akiangalia mionzi ya x, ambayo ina wavelength ndogo na kasi kubwa kiasi, kuingiliana na chembe nyepesi zaidi, elektroni.

Mfano\(\PageIndex{1}\): Electron and Photon Momentum Compared

Tumia kasi ya photon inayoonekana ambayo ina wavelength ya nm 500.
Kupata kasi ya elektroni kuwa na kasi sawa.
Nishati ya elektroni ni nini, na inalinganishaje na nishati ya photon?

Mkakati

Kupata kasi ya photon ni matumizi ya moja kwa moja ya ufafanuzi wake:\(p = \frac{h}{\lambda}\).

Ikiwa tunapata kasi ya photon ni ndogo, basi tunaweza kudhani kwamba elektroni yenye kasi sawa itakuwa isiyo ya kawaida, na iwe rahisi kupata kasi yake na nishati ya kinetic kutoka kwa formula za classical.

Suluhisho kwa (a)

Photon kasi hutolewa na equation:\[p = \dfrac{h}{\lambda}. \nonumber\]

Kuingia mavuno ya wavelength ya photon

\[p = \dfrac{6.63 \times 10^{-34} \, J \cdot s}{500 \times 10^{-0} \, m} = 1.33 \times 10^{-27} \, kg \cdot m/s. \nonumber\]

Suluhisho kwa (b)

Kwa kuwa kasi hii ni ndogo sana, tutatumia\(p = mv\) kujieleza classical kupata kasi ya elektroni na kasi hii. Kutatua\(v\) na kutumia thamani inayojulikana kwa wingi wa elektroni inatoa

\[v = \dfrac{p}{m} = \dfrac{1.33 \times 10^{-27} \, kg \cdot m/s}{9.11 \times 10^{-31} \, kg} = 1460 \, m/s \approx 1460 \, m/s. \nonumber\]

Suluhisho kwa (c)

Electron ina nishati ya kinetic, ambayo hutolewa kwa kawaida

\[KE_e = \dfrac{1}{2} mv^2. \nonumber\]

Hivyo,

\[KE_e = \dfrac{1}{2} (9.11 \times 10^{-31} \, kg)(1455 \, m/s)^2 = 9.64 \times 10^{-25} \, J. \nonumber\]

Kubadili hii kwa eV kwa kuzidisha kwa\((1 \, eV)/(1.602 \times 10^{-19} \, J)\) mavuno

\[KE_e = 6.02 \times 10^{-6} \, eV. \nonumber\]

Nishati ya photon\(E\) ni

\[E = \dfrac{hc}{\lambda} = \dfrac{1240 \, eV \cdot nm}{500 \, nm} = 2.48 \, eV, \nonumber\]

ambayo ni juu ya amri tano ya ukubwa mkubwa zaidi.

Majadiliano

Photon kasi ni kweli ndogo. Hata kama tuna idadi kubwa yao, kasi ya jumla wanayobeba ni ndogo. Electron yenye kasi sawa ina kasi ya 1460 m/s, ambayo ni wazi isiyo ya kawaida. Chembe kubwa zaidi yenye kasi sawa ingekuwa na kasi ndogo hata. Hii inatokana na ukweli kwamba inachukua nishati ndogo sana kutoa elektroni kasi sawa na photon. Lakini kwa kiwango cha quantum-mitambo, hasa kwa photoni za juu-nishati zinazoingiliana na raia wadogo, kasi ya photon ni muhimu. Hata kwa kiwango kikubwa, kasi ya photon inaweza kuwa na athari ikiwa kuna kutosha kwao na ikiwa hakuna kitu cha kuzuia upungufu wa polepole wa jambo. Mikia ya kimondo ni mfano mmoja, lakini pia kuna mapendekezo ya kujenga sails za nafasi ambazo hutumia vioo vikubwa vya chini (vilivyotengenezwa kwa Mylar ya aluminized) ili kutafakari jua. Katika utupu wa nafasi, vioo bila hatua kwa hatua recoil na inaweza kweli kuchukua spacecraft kutoka sehemu kwa mahali katika mfumo wa jua (Kielelezo\(\PageIndex{3}\)).

(a) payload kuwa mwavuli umbo nishati ya jua meli masharti yake ni umeonyesha. Mwelekeo wa harakati za malipo na mwelekeo wa photons za tukio huonyeshwa kwa kutumia mishale. (b) Picha ya mtazamo wa juu wa meli ya nafasi ya utulivu. — Kielelezo\(\PageIndex{3}\): (a) Space sails have been proposed that use the momentum of sunlight reflecting from gigantic low-mass sails to propel spacecraft about the solar system. A Russian test model of this (the Cosmos 1) was launched in 2005, but did not make it into orbit due to a rocket failure. (b) A U.S. version of this, labeled LightSail-1, is scheduled for trial launches in the first part of this decade. It will have a 40-m2 sail. (credit: Kim Newton/NASA)

Relativistic Photon Momentum

There is a relationship between photon momentum \(p\) and photon energy \(E\) that is consistent with the relation given previously for the relativistic total energy of a particle as

\[E^2 = (pc)^2 + (mc^2)^2. \label{photon1}\]

We know \(m\) is zero for a photon, but \(p\) is not, so that Equation \ref{photon1} becomes

\[E = pc,\] or \[p = \dfrac{E}{c} \text{(for photons)}. \nonumber\]

To check the validity of this relation, note that \(E = hc/\lambda\) for a photon. Substituting this into \(p = E/c\) yields

\[p = (hc/\lambda) / c = \dfrac{h}{\lambda},\]

as determined experimentally and discussed above. Thus, \(p = E/c\) is equivalent to Compton’s result \(p = h/\lambda.\) For a further verification of the relationship between photon energy and momentum, see Example \(\PageIndex{3}\).

Photon Detectors

Almost all detection systems talked about thus far—eyes, photographic plates, photomultiplier tubes in microscopes, and CCD cameras—rely on particle-like properties of photons interacting with a sensitive area. A change is caused and either the change is cascaded or zillions of points are recorded to form an image we detect. These detectors are used in biomedical imaging systems, and there is ongoing research into improving the efficiency of receiving photons, particularly by cooling detection systems and reducing thermal effects.

Example \(\PageIndex{2}\): Photon Energy and Momentum

Show that \(p = E/c\) for the photon considered in the Example \(\PageIndex{2}\).

Strategy

We will take the energy \(E\) found in Example\(\PageIndex{2}\), divide it by the speed of light, and see if the same momentum is obtained as before.

Solution

Given that the energy of the photon is 2.48 eV and converting this to joules, we get

\[p = \dfrac{E}{c} = \dfrac{(2.48 \, eV)(1.60 \times 10^{-19} \, J/eV)}{3.00 \times 10^8 \, m/s} = 1.33 \times 10^{-27} \, kg. \nonumber\]

Discussion

This value for momentum is the same as found before (note that unrounded values are used in all calculations to avoid even small rounding errors), an expected verification of the relationship \(p = E/c\). This also means the relationship between energy, momentum, and mass given by \(E^2 = (pc)^2 + (mc)^2 \) applies to both matter and photons. Once again, note that \(p\) is not zero, even when \(m\) is.

PROBLEM-SOLVING SUGGESTIONS

Note that the forms of the constants \(h = 4.14 \times 10^{-15} \, eV \cdot s\) and \(hc = 1240 \, eV \cdot nm\) may be particularly useful for this section’s Problems and Exercises.

Summary

Photons have momentum, given by \(p = \frac{h}{\lambda}\), where \(\lambda\) is the photon wavelength.
Photon energy and momentum are related by \(p = \frac{E}{c}\), where \(E = hf = hc/\lambda\) for a photon.

Glossary

photon momentum: the amount of momentum a photon has, calculated by \(p = \frac{h}{\lambda} = \frac{E}{c}\)

Compton effect: the phenomenon whereby x rays scattered from materials have decreased energy