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

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

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.

Table & Fig. 18.1. Hermite Polynomials

Mathematica

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

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

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

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

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

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

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

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

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

Fig.18.2. n

Mathematica

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 :

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

, 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

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

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

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

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

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

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

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

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

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

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