18. More Special Functions
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Transcript of 18. More Special Functions
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18. More Special Functions
1. Hermite Functions
2. Applications of Hermite Functions
3. Laguerre Functions
4. Chebyshev Polynomials
5. Hypergeometric Functions
6. Confluent Hypergeometric Functions
7. Dilogarithm
8. Elliptic Integrals
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1. Hermite Functions
2 2 0n n nH xH nH Hermite ODE :
Hermite functionsHermite polynomials ( n = integer )
2 2n
n x xn n
dH x e e
d x
2 2
2 0x xn ne H n e H
Rodrigues formula
exp 2w d x x 2xe
0
0
1
qd x
p
nn
n n
p y q y y
wp wq w pe
w p y w y
dy w p
w d x
2 2
0
,!
nt t x
nn
tg x t e H x
n
Generating function :Assumed starting point here.
Hermitian form
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Recurrence Relations 2 2
0
,!
nt t x
nn
tg x t e H x
n
2
12
1
21 !
nt t x
nn
g tt x e H x
t n
10 0
2! !
n n
n nn n
t tt x H x H x
n n
0 11
2 2 2!
n
n nn
txH x nH x xH x
n
1 02H x xH x 1 12 2n n nH x nH x xH x 1n
2 2
0
2!
nt t x
nn
g tte H x
x n
1
0 1
2 2! 1 !
n n
n nn n
t tt H x H x
n n
1 11 !
n
nn
tH x H x
n
0 0H x 12n nH x nH x 1n
0,0 1g x H x All Hn can be generated by recursion.
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Table & Fig. 18.1. Hermite Polynomials
Mathematica
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Special Values 2 2
0
,!
nt t x
nn
tg x t e H x
n
2
0
0, 0!
nt
nn
tg t e H
n
2
0 !
nn
n
t
n
2 1 2
2 1 20 0
0 02 1 ! 2 !
n n
n nn n
t tH H
n n
2 1 0 0nH 2
2 !0
!n
n
nH
n 0n
2 2
0
,!
nnt t x
nn
tg x t e H x
n
0 !
n
nn
tH x
n
n
n nH x H x
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Hermite ODE 1 12 2n n nH x nH x xH x
12n nH x nH x
1 2n n nH H xH
1 2 2n n n nH H H xH 2 1 2 2n n nn H H xH
2 2 0n nH xH nH Hermite ODE
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Rodrigues Formula 2 2
0
,!
nt t x
nn
tg x t e H x
n
2 2t t xge
t t
22 t xxe et
22 t xxe e
x
1
1 1 !
n
nn
tH x
n
2 2
0
n nn x x
n n
t
ge e
t x
nH x 0n
22n n
t xxn n
ge e
t t
22
nn t xx
ne e
x
!m n
mm n
tH x
m n
Rodrigues Formula
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Series Expansion 2 2
0
,!
nt t x
nn
tg x t e H x
n
2 2
0
2
!
mmt t x
m
t t xe
m
0 0
2!
m mm j jm
jm j
tC x t
m
0 0
2! !
jmm j m j
m j
x tm j j
n m j
k j
0, ,
0, ,
n
k m
2j m n m 0, ,2
nk
consistent only if n is even
For n odd, j & k can run only up to m 1, hence &
/22
0 0
22 ! !
knn k n
n k
g x tn k k
2 1n m max
11 / 2
2k n n
/22
0
22 ! !
knn k
nk
H x xn k k
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Schlaefli Integral
1
0
1
1
1 !
2
nn
n n
n
n nC
p y q y y
qw d x
p p
dy w p
w d x
n w py x d z
w i z x
2
2
1
!
2
tx
n nC
n eH x e d t
i t x
2 2 0n nH xH nH
2
1 xp w e
2
2
1
!
2
s xx
nC
n ee d s
i s
s t x
2 2
1
!
2
s x s
n nC
n eH x d s
i s
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Let
Orthogonality & Normalization2 2 0n nH xH nH
2
1 xp w e
2xn m n nmd x e H x H x c
n m n nmd x x x c
2 / 2xn nx e H x
2 / 2xn nH e 2 / 2x
n n nH e x
2 / 2xn n n n n nH e x x x 2 / 2 22 1x
n n ne x x
22 1 2 2 0n n n n n nx x x x n
22 1 0n nn x
Orthogonal
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2 2
0 0
, ,! !
n mx x
n mn m
s td x e g s x g t x d x e H x H x
n m
2 2
0
,!
nt t x
nn
tg x t e H x
n
2 2 22 2
0 0 ! !
n mx s s x t t x
nm nn m
s td x e e e c
n m
22 2 2
20 !
n nx s t xs t
nn
s te d x e c
n
2ste
0
2
!
nn n
n
s tn
2 !n
nc n
2
2 !x nn m nmd x e H x H x n
2xn m n nmd x e H x H x c
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Set
Let
2. Applications of Hermite Functions
Simple Harmonic Oscillator (SHO) :2
21
2 2
pH k z
m
2 2
22
1
2 2
dk z z E z
m d z
x z2 2
22
1
2 2k x E
m
22 1 0n nn x
d
d x
22 2 2 4
20
m mE k x z
2 41
mk
1/4
2
m k
2 2
2mE
2 0x
2 mE
k
m
k
m
2E
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n
mz
n nz x
2 0x 2E
mx z z
22 1 0n nn x
1
2nE n
2 / 2xne H x
2
2
mz
n
me H z
2 1n
2
2
mz
n n
mz e H z
Eq.18.19
is erronous
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Fig.18.2. n
Mathematica
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Let
Operator Appoach2
2 21
2
pH m x E
m
,x p f x p px f r
,x p i
i x x fx x
f fi x x f
x x
i f
see § 5.3
2
2 2 2 21 1 pm x i p m x i p m x i x p px
mm m
2
2
1
1
b m x i pm
b m x i pm
4bb b b H
2
2 2 2 21 1 pm x i p m x i p m x i x p px
mm m
2 ,bb b b i x p 2
Factorize H :
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Set 2
ba
, 1a a
1
2 2
1
2 2
ma x i p
m
ma x i p
m
1
2H aa a a 1
2a a
2
2
1
1
b m x i pm
b m x i pm
4bb b b H
2bb b b
1
2
1
2
ma x i p
m
ma x i p
m
or
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, 1a a 1
2H a a
, , 0A A A c
, , ,A BC B A C A B C
c = const , ,a H a aa ,a a a a
n nH a aH a n nE a
1n na with 1n nE E
, ,a H a aa ,a a a a
n nH a a H a n nE a
1n na with 1n nE E
i.e., a is a lowering operator
i.e., a+ is a raising operator
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Since
n n n na a a a 1
2n n
H
1
2nE
0ma 0j j m we have ground state 1
2mE
1
2m nE n
1n nE E
Set m = 0 1
2nE n
with ground state 0
1
2E
1
2
ma x i p
m
1
2
ma x i p
m
1n na
1n na
n na a n
Excitation = quantum / quasiparticle :a+ a = number operator
a+ = creation operator a = annihilation operator
1n na n
n na a n
11n na n
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ODE for 0
0 0a
1
2
ma x i p
m
0 0d
xm d x
0
0
d mx d x
20ln
2
mx C
20 exp
2
mA x
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Molecular Vibrations
Born-Oppenheimer approximation :
intelec nuclH H H H
nucl transl vib rotH H H H
For molecules or solids :
For molecules :
e nm m ;elecH E r R r R r R treated as parameters
vibH E R R R
nucl vibH HFor solids : R = positions of nucleir = positions of electron
Harmonic approximation : Hvib quadratic in R.
Transformation to normal coordinates Hvib = sum of SHOs.
Properties, e.g., transition probabilities require m = 3, 4 2
1j
mx
m nj
I d x e H x
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for
Example 18.2.1. Threefold Hermite Formula
23
31
j
xn
j
I d x e H x
0 are integersjn
n
n nH x H x 3 0I oddj
j
n
deg nH x n for3 0I i j kn n n i,j,k = cyclic permuation of 1,2,3
Triangle condition
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2 2
0
,!
nt t x
nn
tg x t e H x
n
23
31
,xj
j
Z d x e g x t
Consider
2
3 3 32 2
1 2 2 3 3 11 1 1
2 2j j jj j j
x t t t t t t t t x x t
23
2
1
exp 2xj j
j
d x e t t x
2
3
3 1 2 2 3 3 11
exp 2jj
Z d x x t t t t t t t
1 2 2 3 3 1exp 2 t t t t t t
1 2 2 3 3 10
2
!
NN
N
t t t t t tN
3 1 2
1 2 3
1 2 2 3 3 10 1 2 3
2 !
! ! ! !
Nn n n
N n n n N
Nt t t t t t
N n n n
3 1 1 22 3
1 2 3
1 2 30 1 2 3
2
! ! !
Nn n n nn n
N n n n N
t t tn n n
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3 1 1 22 3
1 2 3
3 1 2 30 1 2 3
2
! ! !
Nn n n nn n
N n n n N
Z t t tn n n
2 2
0
,!
nt t x
nn
tg x t e H x
n
23
31
,xj
j
Z d x e g x t
1 2 3
2
1 2 3
1 2 3
1 2 33
, , 0 1 2 3! ! !
m m m
xm m m
m m m
t t tZ d x e H x H x H x
m m m
1 2 3
2 3 1
3 1 3
1 2 3
m n n
m n n
m n n
n n n N
23
31
j
xm
j
I d x e H x
1 2 3 /2 1 2 33
1 2 3
! ! !2
! ! !
m m m m m mI
N m N m N m
1 1
2 2
3 3
n N m
n N m
n N m
1 1
2 2
3 3
1 2 3 2
m N n
m N n
m N n
m m m N
1 2 3m m m even
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Hermite Product Formula 2 2
0
,!
nt t x
nn
tg x t e H x
n
2 21 2 1 2 1 2, , exp 2g x t g x t t t t t x
1 2
1 2
1 2
1 2
, 0 1 2! !
m m
m mm m
t tH x H x
m m
1 22 2
1 2 1 2exp 2 t tt t t t x e 1 2 1 2
0 0
2
! !
n
nn
t t t tH x
n
1 2 1 2
0 0 0
2
! ! !
s n sn
nn s
t t t tH x
s n s
1 2
0 0 0
2
! ! !
s n sn
nn s
t tH x
s n s
1 21 2
2 1
1 2
min ,1 2
1 2 20 0 0 1 2
2, ,
!! !
m mm m
m mm m
t tg x t g x t H x
m m
1
2
m s
m n s
1
2 1 2
s m
n m m
Set
Range of set by q! q 0
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1 21 2
2 1
1 2
min ,1 2
20 0 0 1 2
2
!! !
m mm m
m mm m
t tH x
m m
1 2
1 2
1 2
1 21 2
, 0 1 2
, ,! !
m m
m mm m
t tg x t g x t H x H x
m m
1 2
1 2 2 1
min ,1 2
20 1 2
! ! 2
!! !
m m
m m m m
m mH x H x H x
m m
1 2
1 2
2 1
min ,
20
2 !m m
m m
m mH x C C
i jH H
i jH H
Mathematica
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Example 18.2.2.Fourfold Hermite Formula
24
41
j
xm
j
I d x e H x
1Integers 0j jm m j
1 2
1 2
1 2 2 1
min ,
20
2 !m m
m m
m m m mH x H x H x C C
2 4 2
1 2 3 4
2 1 4 32 20 0
2 ! 2 !m m
m m m m xm m m mC C C C d x e H x H x
2 41 2 3 4 4 3
2 1 4 3
2
2 , 2 4 30 0
2 ! 2 ! 2 2 !m m
m m m m m m
m m m mC C C C m m
2
2 !x nn m nmd x e H x H x n
2 1 4 32 2m m m m 4 3 2 1
1
2m m m m
4 3 4 3 2 1
12
2m m m m m m 2 :p
4 1min ,
4 3 1 2 3 4
40 4 3 1 2 3 4
2 2 ! ! ! ! !
! ! ! ! ! !
Mm M m m m m m m mI
M m m M m M m m m
Mathematica
M
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Product Formula with Weight exp(a2 x2)
2 2 / 221
min , 2
20
2 11
2
1 2 1! !2
m nm na x
m n m n
m n
m nd x e H x H x a
a
m n am n a
Ref: Gradshteyn & Ryzhik, p.803