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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