UKM 1/23-2-2011

Pwc ja gÐnei to mjhma.

Na steÐlete m numa sth dieÔjunsh:massen@physics.auth.grme to ìnoma, AEM, kai to E-mail sac tou APJ.Arijmì Thlef¸nouSubject: kwdikos YKF-II

https://blackboard.lib.auth.gr

http://users.auth.gr/∼massen

Kat th dirkeia twn majhmtwn ja dÐnontaiorismènec ask seic pou ja prèpei na paradÐdontailumènec se orismènh hmeromhnÐa.H pardosh twn ask sewn kai h parakoloÔjhsh twndialèxewn lambnontai polÔ sobar upìyh stontelikì bajmì.

UKM 1/23-2-2011

Upologistik Kbantomhqanik

To 1o mèroc tou maj matoc perilambnei:

1 Parrthma A

Qr simec sqèseic kai orismoÐSfairikèc armonikècPolu¸numa LegendreSunart seic Bessel

2 SÔntomh epanlhyh thc Kbantomhqanik c3 StoiqeÐa JewrÐac skèdashc

Efarmog thc Mathematica sth skèdash hlektronÐwn apìtomaEfarmog thc Fortran sth skèdash hlektronÐwn apì toma

UKM 1/23-2-2011

Ekfrseic twn telest¸n ~∇ kai ∇2

O telest c ~∇:

~∇ = x0∂

∂x+ y0

∂

∂y+ z0

∂

∂z= r0

∂

∂r+ ϑ0

1

r

∂

∂ϑ+ ϕ0

1

r sinϑ

∂

∂ϕ(1)

x = r sinϑ cosϕ, y = r sinϑ sinϕ, z = r cosϑ

r0 = x0 sinϑ cosϕ + y0 sinϑ sinϕ + z0 cosϑ

ϑ0 = x0 cosϑ cosϕ + y0 cosϑ sinϕ− z0 sinϑ

ϕ0 = −x0 sinϕ + y0 cosϕ

O telest c ~∇2:

∇2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2

=1

r2

∂

∂r

(r2 ∂

∂r

)+

1

r2 sinϑ

∂

∂ϑ

(sinϑ

∂

∂ϑ

)+

1

r2 sin2 ϑ

∂2

∂ϕ2(2)

UKM 1/23-2-2011

Ekfrseic tou telest l = r × p

Oi sunist¸sec tou telest thc stroform c kai otelest c l2:

lx = y pz − zpy = −i~(

y∂

∂z− z

∂

∂y

)= i~

(sinϕ

∂

∂ϑ+ cotϑ cosϕ

∂

∂ϕ

)

ly = zpx − xpz = −i~(

z∂

∂x− x

∂

∂z

)= i~

(− cosϕ

∂

∂ϑ+ cot ϑ sinϕ

∂

∂ϕ

)

lz = xpy − y px = −i~(

x∂

∂y− y

∂

∂x

)= −i~

∂

∂ϕ(3)

l2 = l2x + l2y + l2z = −~2

[1

sinϑ

∂

∂ϑ

(sinϑ

∂

∂ϑ

)+

1

sin2 ϑ

∂2

∂ϕ2

](4)

Gia ton telest thc stroform c isqÔoun oi sqèseicantimetjeshc:

l× l = i ~l, [lx , ly ] = i ~lz , [ly , lz ] = i ~lx , [lz , lx ] = i ~ly

UKM 1/23-2-2011

Oi sfairikèc armonikèc

Y ml (ϑ, ϕ) : EÐnai idiosunart seic twn telest¸n l2 kai lz pou

an koun stic idiotimèc l(l + 1)~2 kai ~m, antÐstoiqa.

l2Y ml (ϑ, ϕ) = l(l + 1)~2Y m

l (ϑ, ϕ), l = 0, 1, 2, . . .

lzYml (ϑ, ϕ) = m~Y m

l (ϑ, ϕ), |m| ≤ l (5)

OrÐzontai me th bo jeia twn prosarthmènwn sunart sewnLegendre Pm

l (cosϑ)

Y ml (ϑ, ϕ) =

√2l + 1

4π

(l −m)!

(l + m)!(−1)meimϕPm

l (cosϑ) (6)

Pml (cosϑ) =

(−1)l+m

2l l!

(l + m)!

(l −m)!sin−m ϑ

dl−m

d(cosϑ)l−msin2l ϑ, −l ≤ m ≤ l

`Opwc orÐsthkan eÐnai orjokanonikèc sunart seic

〈Ylm|Yl ′m′〉 =

∫ 2π

0

∫ π

0Y ∗

lm(ϑ, ϕ)Yl ′m′(ϑ, ϕ)dΩ = δll ′ δmm′

UKM 1/23-2-2011

Oi pr¸tec sfairikèc armonikèc (gia l = 0, 1, 2) eÐnai thc morf c:

Y 00 =

1√4π

Y 01 =

√3

4πcosϑ =

√3

4π

z

r=

√3

4πP1(cosϑ)

Y±11 = ∓

√3

8πsinϑ e±iϕ =

√3

8π

x ± iy

r

Y 02 =

√5

16π(3 cos2 ϑ− 1) =

√5

16π

2z2 − x2 − y2

r2=

√5

16πP2(cosϑ)

Y±12 = ∓

√15

8πcosϑ sinϑ e±iϕ =

√15

8π

(x ± iy)z

r2

Y±22 = ∓

√15

32πsin2 ϑ e±i2ϕ =

√15

32π

(x ± iy)2

r2(7)

IsqÔoun epÐshc oi sqèseic:

Y 0l (ϑ, ϕ) =

√2l + 1

4πPl(cosϑ), Y m

l (0, ϕ) =

√2l + 1

4πδm0 (8)

UKM 1/23-2-2011

Polu¸numa Legendre, Pn(x), | x |≤ 1.

TÔpoc tou Rodriguez : Pn =(−1)n

2nn!

dn

dxn

[(1− x2)n

]=

1

2nn!

dn

dxn

[(x2 − 1)n

]

Genn tria sunrthsh : g(x , t) =1√

1− 2xt + t2=

∞∑

n=0

Pn(x)tn,

| x |≤ 1, | t |< 1

Anadromikèc Sqèseic : (n + 1)Pn+1(x) = (2n + 1)xPn(x)− nPn−1(x)xP ′n − P ′n−1 = nPn, n = 1, 2, 3, . . .

Diaforik exÐswsh:

(1−x2)P ′′n (x)−2xP ′n(x)+n(n+1)Pn(x) = 0 , n = 0, 1, 2, . . . , x ∈ [−1, 1]

UKM 1/23-2-2011

Orjogwniìthta: (Pn, Pm) =2

2n + 1δnm

Qr simec idiìthtec:

Pn(−x) = (−1)nPn(x), Pn(1) = 1, Pn(−1) = (−1)n

Anptugma sunrthshc se seir poluwnÔmwn Legendre.

f (x) =∞∑

n=0

cnPn(x), −1 < x < 1, cn =2n + 1

2

∫ 1

−1f (x)Pn(x)dx

PÐnakac 1.3: Polu¸numa Legendre

P0(x) = 1P1(x) = xP2(x) = 1

2(3x2 − 1)P3(x) = 1

2(5x3 − 3x)P4(x) = 1

8(35x4 − 30x2 + 3)P5(x) = 1

8(63x5 − 70x3 + 15x)

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

P4

P3

x

P2

P1

P0

Pn(x)

UKM 1/23-2-2011

Sunart seic Bessel

H D.E. Bessel

z2w ′′(z) + zw ′(z) + (z2 − ν2)w = 0, ν = parmetroc (9)

Mia merik lÔsh aut c eÐnai h sunrthsh Bessel 1ou eÐdouctxhc ν.

Jν(z) =∞∑

k=0

(−1)k

k!Γ(k + ν + 1)

(z

2

)2k+ν, |z | < ∞ (10)

An to ν 6= akèraioc mia deÔterh lÔsh, grammik anexrthth thcJν(z), mporeÐ na prokÔyei apì th (10) jètontac ν → −ν.

Genn tria sunrthsh: g(z , t) = ez2(t− 1

t) (11)

g(z , t) = exp

[z

2

(t − 1

t

)]=

∞∑n=−∞

Jn(z)tn, 0 < |t| < ∞ (12)

UKM 1/23-2-2011

Oi suntelestèc Jn(z) dÐnontai pli apì thn sqèsh (10).H sunrthsh Bessel eÐnai talantoÔmenhc morf c, ìqi ìmwcperiodik . To pltoc thc Jn(x) den eÐnai stajerì kai gia meglax sumperifèretai ìpwc h sunrthsh 1/

√x .

• Oi sunart seic Bessel eÐnai eswterikèc sunart seic thcMathematica. To Sq ma 1 ègine me th Mathematicaqrhsimopoi¸ntac thn eswterik sunrthsh BesselJ[n, x] kai thnentol Plot

2 4 6 8 10x

-0.4

-0.2

0.2

0.4

0.6

0.8

1

JnHxLJ0

J1

J2Sq ma: H grafik parstash twnsunart sewn Bessel J0(x), J1(x) kai J2(x).

UKM 1/23-2-2011

• Gia tic sunart seic Bessel akèraiac txhc n isqÔoun oi sqèseic

J−n(z) = (−1)nJn(z) kai Jn(−z) = (−1)nJn(z) , n = akèraioc

Anadromikèc sqèseic

Jn−1(z) + Jn+1(z) =2n

zJn(z), Jn−1(z)− Jn+1(z) = 2J ′n(z)

n = akèraioc mh akèraioc

An dojoÔn oi sunart seic J0(z) kai J1(z) mporeÐ na brejeÐ hJ2(z) kaj¸c kai kje llh an¸terhc txhc.

UKM 1/23-2-2011

Sunart seic Neumann

Oi Jν(z) kai J−ν(z) ìtan ν 6= akèraioc eÐnai grammikanexrthtec kai h genik lÔsh thc D.E. Bessel eÐnai

w(z) = AJν(z) + BJ−ν(z) (13)

Me th bo jeia twn Jν(z) kai J−ν(z) mporoÔme na orÐsoume kaillec merikèc lÔseic thc D.E. Bessel. H sunrthsh

Nν =Jν(z) cos(νπ)− J−ν(z)

sin(νπ)(14)

pou lègetai sunrthsh Neumann sunrthsh Bessel 2oueÐdouc txhc ν, eÐnai mia tètoia merik lÔsh.An ν 6= akèraioc arijmìc, h genik lÔsh thc D.E. Bessel mporeÐna ekfrasteÐ wc o grammikìc sunduasmìc thc (13) all mporeÐna ekfrasteÐ kai wc o grammikìc sunduasmìc twn sunart sewnJν(z) kai Nν(z).

UKM 1/23-2-2011

`Otan ν = n = akèraioc (oi Jn(z) kai J−n(z) eÐnai grammikexarthmènec) h genik lÔsh thc D.E. Bessel eÐnai ènac grammikìcsunduasmìc twn Jn(z) kai Nn(z).• H sunrthsh Neumann gia ν = n =akèraioc orÐzetai an kai hsqèsh (14) odhgeÐ sthn aprosdiìristh morf 0/0.• H Nn(z) gia z → 0 teÐnei sto −∞.H morf pou èqoun oi treic pr¸tec sunart seic Neumannakèraiac txhc faÐnetai sto Sq ma 2.

2 4 6 8 10x

-1.5

-1

-0.5

0.5

1NnHxL

N0N1 N2

Sq ma: H grafik parstash twnsunart sewn Neumann N0(x), N1(x) kaiN2(x).

UKM 1/23-2-2011

Sunart seic Hankel

Ektìc apì tic sunart seic Jn(z) kai Nn(z) kai llec eidikècmorfèc lÔsewn thc D.E. Bessel qrhsimopoioÔntai stic efarmogèc.Tètoiec eÐnai oi sunart seic Hankel H

(+)ν (z) kai H

(−)ν (z), pou

orÐzontai apì th sqèsh

H±(z) = Jν(z)± iNν(z) (15)

O orismìc autìc eÐnai anlogoc me ton orismì thc ekjetik csunrthshc

e±iθ = cos θ ± i sin θ

• Oi sunart seic Neumann kai Hankel ikanopoioÔn tic Ðdiecanadromikèc sqèseic pou ikanopoioÔn kai oi sunart seic Bessel.

UKM 1/23-2-2011

Sfairikèc sunart seic Bessel

H D.E. z2w ′′(z) + 2zw ′(z) + (k2z2 − l(l + 1))w(z) = 0 (16)

me to metasqhmatismì w(z) = z−1/2u(z) gÐnetai h D.E.

z2u′′(z) + zu′(z) + (k2z2 − (l + 1/2)2)u(z) = 0 (17)

Aut pli jètontac t = kz metasqhmatÐzetai sth D.E. Besseltxhc l + 1

2 . `Etsi h genik lÔsh thc (16) eÐnai h sunrthsh

w(z) = Az−1/2Jl+ 12(kz) + Bz−1/2Nl+ 1

2(kz)

Aut h genik lÔsh grfetai pio komy me th morf

w(z) = A1jl(kz) + B1nl(kz)

ìpou A1 =√

2k/πA kai B1 =√

2k/πB kai

jl(z) =

√π

2zJl+ 1

2(z) kai nl(z) =

√π

2zNl+ 1

2(z) (18)

UKM 1/23-2-2011

Oi jl(z) kai nl(z) lègontai sfairikèc sunart seic Bessel kaiNeumann txhc l .Apì autèc orÐzontai kai llec merikèc lÔseic thc D.E. (16) ìpwcoi sfairikèc sunart seic Hankel 1ou kai 2ou eÐdouc

h(±)l (z) = jl(z)± inl(z) (19)

• Oi sfairikèc sunart seic akèraiac txhc mporoÔn naekfrastoÔn me th bo jeia stoiqeiwd¸n sunart sewn. Giapardeigma oi jl(z) kai nl(z) gia l = 0 kai l = 1 kai h h

(±)0 (z)

faÐnontai paraktw

j0(z) =sin z

zn0(z) = −cos z

z

j1(z) =sin z

z2− cos z

zn1(z) = −cos z

z2− sin z

z

h(±)0 (z) =

∓i

ze±iz

UKM 1/23-2-2011

An sumbolÐsoume me fl(z) tic sunart seic jl(z), nl(z) kai hl(z)tìte isqÔoun oi paraktw anadromikèc sqèseic

2(l + 1)f ′l (z) = lfl−1(z)− (l + 1)fl+1(z)

2l + 1

zfl(z) = fl−1(z) + fl+1(z)

UKM 1/23-2-2011

Asumptwtikèc ekfrseic twn sfairik¸n sunart sewn

Gia |z | ¿ 1 kai l =akèraioc isqÔoun oi sqèseic

jl(z) ' z l

(2l + 1)!!, nl(z) ' −(2l − 1)!!

z l+1

Gia z →∞ kai l =akèraioc isqÔoun oi sqèseic

jl(z) =1

zcos

(z − lπ

2− π

2

)=

1

zsin

(z − lπ

2

)

nl(z) = −1

zsin

(z − lπ

2− π

2

)= −1

zcos

(z − lπ

2

)

h(±)l (z) = ∓ i

zexp

[±i

(z − lπ

2

)]

UKM 1/23-2-2011

Askhsh A-1. Me th bo jeia thc Mathematica na deqteÐ ìti hDE (16) me to metasqhmatismì w(z) = z−1/2u(z)metasqhmatÐzetai sth DE (17). Aut pli jètontac t = kzmetasqhmatÐzetai sth D.E. Bessel txhc l + 1

2 .

UKM 1/23-2-2011

Arijmhtikìc upologismìc twn Eidik¸n sunart sewn

Askhsh A-2. a) Me th bo jeia twn anadromik¸n sqèsewn twnpoluwnÔmwn Legendre:

(n + 1)Pn+1(x) = (2n + 1)xPn(x)− nPn−1(x), n = 1, 2, 3, · · ·(20)

(1− x2)P ′n(x) = −nxPn + nPn−1 (21)

kai twn gnwst¸n tim¸n P0(x) = 1, P1(x) = x , P ′0(x) = 0 kaiP ′1(x) = 1 na brejoÔn oi timèc twn poluwnÔmwn Legendre P10(x),P20(x) sta shmeÐa x = −1 ,−0.8, . . . , 0.8, 1 kai twn pr¸twnparag¸gwn touc sta shmeÐa x = −0.999 ,−0.8, . . . , 0.8, 0.999.b) Na sugkrÐnete autèc tic timèc me tic timèc pou sac dÐnei hMathematica.

UKM 1/23-2-2011

Askhsh A-3. H Mathematica dÐnei me kal akrÐbeia tic timèctwn sunart sewn Jν(z) kai Nν(z). H Mathematica apì 6 kaimet dÐnei me kal akrÐbeia kai tic timèc twn sfairik¸nsunart sewn Bessel jν(z) kai nν(z).a) Me th bo jeia thc Mathematica na breÐte tic timèc twnsunart sewn j0(z), n0(z) kai j1(z), n1(z) gia z = 0.5, 1., 1.5 kai2.b) Qrhsimopoi¸ntac thn anadromik sqèsh twn sfairik¸nsunart sewn Bessel:

2l + 1

zfl(z) = fl−1(z) + fl+1(z)

kai tic timèc twn j0(z), n0(z) kai j1(z), n1(z) gia z = 0.5, 1., 1.5kai 2 na brejoÔn oi timèc twn jl(z) kai nl(z) gia l = 2 èwc kail = 20 sta Ðdia shmeÐa.g) Na sugkrÐnete tic timèc autèc me tic antÐstoiqec timèc pou dÐneih Mathematica.