TheRigidRotorProblem:Aquantumpar7cleconfinedtoasphere

SphericalHarmonicsReading:McIntyre7.6

Legendre’sequa7on(m=0)Theseriesmustbefinite!

• Iftheseriesisnotfinite,thepolynomialblowsup(checkra7oforlargen)

A = nmax

nmax

+1( )A = ! ! +1( )

an+2

=n(n +1) ! A

(n + 2)(n +1)a

n

• ThesespecialvaluesofAgiveLegendrepolynomials.

An7cipatedthis!

P0(z) = 1

P1(z) = z

P2(z) =

1

2(3z

2!1)

P3(z) =

1

2(5z

3! 3z)

P4(z) =

1

8(35z

4! 30z

2+ 3)

P5(z) =

1

8(63z

5! 70z

3+15z)

Theenergyfortherigidrotorproblem

• GobacktotheoriginalstatementoftherigidrotorproblemtoseehowAisrelatedtoE.

• elisaninteger(energyisquan7zed)

A = ! ! +1( )! E!=! ! +1( )"

2I

2

Legendrepolynomials• Rodrigues’formula

• Orthogonal(Hermop)!

• Ofdegree

• Eitherevenorodd

P!(z) =

1

2! !!

d!

dz!

z2!1( )

!

!1

1

"Pk

*(z)P!(z)dz =

#k!

! +1

2

!

Legendrepolynomials• Orthogonalmeanswecanexpressanyfunc7ondefinedonthatintervalasasuperposi7onofLegendrepolynomials:HW

• Thismeanswecanprojectandfindprobabili7esjustaswithsines,cosinesandexp(imφ)

! + 1

2( )!1

1

"Pk

*(z)P!(z)dz = #

k!

AssociatedLegendrepolynomials

• m≠0

(1! z2 )

d2

dz2! 2z

d

dz+ !(! +1) !

m2

(1! z2 )

"

#$%

&'P(z) = 0

P!

m(z) = P!

!m(z) = (1! z2 )m/2 d

m

dzm

P!(z)( )

=1

2! !!(1! z

2 )m/2 dm+!

dzm+!

(z2!1)!( )

AssociatedLegendrepolynomials

• m≠0,integer !

P0

0= 1

P1

0= cos!

P1

1= sin!

P2

0=

1

2(3cos2

! "1)

P2

1= 3sin! cos!

P2

2= 3sin2

!

P3

0=

1

2(5cos3

! " 3cos!)

P3

1=

3

2sin!(5cos2

! "1)

P3

2= 15sin2

! cos!

P3

3= 15sin3

!

AssociatedLegendrepolynomials

• restrictsmto±eldetermines#oforbitals

• oppositeprojec7onshavesamespa7alform

• yzplaneisanode

• parity(selec7onrules)

• Orthogonal(forgivenm)

P!

m(z) = 0 if m > !

P!

!m(z) = P!

m(z)

P!

m(±1) = 0 for m ! 0

P!

m(!z) = (!1)!!mP!

m(z)

!1

1

"P!

m(z)Pq

m(z)dz =2

(2! +1)

(! + m)!

(! ! m)!#!q

Finally…..

• Thenormalizedpolynomialthatsolvesthethetaequa7onis(typoinbookEq.7.156):

!!

m(") = #1( )m (2! +1)

2

(!# | m |)!

(!+ | m |)! P!

m(cos")

Summary• Thefactthattheseriesmustbefinitepolynomialofdegreeeliswhat(mathema7cally)leadstoamaximum(andinteger)valueofel!

• Thefactthattheseriesisfinitemeansthatwecandifferen7ateitonlyafinitenumberof7mes(el)andthatnumberturnsouttobem!!Thusmislimitedto–el…el.

• Legendrefunc7onsareorthogonalon[‐1,1]• LegendreandassociatedLegendrefunc7ons.

Sphericalharmonics:

• Originalproblemwasseparatedintoangularandradialparts.Wefurtherseparatedangularpartsintothetaandphi.

• Thesefunc7onsaretheangularpartofthesolu7ontothehydrogenatomproblem,andindeedANYcentralforceproblem!!

Y!

m !,"( ) =#!

m !( )$m"( )

Doyouknowwhatthisis?

Thecosmicmicrowavebackgroundisanalyzedwithsphericalharmonics!

SphericalHarmonics

Y!

m(! ,") = (#1)(m+ |m|) / 2 (2! +1)

4$

(!# | m |)!

(!+ | m |)!P!

m(cos!)eim"

! m Y!

m !,"( )

0 0 Y0

0= 1

4#

1 0 Y1

0= 3

4#cos!

±1 Y1

±1= " 3

8#sin!e± i"

2 0 Y2

0= 5

16#3cos

2! $1( )

±1 Y2

±1= " 15

8#sin! cos!e± i"

±2 Y2

±2= 15

32#sin

2!e± i2"

3 0 Y3

0= 7

16#5cos

3! $ 3cos!( )

±1 Y3

±1= " 21

64#sin! 5cos

2! $1( )e± i"

±2 Y3

±2= 105

32#sin

2! cos!e± i2"

±3 Y3

±3= 35

64#sin

3!e± i3"

SphericalHarmonics

• Whatistheac7onoftheseoperatorsonthesphericalharmonics?

Y!

m(! ,") = (#1)(m+ |m|) / 2 (2! +1)

4$

(!# | m |)!

(!+ | m |)!P!

m(cos!)eim"

L2Y!

m(! ,") = ?

LzY!

m(! ,") = ?

• Dothetwooperatorscommute?

SphericalHarmonics

• Whatistheac7onoftheseoperatorsonthesphericalharmonics?

!m

L2!m = ?

Lz!m = ?

• Dothetwooperatorscommute?

SphericalHarmonics

• Orthonormalonthesphere Y!

m(! ,") = (#1)(m+ |m|) / 2 (2! +1)

4$

(!# | m |)!

(!+ | m |)!P!

m(cos!)eim"

0

2!

"0

!

" Y!1

m1 # ,$( )( )

*

Y!2

m2 # ,$( )sin#d#d$

?

" #$ %$= %

!1!2

%m

1m

2

Doyourememberwherethiscomesfrom?

!1m

1!

2m

2= !

!1!2

!m

1m

2

• Nowwe’llspendsome7mevisualizingprobabilitydistribu7onsofeigenstatesandsuperposi7ons.

SphericalHarmonics

SphericalHarmonics

SphericalHarmonics

SphericalHarmonicExpansion

! (" ,#) =60060

139301$1

4+ cos3 2" + sin2#

%&'

()*

P

E!

= !m !2

m="!

!

#

Aboutthesphericalharmonics

• parity(“even”or“odd”.Thiswillbehelpfulwhenstudyselec7onrulesfortransi7onsbetweenstates.

• oppositeprojec7onshavesamespa7alform

• yzplaneisanode

• Orthonormal

Y!

m(! ,") = #1( )!

Y!

m($ #! ," + $ )

Sphericalharmonicsasabasisset• Theorthonormalityofthesphericalharmonicsmeansthatanyfunc7ondefineonaspherecanbeexpressedasasuperposi7onofsphericalharmonics.

• Electromagne7sm–mul7polesofachargedistribu7on

• Gravita7on–mul7polesofamassdistribu7on• TheCMBexample–mul7polesofthetemperaturefluctua7onsoftheearlyuniverse

Sphericalharmonicsasabasisset

• Findthecoefficientsofthefunc7on

expandedasaLaplaceseries(seriesofsph.har.) f (! ,") =

15

16#sin2! sin"

f (! ,") = c!m

m=#!

!

$!=0

%

$ Y!

m ! ,"( )

Thereal(asopposedtocomplex)sphericalharmonics

• Ingravita7onproblems,thecomplexnumbersinthesphericalharmonicsarerarelyuseful,sootherlinearcombina7onsareused.

• You’llrecognizetheseasthep,d,forbitalformsfromchemistry

d

yz(! ,") ! Y

2

1 ! ,"( ) + ei#Y2

$1 ! ,"( )

Summary• Thefactthattheseriesmustbefinitepolynomialofdegreeeliswhat(mathema7cally)leadstoamaximum(andinteger)valueofel!

• Thefactthattheseriesisfinitemeansthatwecandifferen7ateitonlyafinitenumberof7mes(el)andthatnumberturnsouttobem!!Thusmislimitedto–el…el.

• Legendrefunc7onsareorthonormalon[‐1,1]• LegendreandassociatedLegendrefunc7ons.