hermite functions
TRANSCRIPT
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Mathematical Methods for Physicists
by G. Arfken
Chapter 13: Special Functions
Reporters:
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Hermite Functions
Generating functions - Hermite polynomial
Recurrence relation Special values of Hermite polynomial
Alternate representations
Orthogonality
Normalization
Application
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Generating Functions
Define (1)
Take expand
We have
( ) ( )
=
+ ==0
2
!,
2
n
n
n
txt
n
txHetxg
( )
( )
( )( )
( )
( ) 12016032
124816
12824
2
1
35
24
4
2
3
2
2
1
0
+=
+=
==
=
=
( ) 12072048064 2466
5
+=
xxxxH
xxxH
xxxHxxH
xxH
xH
txty 22 +=
=
=0 !n
ny
n
ye
xxxxH
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Recurrence Relations (1/4)
(2)( ) ( ) ( )xnHxxHxH nnn 11 22 + +=
( )
( )
( )
( )
( ) ( ) ( )
( )( ) ( ) ( )( )
=
=
=
+
=
=
=
+
=
+
=+
=+
=+
=
1
1
00
1
0
1
0
0
12
0
2
!1!2
!2
!1!22
!122
!
2
2
n
nn
n
nn
n
nn
n
nn
n
nn
n
nntxt
n
nntxt
tn
xHt
n
xHxt
n
xH
tn
xHt
n
xHxtx
tn
xH
extx
tnxHe
dtd
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Recurrence Relations (2/4)
The coefficient of
The coefficient of
0t
( ) ( )xHxxH 102 =( )0ntn
( )( )
( ) ( )
( ) ( ) ( )xHxxHxnH
n
xHxn
xHxn
xH
nnn
nnn
11
11
22
!2
!2
!12
+
+
=+
=+
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Recurrence Relations (3/4)
(3)
Differentiate the generating function with respectto
( ) ( )xnHxH nn 12 =
( )
( )
( ) ( )
( ) ( )
+
=
=
=
+
=
+
=
=
=
=
1
00
0
2
0
2
2
!!2
!2
!2
2
nnnn
n
nn
n
nn
n
nntxt
n
nntxt
txHtxH
tn
xHt
n
xHt
tn
xHte
t
n
xHe
dx
dx
== 00 !! nn nn
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Recurrence Relations (4/4)
The coefficient of
( ) ( ) 0!0
00 == xHxH
nt
0=n
0>
n ( )( ) ( )
( ) ( )xnHxHnxH
nxH
nn
nn
1
1
2
!!12
=
=
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Value at 0=x
( )( ) ( )
( )
( ) ( )
( ) ( ) ( )
( ) 00:12
!
!210:2
!!1
!
1
!2!1
1
!2!1
1
,
12
2
00
20
242222
2
2
2
=+=
==
=
=+=+
+
+=
=
+
=
=
=
+
k
k
k
n
n
n
k
kk
k
kkt
txt
Hkn
k
kHkn
n
txH
k
tk
ttttte
etxg
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Parity Relation
Expand the generating function We have
( ) ( ) ( )xHxH nk
n = 1
( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
xHxxxxH
xHxxxxH
xHxxxH
xHxxxH
xHxxH
xHxxH
xH
66246
6
5
535
5
4
424
4
3
32
3
2
22
2
1
1
1
0
112072048064
112016032
1124816
1128
124
12
1
=+=
=+=
=+=
==
==
==
=