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8/9/2019 Revision No Exam wwStatistics

1/14

PHAS2222 Revision Lecture 2008

PHAS2222 Revision Lecture

The plan: First hour:

Summary o main points an! e"uations o course

#pportunity to re"uest particular topics

Secon! hour: Specially re"ueste! topics

The 200$ e%am paper &mainly Section '(

)ote: mo!el ans*ers to e%ams not availa+le on*e+site, +ut - am happy to .ive ee!+ac/ onattempts at past e%amination papers


2/14

Photoelectric eect, 1ompton

scatterin.

avisson3ermer e%periment,

!ou+leslit e%periment

Particle nature o li.ht in

"uantum mechanics

4ave nature o matter in

"uantum mechanics

4aveparticle !uality

Time!epen!ent Schr5!in.er

e"uation, 'orn interpretation

2267 athsetho!s ---

Timein!epen!ent Schr5!in.er

e"uation

9uantum simple

harmonic oscillator

102

( )nE n = + h

Hy!ro.enic atom pro+lems

Ra!ial solution2

2

1,

2

nl

ZR E

n

=

An.ular solution

( , )mlY

Postulates:

#perators,ei.envalues an!

ei.enunctions, e%pansionsin complete sets,

commutators, e%pectation

values, time evolution

An.ular momentum

operators2 ,zL L

E h=

hp

=

2267

Frobenius

method

Separation ofvariables

Legendre

equation


3/14

Section Failure o classical

mechanics (1.1)E h=

max (1.2)K h W=

hp

=

The photoelectric eect:

e 'ro.lie;s relationship or matter*aves:

1ompton scatterin.: *hen photon !electe! throu.h

an.le


4/14

Section 2 A *ave e"uation or

matter *aves

( ) exp( i / )T t Et = h

2 2

2i ( , ) (2.3)

2V x t

t m x

= +

h

h

2 2

2( , ) (2.4)

2

dV x t H

m dx

+

h

( , ) ( ) ( ) (2.9)x t x T t =

Time!epen!ent Schr5!in.er e"uation:

2 2

2i (2.2)

2t m x

=

h

h

(for matter waves in free space)

&.enerally(Hamiltonian operator &represents ener.y oparticle(:

2

( , ) (2.6)x t x

Born interpretation: proai!it" of fin#in$ partic!e in a sma!! !en$t% &xat

positionxan# time tis ea! to

- Hamiltonian is in!epen!ent o time,can try solution

(2.13)H E =Fin!&time

in!epen!ent

S?(

@ncertainty principle: (strict version)2

x p h


5/14

V(x

)

a

V0

Section ?%amples o the time

in!epen!ent S? in !imension2 2

2# ( ) (2.12)

2 #V x E

m x + =h

1.*e a continos an# sin$!e+

va!e# fnction of ot%xan# t

2.ave a continos first #erivative(n!ess t%e potentia! $oes to infinit")

3.ave a finite norma!i-ation

inte$ra!.

%e wavefnction mst:Free particle

-ninite s"uare *ell

Finite s"uare *ell

Potential step

Rectan.ular +arrier

V = V =

V(

x)

-a a

0V =

V(x)

V0

9uantiBation o ener.y

Travellin. *aves,

ar+itrary value o

ener.y

atchin. o solutions:

travellin. *aves &sines or

cosines( in *ell, e%ponentials

in +arriers

Transmission an! relection

V(x)

x

V0 Tunnellin.


6/14

Section cont!

Particle lu% &lo* opro+a+ility(:

artic!e f!x at position

i( )

2

x

xm x x

=

h

Simple harmonic oscillator:1/ 2

0m

y x =

h 2stitte ( ) ( )exp( / 2)y H y y =

Series solution or H(y) must

terminate, so H is a inite po*er

series &polynomial( calle! a Hermite

polynomialC

Termination con!ition0

1ner$" ( )

2E n = + h

2 2

2 21022#ami!tonian 2 #

H m xm x

= +h


7/14


8/14

Section 6 cont!

7 wit%n n n m m m n m' ' = =

%e ei$enfnctions of a ermitian operator e!on$in$ to#ifferent ei$enva!es are orthogonal.

# 0n m

x

=f t%en%e ei$enfnctions nof a ermitian operator

form a complete set, meanin$ t%at an" ot%er

fnction satisf"in$ t%e same on#ar"

con#itions can e expan#e# as

( ) ( )n nn

x a x =

( ) ( ).n nn

x a x =

Postulate 4.4: sppose a measrement of t%e antit" 'is ma#e,an# t%at t%e (norma!i-e#) wavefnction can e expan#e# in terms of

t%e (norma!i-e#) ei$enfnctions nof t%e correspon#in$ operator as

%en t%e proai!it" of otainin$ t%e ei$enva!e nas t%emeasrement res!t is

2

na

8ote t%at2

1n

n

a = if *norma!i-e#, ort%onorma!.


9/14

Section 6 cont!

,' R 'R R' 1ommutator:1ommutin. operators have same ei.enunctions,

can have *ell!eine! values simultaneously

&Dcompati+le;(

2

.n nn

' a (= ?%pectation value:

Postulate 4.5: *etween measrements (i.e. w%en it is not #istre# "

externa! inf!ences) t%e wave+fnction evo!ves wit% time accor#in$ to

t%e time+#epen#ent c%r#in$er eation.

n n nH E = ( ,0) ( )n n

n

x a x =

Time !evelopment in terms o ei.enunctions o Hamiltonian:

- an!

then ( , ) exp( / ) ( )n n nn

x t a iE t x = h


10/14

Section E An.ular momentum

2 2 ( , ) ( 1) ( , )m ml lL Y l l Y = + h

22 2

2 2

1 1

sinsin sinL

= +

h

, ;x y zL L i L= h

ierent components !o not commute:

Lz

Lx

Ly

+ut2 , 0zL L =

-n spherical polar coor!inates:

zL i

= h

Their simultaneous ei.enunctions

are spherical harmonics:

( , ) ( , ),m m

z l lL Y m Y l m l = h

1onserve! or pro+lems *ith

spherical symmetry


11/14

Section 7 The hy!ro.en atom

2 2

eff 20

( 1)

( ) 4 2e

Ze l l

V r r m r

+

= +

h

2 2

eff2

#( )

2 #eV r E

m r

+ =

h

2 22

0

2 4e

ZeH

m r= h

8ow !oo5 for so!tions in t%e form

( , , ) ( ) ( , )r R r Y =

An.ular parts are spherical harmonics

( )( )

rR r

r

=Ra!ial part:

*ith

2 1

23 2

0

!anc5 constant 1 (#imensions ;)

!ectron mass 1 (#imensions ;)


12/14

Section 7 cont!

Put ( ) ( )exp( )r , r r =

2

wit%2

E =

Series solution or Fmust terminate:

possi+le only iw%ere is an inte$er : 1, 2

Zn n l n l l

= > = + + K

2

2(in atomic nits)

2n

ZE

n=

, ( 1), 0, ( 1),m l l l l = K K

l=0,1,2,>,n-1

n is principal "uantum num+er

l0 l. l/ l

0

2

in nits2

h

ZE E

+1


13/14

Section $ Spin

1(for e!ectron)7 !i5e in or#inar" an$!ar momentm

2

1(for e!ectron)7 !i5e in or#inar" an$!ar momentm

2%

% l

m % % m

=

= = K

0

( )

2 (?irac's re!ativistic t%eor")

2.0023193043@ (Aantm !ectro#"namics)

1

H H "

"

"

= + +==

B L Sh-nteraction *ith ma.netic iel!:

9uantum num+ers !escri+in. spin:

Lx

Ly

Lz

L

2

LB2

CL+2C

(t%e *o%r ma$neton).2

1

e

e

m =

h

1ouplin. o spin to or+ital

an.ular momentum:

Stern3erlach e%periment: atoms*ith sin.le outer electron !ivi!e into

two.roups *ith opposite ma.netic

momentsC


14/14

Section $ cont!

Total an.ular momentum &or+ital

spin( = +J L S

L

S

escri+e! +y t*o "uantum num+ers:

&!eterminin. quantityo total an.ular momentum

present(G ran.es rom l!s to l"sin inte.er steps

m&!eterminin. proIection o total an.ular

momentum alon. B(, ran.es rom#to "in inte.er

steps

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