á þ á ñ ü ½ á þ á á àþã þ ë

 )
x0
* $
+,- ./ 0 1
:
:
x0
V (x)
+;
!"#
$%
& '$ (
23
45 2#6 V (x)
m
78 9
<# & : & 2= !#6 1
V (x) ≈ V (x0 ) + V ”(x0 )(x − x0 )2 + · · ·
2
x0
./ @ # A 4 B C V (x0 )
:/ 9F 7
>1 ?
2 # 6 D E . / + / mω 2 := V ”(x0 )
'$ ( 2#6 $ ./ +; 25 $ : '$ ( 2#6 %- 1 V = 12 mω 2 x2 G2#6
DH - = $ ! +,- ./&
: +,-
+; $- +5
9 - +3
I JE 9& 8 :,- 45 # 9 KL
M 0 $ : )5,- ./ 0 0 :
8 @= 9 N: )5,
,L
# 9& !@1 # O 0P # 9 5 : $ 0
,5.
'$ ( Q1 2; I ,5.
'$ ( 2AD- & , & ; 2
= : O: B4 6 R!4
9 R!4
9 @8
6 5 - PH +# +5
$ & ::6
6 9( S8 ,1 9 & T, 9 S$8
& 2= :, H=
ω
,@U VP 1
P2
+ mω 2 X 2 .
2m 2
W
m
78 '$( !
>2 ?
X L
$ K
mω
+U
78
m
ω −1
ω
ω
1 m
-
$ 2:@- XL YW +:8
7C 9 #4 0
&
9F : !
9F RU B4 O:
:; Z4 [ 9 & - & 78 9:; Y: & 2= JP !
\ E :=
H=
9 :; H +U #$
1 2 1 2
P + X
2
2
>3 ?
mω 78 #$
+:8 Q.
X
X L ZL - 1 m
1
N
& #$ ω : $ 9:I :/
Z4 9::8
: : 9 :; $ :/
]
R ,- & 2= (!$I - +;
1
a := √ (X + iP )
2
1
†
a := √ (X − iP ).
2
Y
>4 ?
= & 2= ( !
- (!$I X#;
1
H = a† a + .
2
9 := : :6 0
H
RU :6 (!$I RU H & < @^
Y : a
†
a
>5 ?
@3
(!$I N := a† a
9(!$I (!$I _ (!$I
% Y ]
:
[a, a† ] = 1
@3
[N, a† ] = a† .
[N, a] = −a
# 9:/
` 7$- @3
(!$I N = a† a
>6 ?
(!$I 8-
#4
H ) - ; a$L :=
N |ψ = λ|ψ,
>7 ?
(
N (a† |ψ) = (λ + 1)(a† |ψ).
N (a|ψ) = (λ − 1)(a|ψ),
a
b := = 8 6 ` >8 ?
# C 6 :/
` a $"
)
|ψ0 a
(!$I ` @ DE a|ψ0 = 0 −→ N |ψ = 0.
>9 ?
Y:= 5; 9 a†
b ` /
|ψn := (a† )n |ψ0 , n = 1, 2, · · · .
, N
n
|ψn >10?
G 5; :/
` = 7!,
N |ψn = n|ψn .
cL
a† |ψn = |ψn+1 .
|ψn−1 X
: 8 5; d e
n
f
n−1
.
8 . 8-
>12?
#
|ψn 9 a (!$I b 6 +;
Y: ) ; :- : D & . & [a, a† ] = na†
D
>11?
9 :=
8 . &
: X- X L >13?
Y ) - . & D
n
n
a|ψn = aa† |ψ0 = (a† a + na†
n−1
)|ψ0 = n|ψn−1 .
Y Z#; >14?
|ψn 9 & 7:
n
&: - +;
n
ψn |ψn = ψ0 |an a† |ψ0 = ψ0 |an−1 (aa† )|ψ0 n
ψ0 |an−1 (a† a + na†
n−1
)|ψ0 = nψn−1 |ψn−1 .
>15?
ψn |ψn 2 = - : [ ) |n
$ |ψn g : = C % = n!
C
5 ; 9 H #& &
a|n =
√
a† |n =
n|n − 1,
√
n + 1|n + 1,
>16?
1
1
Ĥ|n = (n + )|n −→ H|n = ω(n + )|n.
2
2
Y n|a|m =
√
√
mn|m − 1 = mδn,m−1 ,
Y
⎛
0 1
⎜
0
⎜
⎜
⎜
⎜
a=⎜
⎜
⎜
⎜
⎜
⎝
: := = : 9 :; ( 7 9#-
a†
n|a† |m =
& \ E √
2 √
0
3 √
0
4
0
h
>17?
a†
a 9(!$I # -
√
m + 1n|m + 1 =
a 9:,
EI - √
16 . &
m + 1δn,m+1 .
>18?
% 9 V -
c E 2=
⎞
·
0
·
0
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
· ⎠
>19?
⎛
0
⎜ 1
⎜
⎜
⎜
⎜
†
a =⎜
⎜
⎜
⎜
⎜
⎝
0
√0
2
⎞
√0
3 √0
4 0
·
0
· 0
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
>20?
8- 9 : 9F 6 9(i6 :)
C P
X
=
P
=
1
√ (a + a† )
2
1
√ (a − a† ).
2i
a
X
†
# -
EI - a 9(!$I R ,-&
>21?
= C
n|X|n =
n|P |n =
1
n| √ (a + a† )|n = 0
2
1
n| √ (a − a† )|n = 0.
2i
>22?
n|X 2 |n =
n|P 2 |n =
1
1
n| (a + a† )2 |n = n +
2
2
−1
1
(a − a† )2 |n = n + .
n|
2
2
>23?
= : 9:; (
1
(n + )
mω
2
1
mω(n + ).
2
n|X 2 |n =
n|P 2 |n =
j
>24?
Y
= )
n|T |n =
n|V |n =
V
2#6 9F T
)@8 9F H
1
1
ω
n|P 2 |n =
(n + )
2m
2
2
1
ω
1
mω 2 n|X 2 |n =
(n + ).
2
2
2
>25?
:# 9#
(: 2#6 9F <
)@8 9F <
Y
9F 5; ` - ,.1 7:I 0
: ) . 1
ΔXΔP = (n + ).
2
1
k 9 V -
- : 2 , :/
$ 6
>26?
5;
,.1 7:I 0
# 9F 6 2
U P
X
9(!$I
1
1
n|X|m = √ n|a + a† |m = √ (mδn,m−1 + (m + 1)δn,m+1 )
2
2
1
1
†
n|P |m = √ n|a − a |m = √ (mδn,m−1 − (m + 1)δn,m+1 ).
i 2
i 2
>27?
!
1 :6 \A4
6 9F S- ` 2= B4 & ψn (x) := x|n.
H C
x|X + iP |0 = 0
>28?
a=
√1 (X
2
+ iP ) a|0 = 0 &
>29?
(x +
d
)ψ0 (x) = 0
dx
l
>30?
f
1
f4
( 9
φ4
φ1 φ0
9
5; ` YW 2=
= 7!,
; φ0 (x) = Ae−
:/
: :
∞
−∞
ψ0 (x)2 dx = 1
x2
2
a 5,
2;
>31?
.
9L/- & *C%
b *C%
b A
=π
− 14
A
= †
a =
√1 (X
2
− iP ) |n =
n
√1 a† |0 n!
Y& :-@I + 4(
1
1
4
xe−
: - 0 m
S- / +;
d n
1
1
1
d n − x2
1
)] φ0 = √ n (x −
) e 2.
φn (x) = √ [ √ (x −
dx
dx
n! 2
n! π 2 2
φ1 (x) =
>32?
5; : m
S- 2=
x2
2
π
1 2x2 − 1 − x2
e 2
φ2 (x) = 1 √
6
π4
x2
1 1 √
φ3 (x) = 1 x 3(2x2 − 3)e− 2
π4 6
···.
>33?
:I m
S- X-- 0 f ] 9 2= : := ( 5 m
S- : W 2=
: )
Y
& \ E S- ! 2= φn (x) =
b XL :8 x2
1
√
Hn (x)e− 2 ,
n
2 n!
π
1
1
4
o
+$; 5( S n
(x −
d
dx ) (!$IG5
b
>34?
( 9
:# O ,
9 9 !$8 : 9 !$8 : ( 9F S- ` 2=
5; ` Y] 2=
x & n
8 9 !$8 : Hn (x)
7&n & 2@1
,5.
_ & : @58# _
#/- +U :, 6 :, 9
φ30
x
p
8
# d
9 # : 9 :;
# m
S- 8- 7C Nξ
Y& : \@I m
S- % 2= C ZL
φn (x) =
ξ
1
2
=
mω
# φn
x −→ ξx
0(8 ,
:= +U i8 VI
1
(ξx)2
1
√
ξ 2 Hn (ξ x)e− 2 .
2n n!
π
1
1
4
>35?
Y:= R ,- & \ E 9 !$8 : - n < 8-
(x −
x2
d n − x2
) e 2 = Hn (x)e− 2 .
dx
>36?
Hn (x) = e
x2
2
(x −
d n − x2
) e 2.
dx
& . $ 36 . PU (x −
Hn+1 = 2xHn (x) − Hn (x)
q
>37?
d
dx ) (!$I +$I
>38?
( 9
5; ` 9 +$; 5( Yf 2=
φ30
Hn+1 = (2x −
Y: d
)Hn (x).
dx
9 9 !$8 : NH0 (x)
: & \ E Hn (x) = (2x −
Y : & \ E >39?
=1
8- d n
) 1.
dx
9 9 !$8 :
>40?
I# - . & D H0 = 1,
H1 = 2x,
H2 = 4x2 − 2
H3 = 8x3 − 12x,
H4 = 16x4 − 48x2 + 12,
···.
>41?
H C :# :
,
m|n ≡
|n 9
5; C&
dxm|xx|n =
r
dxφn (x)φm (x)
>42?
. √
2
dxHn (x)Hm (x)e−x = δm,n π2n n!.
$#1 R ,-
g(t, x)
g(t, x) =
# D
x2
2
>43?
23
:5
S- 9 !$8 : 9 - ∞ n
t
Hn (x).
n!
n=0
# D
g(t, x) = e
34 9 \@I 9i(8 >44?
37 . &
:6 :5
S- 7P 9
∞ n
x2
d
x2
d n − x2
t
(x −
) e 2 = e 2 et(x− dx ) e− 2 .
n!
dx
n=0
:; X
[A, B]
(!$I 9
t2
>45?
1
eA eB = eA+B+ 2 [A,B]
^-& et(x− dx ) = e 2 etx e−t dx .
d
d
>46?
Y: & 2= :5
S- C
g(t, x) =
e
x2
2
=
e
x2
2
t2
e 2 etx e−t dx e−
d
t2
1
x2
2
2
2
e 2 etx e− 2 (x−t) = e−t
+tx
>47?
.
) 2
e−t
.
$
_ & T, $$L +tx
=
∞ n
t
Hn (x).
n!
n=0
PH C - >48?
9 9 !$8: & 9) _ . &
# :P & 5 % # 5 := =: s k \@^
/#
_ & T, s !"! #$ :
H d : Y
& 2= !
ω
VP B5,- ./ +; q
5 m
78 9 9F /
9F S- ` P O: 1
Wt
E
5
H=
1
P2
+ mω 2 X 2 − qEX.
2m 2
>49?
Y
H=
:= C8
qE
$ mω 2
= & 2= - !
1
P2
qE 2
q2 E 2
+ mω 2 (X −
) −
,
2
2m 2
mω
2mω 2
>50?
&: +,- ./ ( !
08
& 2= E0
ξ :=
ξ
9
6 R ,- qE
,
mω 2
E0 :=
# 90 := b:/
&: 0 2 9F q2 E 2
,
2mω 2
>51?
Y: 2= !
H=
1
P2
+ mω 2 (X − ξ)2 − E0 .
2m 2
>52?
Y H d & (!$I
−i
ξP
Tξ := e
>53?
.
Y
Tξ |x = |x + ξ ,
H
= Tξ H0 Tξ−1 − E0
E0
: / % -
= 7!,
N )
H
H0
& 9
Tξ XTξ−1 = X − ξ.
>54?
5 :
Zu( !
H R
G U N = 7 ! , ) H1
A H1
Tξ H0 Tξ−1
H0 |n = En |n
H , N:=
|φn = Tξ |n,
WW
R U H C RU :6 :=: P
H1 = Tξ H0 Tξ−1 .
R ,- ( N
E 9 (!$I H1
9 9F &
>55?
H0
G 5; ` |n
H \ E >56?
= H1 |φn = Tξ H0 Tξ−1 (Tξ |n) = Tξ H0 |n = En |φn .
NEn
= ω(n + 12 ) 8-
v- : 9F $
>57?
H1
G 5; ` 0
Y
|φn ,
= a%
1
H|φn = [ω(n + ) − E0 ]|φn .
2
>58?
Y 8- & . ,- \A4
9TP |φn m
S-
2= 9
G
φn (x) = x|φn = x|Tξ |n = x − ξ|n = ψn (x − ξ),
:=C8
$ ξ
&: H0
v
S- ` 08
>59?
# 90
H
S- ` ,
%&
C& :=
a
g : 6 (!$I ` = DH 5; ` <!4
:I ,
z
:/
`
Coherent State s:$
# //; a
050 :/
`
%8 $ 5; L d&
# # )
a|z = z|z.
XL #/
<# 6 9
5; X#; Y
|z †
|z G
+; = †
|z := Aeza |0.
W]
:6 ` 9 e−za aeza = a + z.
Y R ,- & \ E |z a(!$I
>60?
& ^- & D % †
:% n . OU <#
>61?
b O : 5 & D ; ^- >62?
: D
61 . &
b
(!$I OU 9 a
†
*C% X L †
a|z = Aaeza |0 = Aeza (a + z)|0 = z|z.
9F 6 9
5; X#; & <#
|z := A
|z cL A >63?
s:$
5; ∞
zn
√ |n.
n!
n=0
|z >64?
$%P <# & D
1 = z|z = A2
∞
|z|2n
n=0
2
1
= A2 e|z| .
n!
>65?
2
|z|
Y Z4 e− 2
|z = e−
|z|2
2
A := C% s:$
5; 9 †
eza |0.
@^
$ - S1 :# $I (: s:$ 9
G
5; >66?
8- : $%
: ) 7C *C%
z|w = e−
|z|2
2
Y= 7!,
; e−
|w|2
2
b ,- 9 9
∞
|z|2
|w|2
1
(zw)n = e− 2 − 2 +zw .
n!
n=0
:/ - <
>67?
5; 6 & +;
1
1
1
1
z|X|z = z| √ (a + a† )|z = √ (z + z), z|P |z = z| √ (a + a† )|z = √ (z − z).
2
2
i 2
i 2
# & \ E 1
z = √ (x + ip)
2
>68?
z
H >69?
= (
z|X|z = x,
z|P |z = p.
Wf
>70?
<!4
^DE s:$
p x 5; : Yh 2=
: X-- - <
5;
|z = | √12 (x + ip)
5; Z#; - / 7:I 0
+;
1
1
1
1
z|(a + a† )2 |z = (z + z)2 + = z|X|z2 +
2
2
2
2
−1
−1
1
1
z|(a − a† )2 |z =
(z − z)2 + = z|P |z2 + .
z|P 2 |z =
2
2
2
2
z|X 2 |z =
Y:1 & < $ s:$
(ΔX)2 =
1
,
2
(ΔP )2 =
1
,
2
, : / $ % / 7: I 0 $ ) s: $ 1
.
2
5 ; s: $ 9 >72?
5 ; : / % < < 1
5 ; : - $ = U h 2 = 1 2 : 9 : ; 2
# 7 $ - K , = : ) 5; 9 :6 V6
ΔXΔP =
& 2 / # : / 7: I 0 $ : # >71?
:/ % - <
:/
& 2/#
R ,- & 9 5 ; P
# - \A4
9TP
X
< : / 0 & ! E P : 5; $ / 7:I |zGs:$
: ) 5; m
S- + +;
R ,- Q.
,8
ψz (x) := x|z.
Wh
>73?
$%P JP . √1 (X
2
a=
+ iP ) KL
a|z = z|z 8-
1
√ (x + ∂x )ψz (x) = zψz (x).
2
ψz (x) = Ae
Y: :6 H 2= m
S- #
z=
√
√1 (x0
2
2zx− 12 x2
>74?
: = 2; H# 2#D 5,
>75?
,
+ ip0 ) \ E
z
H b A 2
1
ψz (x) = A eip0 x− 2 (x−x0 ) ,
; +;
p0
- W 9%6 R ,-& D P 2:@- & D ? >76?
x0
0
H m
# : ) : - 9TP m
S- 2= - Y
& \ E C >
2
1
ψ̃z (p) = A e−ix0 p− 2 (p−p0 ) ,
W 9%6
d^5 : wP e: :6 +^- ( s:$
Y& : \@I 5;
p0
>77?
0
H S- : ) & 5; N
( !
t & &:, \ E
:= PH1
|z 23
^- &
s:$
|ψ(t) = e−iHt |ψ(0) = e−iHt |z.
Y
s:$ +; |ψ(t)
+^- %
#
a $,
&
m
S- n
E : 5;
=
e−i(
ωt
2
=
e−i(
ωt
2
ω
A
64 . ,
s:$
>79?
5; <# H +;
∞
(ze−iωt )n
√
|n
n!
n=0
|ze−iωt .
9 &
a 8- X58 1
5; DE
OU @^
9
∞
∞
1
zn
zn
√ |n = A
√ e−i(ω(n+ 2 ))t |n
n!
n!
n=0
n=0
OU N
=iH
>78?
<# !
9 ` X#; s:$
|ψ(t) = e−iHt A
I g:@
+; & +U s:$
>80?
5; E : $ p- 0 - H:=6 0
Wj
( !
^- s:$
9TP m
S- H:= %6 & := %6 &
5; +^- Yj 2=
=iH \A4
9TP N- 9TP
a @1 : $- E : N\A4
: ) s:$
5; +^- j 2=
"#$ %&'
,5.
s:$ 9
. 2 3 9 . e: # 2 s: $ 9 \ 08 b
Y
|n 6
5; 2
.
# B4 6 5 ; : @ $ I ( : s: $ 9 5; 9 x OU(!$I N. Q/^- 9 e
1 & 2= 2
. 1
π
d2 z|zz| = I
>81?
h 2 =
G
5; ) 9 / 7:I /
/ 7:I P &% /- :# 5; )P 9
//;
5; 9 :
1
5; 5; 5; & Z4
PH1 $ /
U - / 7:I Ns:$ 9
5; % 25 $ 5 ; : d2 z|zz| = I.
<#% X#; 45
)P 9
1
C iH $- :%I @^
%& I$C
& - ,
N: 2)- 2
JP 6 s:$ 9
( )'
ΔXΔP =
2 . U
:= ) P B ( A4
/ 7:I B0P 9% A4
H
Wl
'( % # N I : +,- ./ +; 7: F : 23
$- +5
9 #
Y : & 2= $# !
: =,- 9 0(: :-
H=
1 2 1 2 1 2 1 2 1 2
P + X1 + P2 + X2 + ω (X1 − X2 )2 .
2 1
2
2
2
2
>82?
: 9 # 7$- # 9 = :6 !
RU & # -
2= !
#4 0 : B (: , 9 2#6 9 Y#
H=
1 T
1
P P + X T AX
2
2
>83?
& /
V -
A=
$#1 8
S
1 + ω12
−ω 2
−ω 2
1 + ω2
23
9:
,
V -
, N= 9.1 :
,
2:@- - !
C A = S T DS
= - ST S = I
>84?
.
A A /
V -
9.1 V -
D
S
E ,
SAS T = D
:
,
8-
#& & 2=
H=
1 T T
1
P S SP + X T S T DSX
2
2
>85?
\ E 9p
p-
P̃ = SP,
X̃ = SX,
Y= = & # -
2= !
H=
1
1 T
P̃ P̃ + X̃ T DX̃,
2
2
>86?
D
9.1 8-
>87?
Wo
H=
1 2 1 2 1 2 2 1 2 2
P̃ + P̃2 + ω1 X̃1 + ω2 X̃2 ,
2 1
2
2
2
::8 9p
+ 8- %
# :#
D
>88?
V -
9.1 9 EI
ω22
ω12
, N: J:E 9(T,- < $
[X̃i , X̃j ] = [P̃i , P˜j ] = 0
[X̃i , P˜j ] = iδij .
:=
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25
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-
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6 & # 9F 9
& 2/#
( : ) !
::8 9p
X#; : )
Y R ,- & 9(!$I 1
a1 := √
(ω1 X̃1 + iP̃1 )
2ω1
1
(ω1 X̃1 − iP̃1 )
a†1 := √
2ω1
P 88 .
: - ;
1
a2 := √
(ω2 X̃2 + iP̃2 )
2ω2
1
a†2 := √
(ω2 X̃2 − iP̃2 ).
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Y: J:E & < (!$I [a1 , a†1 ] = [a2 , a2 †] = 1
X#; 9( 2/#
7
>90?
89 < 8-
>91?
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2
2
Y
H|n1 , n2 = En1 ,n2 |n1 , n2 ,
>92?
& \ E !
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1
1
En1 ,n2 = ω1 (n1 + ) + ω2 (n2 + ).
2
2
Wq
>93?
&G
|n1 , n2 = √
5; N|n1 , n2 5; . n1
n2
1
a†1 a†2 |0, 0,
n1 !n2 !
>94?
a1 |0, 0 = a2 |0, 0 = 0.
s 9: |n2 |n1 G 5 ; 9 # - Z L0 8
>95?
# 90 |n1 , n2 5 ; 9 :==%
\ E \ A 4 X # ; 1
|n1 , n2 2 3 9F 5 ; ` # 1 m S Y : &
ψn1 ,n2 (x̃1 , x̃2 ) = x̃1 , x̃2 |n1 , n2 = x̃1 |n1 x̃2 |n2 = Hn1 (x̃1 )e−
x̃2
1
2
Hn2 (x̃2 )e−
x̃2
2
2
>96?
iH $- KL
/ )*+
H a ;.E := :; (!$I X
[A, B]
9^ :# 45(!$I
B
A
: wP y W
:= :I [A, B]
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1
eA eB = eA+B+ 2 [A,B] .
: ) : Z#;
λ >97?
@# Q)
: ( d U (λ) := eλA eλB e−λ(A+B)
d
U (λ) = λ[A, B]U (λ).
dλ
(!$I Y$
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: 2; 5,
Vz
: ) n 2;
- :# :I
[B, [A, B]] [A, [A, B]] !
1
1
1
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[A, B]
: wP +; YZ
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1
1 √
(|0 + |1)G
5;
ω
$ )
At
& X#; 2
Y: Z#; & 9
Xt ,
P t ,
VP
G
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Kt ,
m 78
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2#6 9F >100?
V
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- 5 :#( 9F 1 2 1 2
P + X + λX 4
2
2
5; `
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5; >101?
6
: Z#; <
Z#; V (x) =
⎩
1
2
2 kx
x >≥ 0
∞
x<0
:= 5; 9 ( 9F 1 & 2#6 ⎧
⎨
λ
m 78 9 y h
>102?
::6 9F 9:/
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- <
<
:/
9F 5; ` y Z
:
Y:= RE- & !
:=
H=
D8( yj
1 2 1 2 1 2 1 2 1 2
P + X1 + P2 + X2 + ω (X1 − X2 )2 .
2 1
2
2
2
2
9F & d
: Z#; H:8 ( 9F <
:/
9F 6
7( 9 %)
R ,-
: >103?
5; # 1 YR5
H1 := 12 P12 + 12 X12
\@I $=(
: \A4
9TP 6
5; m
S- Y Z
: 2; # 9 M@0
; \n,
Y m
]t
X1 = 0,
X2 = A,
: ,- O^
+,- 2^
&
:@,- 5; # DE d^5 : wP +; Y
P1 = 0,
P2 = 0.
&: \& A
>104?
5; v
5; 5; #/
2; : p- ( & +U <
5; N9
05 2E Q.
PH1
T
9
ω
VP
m 78 ( y
l
Y= RE- & 5( V -
(
G
ρ=
1 −βH
e
Z
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05
b
k
β=
: Z#; 1
kT b YR5
Z
: Z#; ( 9F <
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Y
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n!
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2πi C
= & 2= - b d
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$
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5; |z : wP Z#;
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C
5; y q
5; 9F @# ,- 7:I Y: Z#; &
2
2
z|H |z − z|H|z
.
z|H|z2
>107?
%$ ) , -
.$/
Y:= R ,- & :5
S- 2
g(x, t) = e−t
+tx
=
]W
9 9 !$8 : : s ∞
Hn (x) n
t .
n!
n=0
>108?
Y:# & {= _ & T, PH C 9 9 !$8 : _ & 9# - R ,- &
H0 (x) = 1
t
= 0 1 N48 . OU
% \@I :# |4)
6 9
H2n+1 = 0,
(:
48 . >109?
−x −t x t
Y: & /
H2n (0) = (−1)n
OU <# 1 DE 9#
P
S- Y Z
Hn (−x) = (−1)n Hn (x).
#/
5 <# 2:@-
YR5
:5
S- P \@b 9
9 9 !$8 : DE ./ Y m
(2n)!
.
n!
>110?
x :/
:5
S- P . \@b 9
#/
Y
1 & .
S- 7$- 9& Y
Hn (x) = 2nHn−1 (x).
#/
PU ( Q)
Y
>111?
x
@#
1 & - .
48 PU& S- 7$- 9& Y
Hn+1 (x) = 2xHn (x) − 2nHn−1 (x).
#/
PU ( Q)
]]
P
>112?
t
@#
48 PU& P
Y: J:E & 2#D 5,
(
d2
d
− 2x
+ 2n)Hn (x) = 0
2
dx
dx
# & 2= >113?
112 G- . 111 &
Hn+1 = 2xHn − Hn −→ Hn = 2xHn−1 − Hn−1
.
9 9 !$8 : Y
: 9i(8
111 G-
1
1 H
Hn = x Hn −
n
2n n
. &
D . \@b 9
>114?
OU S- 98 +;
>115?
: := 2#D 5,
2= X-
& V6 ]f