SHMEIWSEIS GIA TO MAJHMA JEWRHTIKH FUSIKH - physics…ntrac/MATHIMATA/TH_PHYS/19-20/MATHIM… ·...
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SHMEIWSEIS GIA TO MAJHMAJEWRHTIKH FUSIKH
9o EXAMHNOSqol Efarmosmènwn Majhmatik¸n
kai Fusik¸n Episthm¸n EMP
NÐkoc Tr�kac
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Perieqìmena
1 BIBLIOGRAFIA
2 ANTISWMATIDIA
3 HLEKTRODUNAMIKH SWMATIDIWN QWRIS SPIN
4 H EXISWSH DIRAC
5 HLEKTRODUNAMIKH SWMATIDIWN ME SPIN=1/2
6 H DOMH TWN ADRONIWN
7 JEWRIES BAJMIDAS
8 ASJENEIS ALLHLEPIDRASEIS
9 MEGALOENOPOIHSH
10 UPOLOGISMOS THS SUNARTHSHS b
11 LUSEIS TWN ASKHSEWN
12 PARARTHMA
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BIBLIOGRAFIA
Quarks and Leptons: An introductory course in ModernParticle Physics,F. Halzen and A.D. Martin
Gauge Theories in Particle Physics,I.J.R. Aitchison and A.J.G. Hay
Relativistic Quantum Mechanics,J.D. Bjorken and S.D. Drell
Introduction to Elementary Particles,D. Griffiths
Field Theory, A Modern Primer,P. Ramond
Sqetikistik Kbantomhqanik ,S. Traqan�c
Swmatidiak Fusik . Mia Eisagwg sthn Basik Dom thc 'Ulhc,K.E. Bagiwn�khc
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ANTISWMATIDIA
Oi perissìterec allhlepidr�seic, p.q.
e+e− → µ+µ−, eq → eq, γq → e+e−q
apoteloÔn sust mata poll¸n swmatidÐwn kai apì tapeir�mata pou èqoume sth di�jes mac briskìmaste sthnperioq thc sqetikistik c kinhmatik c. EpÐ plèon emfanÐzontaikai antiswmatÐdia pou den apaitoÔntai sthn mh sqetikistik jewrÐa.Qrhsimopoi¸ntac th jewrÐa diataraq¸n ja qrhsimopoioÔme tickumatosunart seic pou perigr�foun eleÔjero swmatÐdio (INkai OUT katast�seic) kai thn allhlepÐdrash metaxÔ twnswmatidÐwn ja th jewroÔme wc diataraq se periorismèno q¸rokai qrìno.
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QrhsimopoioÔme sqetikistikì formalismì thc jewrÐacdiataraq¸n. Shmantikì rìlo paÐzoun ta ��diagr�mmataFeynman��. Me th qr sh twn ��kanìnwn Feynman�� mporoÔme naupologÐsoume fusikèc posìthtec (energèc diatomèc, rujmoÔcmet�bashc k.lp.) qwrÐc na katafeÔgoume k�je for� sthjewrÐa pedÐou. Bèbaia, oi kanìnec autoÐ kajorÐzontai apì tonLagkranzianì formalismì kai th jewrÐa pedÐou.Arqik� ja agno soume to spin twn swmatidÐwn, pou k�pwcperiplèkei thn eikìna, kai ja asqolhjoÔme me ��hlektrìnia��qwrÐc spin.
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Mh sqetikistik Kbantomhqanik Me tic antikatast�seic
E → i~∂
∂t, p→ −i~∇
h klasik sqèsh E = p2
2m gÐnetai (~ = 1)
E =p2
2m→ i
∂Ψ
∂t+
1
2m∇2Ψ = 0
ìpou ρ = |Ψ|2 eÐnai h puknìthta pijanìthtac (|Ψ|2d3x dÐnei thnpijanìthta na broÔme to swmatÐdio ston ìgko d3x). Aut eÐnaih exÐswsh Schrodinger gia eleÔjero swmatÐdio m�zac m.
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An�loga me thn diat rhsh fortÐou ston hlektromagnhtismì, hdiat rhsh thc pijanìthtac mac odhgeÐ sthn exÐswsh
∂ρ
∂t+ ∇ · j = 0 diaforik morf
d
dt
∫Vρ dv +
∮S(V )
j · da = 0 oloklhrwtik morf
ìpou j eÐnai h puknìthta reÔmatoc pijanìthtac. Ac broÔme thmorf tou. Pollaplasi�zoume thn exÐswsh Schrodinger me−iΨ∗ kai thn suzug thc me iΨ kai ajroÐzoume
−iΨ∗(i∂Ψ
∂t+
1
2m∇2Ψ
)+ iΨ
(−i ∂Ψ∗
∂t+
1
2m∇2Ψ∗
)= 0→
Ψ∗∂Ψ
∂t− i
2mΨ∗∇2Ψ + Ψ
∂Ψ∗
∂t+
i
2mΨ∇2Ψ∗ = 0→
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∂|Ψ|2
∂t− i
2m
(Ψ∗∇2Ψ−Ψ∇2Ψ∗
)= 0→
∂ρ
∂t+ ∇ ·
[− i
2m(Ψ∗∇Ψ−Ψ∇Ψ∗)
]︸ ︷︷ ︸ = 0
j
H lÔsh thc ex. Schrodinger gia to eleÔjero swmatÐdioΨ = N exp [i (p · x− Et)] dÐnei ρ = |N|2 kai
j =−i |N|2
2m(∇(ip · x)−∇(−ip · x)) =
|N|2
2m2p =
|N|2
mp
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TetradianÔsmata kai analloÐwta Lorentz
'Askhsh 1 DeÐxte ìti o metasqhmatismìc Lorentz antistoiqeÐme strof kat� gwnÐa iθ ston q¸ro (ict, x)
'O,ti metasqhmatÐzetai ìpwc to (ct, x) kaleÐtai tetradi�nusma.QrhsimopoioÔme ton sumbolismì
(ct, x) = (ct, x1, x2, x3) = (x0, x1, x2, x3) ≡ xµ
EpÐshc, to E/c kai p sugkrotoÔn tetradi�nusma
(E/c ,p) = (E/c , p1, p2, p3) = (p0, p1, p2, p3) ≡ pµ
OrÐzoume to bajmwtì ginìmeno dÔo tetradianusm�twnAµ = (A0,A) kai Bµ = (B0,B)
A · B = A0B0 − A · B
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OrÐzontac to Aµ = (A0,−A) mporoÔme na gr�youme tobajmwtì ginìmeno wc (epanalambanìmenoc deÐkthc �nw kaik�tw ajroÐzetai)
A · B = AµBµ = AµBµ = A0B0 + A1B1 + A2B2 + A3B3 =
A0B0 + A1B
1 + A2B2 + A3B
3 = A0B0 − A1B1 − A2B2 − A3B3
OrÐzoume ton (metrikì) tanust gµν
g00 = 1, g11 = g22 = g33 = −1, oi �lloi ìroi mhdenikoÐ
O antÐstrofìc tou gµν (dhlad gµνgνµ′ = δµµ′) eÔkola faÐnetaiìti èqei touc Ðdiouc ìrouc. To ginìmeno twn dÔotetradianusm�twn mporeÐ na grafeÐ
A · B = gµνAµBν = gµνAµBν
Me to gµν kai to gµν mporoÔme na anebokateb�soume toucdeÐktec enìc tetradianÔsmatoc
Aµ = gµνAν , Aµ = gµνAν
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To di�nusma me �nw deÐkth onom�zetai antalloÐwto(contravariant), en¸ me k�tw deÐkth sunalloÐwto (covariant). Giana sqhmatisteÐ èna analloÐwto, wc proc metasqhmatismoÔcLorentz, mègejoc ja prèpei gia k�je �nw deÐkth na up�rqei oantÐstoiqoc k�tw. EpÐshc, mia sqèsh eÐnai Lorentz sunalloÐwthìtan oi mh epanalambanìmenoi (�nw kai k�tw) deÐktec stic duopleurèc thc isìthtac antistoiqÐzontai ènac proc ènan.
'Askhsh 2 DeÐxte ìti gµνgµν = 4
ParadeÐgmata bajmwt¸n ginomènwn eÐnai
pµxµ ≡ p · x = Et − p · x, pµpµ ≡ p · p ≡ p2 = E 2 − p2
'Askhsh 3 DÔo swmatÐdia me Ðsh m�za M sugkroÔontai stosÔsthma Kèntrou M�zac. H sunolik enèrgeia eÐnai Ecm.DeÐxte ìti
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s ≡ (p1 + p2)µ(p1 + p2)µ ≡ (p1 + p2)2 = E 2cm
An h sÔgkroush gÐnei sto sÔsthma ergasthrÐou ìpou to ènaswmatÐdio eÐnai akÐnhto, tìte h enèrgeia Elab tou �llouswmatidÐou dÐnetai apì th sqèsh (upologÐste to s sto sÔsthmaergasthrÐou)
Elab =E 2
cm
2M−M
Prosoq sto tetradi�nusma(∂
∂t,−∇
)= ∂µ kai
(∂
∂t,∇)
= ∂µ
MporeÐte na deÐxete ìti to pr¸to metasqhmatÐzetai ìpwc to(t, x) en¸ to deÔtero ìpwc to (t,−x).H antikat�stash thc enèrgeiac kai thc orm c me toucantÐstoiqouc telestèc genikeÔetai
pµ → i∂µ
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Tèloc, qrhsimopoeÐtai o sumbolismìc
�2 ≡ ∂µ∂µ
H exÐswsh Klein-GordonQrhsimopoi¸ntac thc sqetikistik exÐswsh E 2 = p2 + m2 kai ticantikatast�seic E → i~ ∂
∂t kai p→ −i~∇, odhgoÔmeja sthn(~ = 1) (
i∂
∂t
)2
φ =((−i∇)2 + m2
)φ
− ∂2
∂t2φ+∇2φ = m2φ
pou apoteleÐ thn exÐswsh Klein-Gordon.
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'Askhsh 3 DÔo swmatÐdia me Ðsh m�za M sugkroÔontai stosÔsthma Kèntrou M�zac. H sunolik enèrgeia eÐnai Ecm.DeÐxte ìti
s ≡ (p1 + p2)µ(p1 + p2)µ ≡ (p1 + p2)2 = E 2cm
An h sÔgkroush gÐnei sto sÔsthma ergasthrÐou ìpou to ènaswmatÐdio eÐnai akÐnhto, tìte h enèrgeia Elab tou �llouswmatidÐou dÐnetai apì th sqèsh (upologÐste to s sto sÔsthmaergasthrÐou)
Elab =E 2
cm
2M−M
(P)LÔshSto sÔsthma Kèntrou M�zac
pµ1 = (E1,p) kai pµ2 = (E2,−p)
Opìtes = (p1 + p2)2 = (E1 + E2, 0)2 = E 2
cm
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Sth deÔterh perÐptwsh pµ1 = (Elab,p1) kai pµ2 = (M, 0)
s = (p1 + p2)2 = (Elab + M,p1)2 = (Elab + M)2 − p21 =
(Elab + M)2 − (E 2lab −M2) = 2ElabM + 2M2
All� to s eÐnai analloÐwto, opìte
E 2cm = 2ElabM + 2M2 → Elab =
E 2cm
2M−M
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Par�rthma
PARARTHMA 1
AntalloÐwto kai sunalloÐwto tetra-di�nusmaO metasqhmatismìc Lorentz, gia kÐnhsh ston �xona x (dhlad x1) eÐnai (c = 1)
t ′ =t − vx1√
1− v2, x ′1 =
x1 − vt√1− v2
en¸ o antÐstrofoc metasqhmatismìc eÐnai
t =t ′ + vx ′1√
1− v2, x1 =
x ′1 + vt ′√1− v2
'Osa tetradianÔsmata metasqhmatÐzontai ìpwc o eujÔcmetasqhmatismìc ta onom�zoume antalloÐwta (contravariant)dianÔsmata kai ta sumbolÐzoume me deÐkth p�nw. Opìte, to tkai to x apoteloÔn antalloÐwto tetradi�nusma
(t, x) = (t, x1, x2, x3) = (x0, x1, x2, x3) = xµ
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Ta tetradianÔsmata pou metasqhmatÐzontai me ton antÐstrofometasqhmatismì onom�zontai sunalloÐwta (covariant) kaib�zoume ton deÐkth k�tw. EÔkola faÐnetai ìto totetradi�nusma (t,−x) eÐnai èna sunalloÐwto tetradi�nusma:
(t,−x) = (t,−x1,−x2,−x3) = (x0,−x1,−x2,−x3) = xµ
T¸ra mporoÔme na deÐxoume ìti to (∂/∂t,∇) metasqhmatÐzetaime ton antÐstrofo metasqhmatismì. Pr�gmati
∂
∂t ′=
∂
∂t
∂t
∂t ′+
∂
∂x1
∂x1
∂t ′=
1√1− v2
(∂
∂t+ v
∂
∂x1
)∂
∂x ′1=
∂
∂t
∂t
∂x ′1+
∂
∂x1
∂x1
∂x ′1=
1√1− v2
(∂
∂x1+ v
∂
∂t
)To (∂/∂t,−∇) metasqhmatÐzetai me ton eujÔ metasqhmatismì.Opìte, gr�foume
(∂/∂t,−∇) = ∂µ, (∂/∂t,∇) = ∂µ
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