PH 413 163 - Ramkhamhaeng Universityold-book.ru.ac.th/e-book/p/PH413/ph413-5.pdf · (5) * ‘2...
Transcript of PH 413 163 - Ramkhamhaeng Universityold-book.ru.ac.th/e-book/p/PH413/ph413-5.pdf · (5) * ‘2...
PH 413 163
dpz=
=
= 0
+a UlflmJfllJ (5.1) Th%~6%=61wu~lIl
,
5.2 eed6we6~6m&m%fJId
=i
(A w>*Y~
--m
164 PH 413
PH 413 165
Vv,(CY, ’ = < v,((aA + PB)v, 1
<WlV*’ = < lu,l(a’A + P’W2 >
166 PH 413
<Y,IDw~ ’ = <y&4&u, > = <BAry,l~, >
5.3 Whl~iarPIPd"er ( Correspondence principle )
PH 413 167
$ <y/l&’ = <+>+;<y((H,A)(y>I I
(5.3)
dxar =t”
3P66RE - = 0at
-$-+lY’ = + < wI(H> f]w ’ (5.4) '
(f,x) = !y*y
Ll = Em ap
(5.5)
PH 413
(5.9)
PH 413 1 6 9
-tt d= --i at
2E <p>+<v> = <H>
2 m
H = &+v(x) = E
*
PH 413
-Ii2 a*y---+V(x)~ = Ety2m ax2
-A2 d2!u--+V(x)ly = +$2 m aX2
Hv, =
PH 413 171
<x> = jv’(4wI(x~
<f(x) 1 = jv’ Wb)Y(x)du
<P’ = J@(P)P@P)dP
<f(P)’ = j@*(P)f(P)@(P)dp
u = v(x) du = -m
dv =-h
e -‘pr/ndrV = _e-‘Px’”
lP
PI-l 413
F = i2-1 h82( 1
-tt* 82-_--- =--2m 2m i& 2m Sk2
PH 413
wABy/ = -&
oYdlJnd 5.3
1 7 4 PH 413
0(P) =
0’(P)&) =
A sin[(p, -P)L/A]$-zl PO-P
h sin2[(po -p)L/ti]
2 h-P)
PH 413 1 7 5
v(x) = Ae- x’/201
P(X) = 1 Al2 e-x2’0’
44=
P(P) =
Be-p+l’/2h’
Id2 e-P’02’h2
PH 413
&i?ofhnd 5.4 F~JMI Fourier transform ~~~$h%‘udilk.h?
s(x) = IdIxI < 1
0,1x1 2 1
9- 036Yll g(k)
t
1= ~[1:
xe’“dx+ [ xem’“dx]
= fYJ 1
4%
2x(e” +eh”jdx2
= -[ xcoshxakJ&ru = x
d u = d x
dv = cos kx dx
sin FoeV-
k
g(k)
= +$++(cosk-1) 1
PH 413 1 7 7
= -/Al2 L(0 -e”) = IAl2 L = 1
. .
‘4 0(P)
r
178 PH 413
21i3J- 1=
7rL3 $+(p. -p)
11;; p,-p=y dp = -dy
1 n --x= - - - -[ ( 11
=1s2 2
\
t&i 0 ( p ) “aJuoimalad
PH 413 1 7 9
sin’ 4px-:pox/heipoxlh
X2h fix=1
2c2hsin2 !i!E
-P X2Ii a?=1
n -
2c2Ap=l
y(x)= 2 sin’yih ,Ip,&h
J--
180 PH 413
&an&hd 5 . 7 RlwlM y(x,o) = Aerpex’Ae-‘x”L
= 1-d [- -]IL L=l j2+2
PH 413 1 8 1
lp,xlhe-l~llLe-lpxlna!x
4PJ-9 = d--g 10; *+(PrP)
2h3=
;;i;j-L [(:,‘+;-py]
PH 413
PH 413 1 8 3
(5)
* ‘2 6aQffGuu , 2 = 2<!Li’ = <y/a’ j < y# = <iyla’ (2)
nIJfl1s (1)I
<y&f> =I I <vlalv> = a <cvlw > (3)
nufl15 (2) <&Y> =I I
< yja’ly > = a* <vv(yl> (4)
(3) - (4) , 0 = (a-a’)<yjyz
&lid .a = a
dila6ouu09~o~6w~656~0kw”~taQs”i6w”u~ hAit&
184 PH 413
3Axe-“1’2
PH 413
Iy’y/dw =
m
I
A2x2emX2& =
-co
6
A21 =
A =
<x> =
1
Y = AC&(x-x,)2 /2a2]
fl) owhwatu A X l y/ QflweiaJQnZi=iQ
u) what3 < x >
n) what-u < (x- <x >)2 >
9) nnry~ileyn7nm~ou~nlu~u~“~ V(x) oailwat-4 < V > thh V = mgx w7z
d1ns’El v = $2
94 Darm
‘#=A-i-
n % A= ’a.a ,+/4a112
U) <x> = IVv’X Vk
-&f= A2 jxe (12 dx
-m
1; xcua
&iil dr=adu
186 PH 413
<X> = A2 1(x0 + au)dadzJ
-m
. C--X0 Y
-m
-a= A2 j(x-x,re ‘* a!~
-m
+ A2 ]2x,xe-M
a* ak- A* jx:e-(x--xo)I
a2 a!x-co -m
= A2 j(x-x,)‘e-(x--x0
a’ &+2x, <x>-A’xi&a-m
fhMU~-IXx-x0
U C - - & I L dx=adua
..w ODa’& =
Ia2u2e-“‘adu--oD
PH 413 1 8 7
&IL <x2 >
iG.4 <(x-<x>)z >
VW,<v>
v(x)
<v>
= A2 a3J;;.----+2x, .x0 - A2$&a2
z <x2>-<x>22
=( 1%+x,2 -x;2
a2=T
,
fl)+--x0) (x-d
VI = Ae”e-2”’
i) Jv:lvl~ = k?!$.*A= Je z0 dr
k%P-__= A2 Je a1 a!x= A2 &a
= 1
ii) <x> = Jvy;w,~-Mz A2 Jm o1 ak
=X0
iii) <(x-<x>)‘> = <X2 >-<<X2
<x2 > = Jt4x2v/&-& *
= A2 Jx2e 0’ dx2
= a+X,22
&lfu2<(x-<x>)‘> = ( 1
“+x,z -x,”2
a2=2
iv) V = mgx
<v> ’ = mg-cx>
= ma
V = $x2 I
189PH 413
<V> = $k<x2 >
ihl3tul
%X x--o -t- - 6 & &=adta
f’+it = (t+;)‘-(i)
zz ue-“4 . J;;
1 9 0 PH 413
-m
= ae -I/4 je-(4,
zc ,-l/4-“,
i) sy/:y/+GfX = C:A2[2&a+2ae-1r4&Lj
=“iia
--2J;;a(l+e-1’4)
= Ci2(1+e-I”)
a d IVv:V+d = 1
C,z
IC+I= &ii-)= [2(1+ em1’4)r”2
IC-1 = [2(1- e-li4)r”2
-M= A2e a1 E
x-&J
a
PH 413 191
1 9 2 PH 413
X--x0
a-48 -2x 2iT 4rr
PH 413 193
<x>=<p>=o m=l~ &=x , &l=p
Ol%J p+-ih$
wi sin2y = 2sin y cos y
<xp> =
=
I<&!$>>( =
<px> =
iliz *I2-a2 J (
sin 2nx XG?X--a/2
a 1
itrn
[
sin(2m I u) cos(2da) a’2-a2 (2x/a) -x (2ala) 1 _!,,
ih-5A2
<b,x]>+<xp>
=ih
-ih+-2
194
2
PH 413
= ‘ 2 nz iiLsinnmcosnmdr- - -( ) I
- -LLi, L L
1 nz ti’=( 1 Isin 2nm a!x- - - -
LLi, L
= 0
In2a2h2=
2mL2= En
PH 413 1 9 5
=
2L=I
. ,nm- xslnLo
--drL
2=-L bn n
= -2!-[$~[O+(nn)i]
< YE. X2 v/E, ’I I =
[usinu(2sinu-2ucos~)]‘~+2”~u~du-2~~sin~udu0 0 0
1
PH 413