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