Fall6Boas3Mod3

8/20/2019 Fall6Boas3Mod3

http://slidepdf.com/reader/full/fall6boas3mod3 1/6

Mathematical Methods of Physics – Fall 2010 – Dr. E.J. Zita

Week 6 – Tuesday 2 Nov. 2010

1) Continue Boas Ch.! "inear #$%e&ra

2) 'odern (hysis Ch.!

. 'atri* o+erations ,+.11-) – euations/ u$ti+$iation &y sa$ar addition o atries

(ratie atri* u$ti+$iation 3.6.2 ,+.121).

C4''5TE 7#B8 9 #B –B# very i+ortant in :uantu 'ehanis

E*! ;eisen&er% 5nertainty (rini+$e! 7*+*8 9 i <

=dentity atries inverse atries oators ,+.120) We did examples in week 4

'atries an &e used or transorations e.%. rotations ,+.120) "orent> transorations/

det #B 9 det B# 9 det# ? detB

=nverse o 'atri* ' is '@1. '@1 ' 9 = 9 =dentity atri*.

,'atri* Aithout inverse is sin%u$ar).C9oator o ' atri* ,+.120). CT 9 trans+ose ,sAith roAs and o$uns)

'@1 9 CT det'

E* 1 or 1- +.122! ind inverse

E*. 1 or 20 +.12! o$ve set o euations &y ethod o indin% inverse o oeiientatri*. ee E*. +.120

;W! (au$i s+in atries 36 +.122

.F "inear o&inations $inear untions $inear o+erations.

Matrices can represent linear operations on functions, or vectors, which can represent states

of physical systems.

"inear o+erations yie$d the sae resu$t Ahether done on a su o untions or done on the

untions individua$$y then added.

"inear$y inde+endent untions – anGt &e strethed shrunk or o&ined to &eoe eah

other.

E*.- +.1H! ind va$ues or Ahih a set o euations has nontrivia$ so$utions!

E=IEN#"5E

Do E*.2 or 2- +.1F



(=:

(2. # homogeneous syste o N euations Aith N unknoAns is so$va&$e i the deterinant o the oeiients is >ero.

(. 'atri* ' an &e dia%ona$i>ed &y a sii$arity transoration on$y i ' is syetri.,=(")

=1. The trae o a atri* set an &e +eruted y$ia$$y.=2. etor s+aes on$y uti$i>e tAo o+erations addition and u$ti+$iation. ,=(")

:5ET=4N

What is an ei%envetorK A natural state of the system, e.g. a natural mode of oscillation.

,#5) What is an ei%enva$ueK A simple solution of the system, e.g. the natural frequency of

oscillation.

What is the si%niiane o dia%ona$i>in% a atri*K The diagonal values are eigenvalues.

Dia%ona$i>ation +rovides a &etter hoie o varia&$es and thus aounts to a si+$iiation o

the +ro&$e. ,Boas 1H0) ,5;24)

? =t is hard or e to see the dierene &etAeen a $inear o+erator and Lust soe ter thatMshrinksM or M%roAsM a %iven atriesK ,Boas 12-ish) ,5;24) inear operators can also

com!ine functions.

? The &ook %ives a very &rie introdution to %rou+s and entions that there are +hysisa++$iations. Can you %ive an e*a+$e o one suh a++$iationK ,Boas 1F2) ,5;24) "nified field theories in particle physics

:1! What are the a++$iations or usin% atries in a o+$e* eu$idean s+ae ,Boas 1-6)K

#scillations$:! What are the ost oon a++$iation in Ahih to use atriesK ,Tea (hysis)

%otations, oscillations, neutrino mixing, any solution of multiple equations in multiple

unknowns

:2. =s there any +ratia$ use or the trivia$ so$utionK no

:. What is the use o the seu$ar deterinant. ,=(") KK

Modern Physics Ch.3: Light as particles ,6-)

evieA o Aave euation! y 9 y0 sin ,k*@ωt). Wave s+eed v 9 reueny?Aave$en%th 9 ωk.

ind d2ydt2 and d2yd*2. e$ate the tAo derivatives Aith the s+eed.



.1 evieA o e$etroa%neti Aaves! Derive E' Aave euations ro 'a*Ae$$ euations

– W4;EET. ;a$ the $ass derive E Aaves ha$ derive B Aaves. &later'

ee i%..1 or diretions o E' Aave! +o$ari>ation and diretion o ener%y trans+ort.

(oyntin% $u* 9 (oAerarea0

1

µ

=S !"

2

0

02

( )ower * +ntensity

Area c µ = = = =S .

u& in ( -c to ind * in ters o the a%neti ie$d a+$itude .

Dou&$e@s$it intererene – %et stri+ed ro+e – derive a*ia at yn9nλDd and dsinθ9nλ

Bra%% atterin%! a*ia at 2dsinθ9nλ

E* 2 ,+.H)! =n E*..1 Ahat an%$e o inidene Ai$$ +rodue the seond@order Bra%% +eakKs

.2 (hotoe$etri eet! ,F1) (redit Ahat youGd e*+et i Ae!

•turn u+ the intensity o the $i%ht on the eta$ kee+in% the o$or the sae• ake the $i%ht &$uer and turn the intensity Aay doAn

• kee+ the intensity hi%h and %radua$$y tune the $i%ht reueny doAn

'ore e$etronsK eAerK 'ore ener%eti e$etronsK "ess ener%etiK No han%eK

Wave theory says the e$etron ener%y O intensity o the $i%ht.

"i%ht +arti$e arries E 9 h ν 9 hλ and oentu + 9 E 9 hλ. ,ass 9 0)

4&served! ener%y o inoin% $i%ht 9 Aork untion o eta$ P o eLeted e$etron

a*h / ν φ = +

,ea$$ h 9 6.6 * 10@- J.s 9 -.1 * 10@1H e.s 9 12-0 e.n)

E*.- +.H! ind the oentu o ,a) a 10 'e %aa ray ,and other +arti$es i youinish ear$y)

E*. 6 +.6! What is the Aave$en%th o %aa ray o 10 'eK ,and other +arti$e i you

inish ear$y)

= a +hoton has Lust &are$y enou%h ener%y to $i&erate an e$etron ro a eta$ Aith soe

Aork untion

=ND the uto reueny λ o the +hoton QQQQQQQQQQ

;oA ou$d you use this re$ationshi+ to easure ($ankGs onstant e*+erienta$$yK ,+.FH)

E*.R! What is the uto Aave$en%th or the +hotoe$etri eet usin% an a$uinu suraeK

E*.10! # eta$ surae has a +hotoe$etri uto Aave$en%th o 2H.6 n. =t is i$$uinatedAith $i%ht o Aave$en%th 2H.R n. What is the sto++in% +otentia$K



E*.12 or a ha$$en%e i you inish ear$y

. B$ak&ody radiation! ,FF)

teanGs "aA! =ntensity 9 σ T-

Ahere σ 9 H.6F * 10@R

W2

-

9 tean@Bo$t>ann onst.

WienGs "aA! λa*T 9 * 10@ .

:1 ,=(") What is the $osest rea$ $ie e*a+$e o a &$ak &odyK *TA%*

ES! keth BB urves or Earth and un.

ES! ;oA uh ore radiation does a star eit i its surae is tAie as hotK

B$ak&ody!

• re$ets no radiation• onsider a hot &o* eittin% radiation ro a ho$e – that is the BB radiation

• &o* is i$$ed Aith E' radiation

• a$$ +ossi&$e Aave$en%ths ontri&ute to the radiation

• radiation is in thera$ eui$i&riu Aith Aa$$s o the &o*

• a$$ inoin% radiation is a&sor&ed

ay$ei%h@Jeans oru$a or radiany 9-

R, )

-

d+ c % kT

d

π λ

λ λ = = $eads to u$travio$et atastro+he.

($ank invented uantu h to so$ve &$ak&ody +ro&$e! -

R 1

, ) - 1hc

kT

c hc

% e λ

π

λ λ λ

= −

Where h 9 6.6 * 10@- J.s ,an%u$ar oentu units) andH -

2

2

1H

k

c h

π σ =

,WeG$$ derive ($ankGs $aA in Cha+ter 10)

E*.16! By dierentiatin% ($ankGs euation hoA that has its a*iu as e*+eted

aordin% to WienGs dis+$aeent $aA En .2.

E*.20! C4BE radiation has T92.F. ind +eak Aave$en%th and ener%y in e. =n Ahat

re%ion o the E' s+etru is this Aave$en%thK

E*.22 – do i you inish ear$y

.- Co+ton Eet! Conservation o oentu &etAeen +hoton@+arti$e and ree e$etron,near$y ree) ,R)

5se Einitia$ 9 Eina$ and (initia$ 9 (ina$ to ind Co+ton Wave$en%ths!



λ 9 Aave$en%th o inident +hoton

λG 9 λ P he ,1@osθ) Ahere θ 9 +hoton satterin% an%$e

Chan%e o Aave$en%th o e$etron 9 he

.H Bresstrah$un% and other *@ray +rodution due to e$etron dee$eration ae$eration,RF)

=! Both the &rehsstrah$un% +roess and +air +rodution reuire the +resene o atonear&y in order to su++$y the neessary reoi$ oentu ,rane R). ,Tea (hysis)

E$etrons Aith 9e strike tar%et and are s$oAed doAn to G %ivin% +hoton ener%y h ν!

Ener%y in 9 Ener%y out! 9 G P h ν.

a$$est +hoton Aave$en%th λ or a*iu e$etron ener%y $oss! h ν 9 hλ 9 9 eAhere is the ae$eratin% +otentia$ or the e$etron.

E* +.RR! Ty+ia$ O 10000 vo$ts yie$ds ontinuous S@ray distri&utions – i% .2- +.R

Bresstrah$un%.

(air +rodution ,R) (hoton disa++ears and +rodues +arti$e P anti+arti$e.

#ssue +airs are +rodued essentia$$y at rest.

E*! e P +ositron eah have rest ass .H11 'ev so +hoton ust have 1.022 'e

What is a +hotonK ,+.0)

• s+eed 9

• ass 9 0

• +hotons have ener%y E 9 h ν 9 hλ and oentu + 9 hλ

• an &e reated or destroyed Ahen radiation ,e.%. +arti$es) are eitted or

a&sor&ed

• an have +arti$e@$ike o$$isions Aith other +arti$es suh as e$etrons

"i%ht a$so e*hi&its Aave@$ike +ro+erties suh as intererene and diration.

:'! Whih is it Aave or +arti$eK = Ae send a $i%ht Aave throu%h a dou&$e@s$it

e*+erient then detet Ahih s$it it Aent throu%h ,e.%. Aith a +hotoe$$) Ae destroy theAave nature o the $i%ht and redue it to +arti$e@$ike &ehavior.

"i%ht &ehaves $ike a Aave even i you send on$y one +hoton at a tie throu%h a devie as$on% as you donGt o&serve the +hoton a$on% the Aay.



4&servation o$$a+ses the Aave untionU and han%es the e*+erient.

The e*+erienter is +art o the e*+erient.

:2! Can it &e +redited Ahether a +hoton Ai$$ at ore Aave@$ike or +arti$e@$ike in any

%iven e*+erientK ,Tea (hysis) +t depends what we choose to measure$

'odern (=:!

The de$ayed hoie e*+erient detai$ed on +a%e 1 o rane shoAs that it is not the ase that

$i%ht is soeties a +arti$e and soeties a Aave. The e*+erient shoAs that $i%ht has +ro+erties o a +arti$e and o a Aave a$$ the tie. 4n +a%e rane Arites that +arti$e and

Aave &ehavior ust soehoA &e taken to%ether to %ive a o+$ete desri+tion o the

+ro+erties o e$etroa%neti radiation.U ,#5)

eent$y e*+erienters Aere a&$e to +eror the dou&$e s$it e*+erient usin% &uky&a$$s,o$eu$es ade out o si*ty ar&on atos). ,see htt+!si.teh@

arhive.net#rhivesi.+hysis200@0-s%00R-2.ht$) ,#5)

'odern :5ET=4N

5nder Ahat irustanes is it onvenient to re%ard a +hoton as havin% a assK ,+a%e F-

rane) + 0#12T %(3#MM(10 +T

Documents

Fall6Boas3Mod3