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November 13, 2012 University Physics, Chapter 37 1
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m a x s inE E x t
,r t
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1 .
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November 13, 2012 University Physics, Chapter 37 2
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x
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37.3 .
( ) ( ) .
November 13, 2012 University Physics, Chapter 37 3
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? dx .
x x+dx I(x)dx .
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x . 0 1 . dx .
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November 13, 2012 University Physics, Chapter 37 4
2
x d x x d x
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1 . .
(x)* (x) , .
( A .) , .
November 13, 2012 University Physics, Chapter 37 5
2
*1x d x x x d x
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(1)
, . .
t=0 x . .
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November 13, 2012 University Physics, Chapter 37 6
c o s s inx C x D x
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(A, B, C, D) .
A B , .
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November 13, 2012 7
i x i x
x A e B e
University Physics, Chapter 37
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(1)
B=0 B .
, .
?
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November 13, 2012 University Physics, Chapter 37 8
i x
x A e
2 2 2 p p
h p h
d
x i xd x
p
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p . x i .
, .
p= , x p .
November 13, 2012 University Physics, Chapter 37 9
(2)
i x i x i xd d
i x i A e i A i e A e x p xd x d x
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? . .
K=p2/2m , .
November 13, 2012 University Physics, Chapter 37 10
2 2 2
2
2
1 1
2 2 2
d dx x i x x
m m d x m d x
K p
(1)
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-x .
.
November 13, 2012 University Physics, Chapter 37 11
2 2 2 2 2 2
2
2 2
2 2
2
2 2 2 2
2 2
i x i x i x
i x i x i x
d d dx A e i A e A e
m d x m d x m d x m
pA e A e K A e
m m
(2)
2 2 2 2
22 2
i x i x i x i x i x i xdA e B e A e B e A e B e
m d x m
K
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November 13, 2012 University Physics, Chapter 37 12
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1 .
U(x) , E .
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November 13, 2012 University Physics, Chapter 37 13
22
22
d xU x x E x
m d x
(2)
K x U x x E x
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0, . .
? .
E 0 . , 0 .
, . .
November 13, 2012 University Physics, Chapter 37 14
i x i x
x A e B e
(3)
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0 .
.
(, ).
0 .
.
November 13, 2012 University Physics, Chapter 37 15
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(1)
. x .
0 a 0.
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November 13, 2012 University Physics, Chapter 37 16
0
0 0
fo r x
U x fo r x a
fo r x a
i x i x
x A e B e
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(0)=0 .
.
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November 13, 2012 University Physics, Chapter 37 17
c o s s inx C x D x
0 c o s 0 s in 0 0C D C
(2)
s in 0
0
x D x fo r x a
x e ls e w h e r e
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, 0 . x=0 sin(0)=0 (0)=0 .
x=a .
.
November 13, 2012 University Physics, Chapter 37 18
s in 0a D a
21, 2 , 3,
aa n fo r n
(3)
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D .
November 13, 2012 University Physics, Chapter 37 19
0 0
s in 1, 2 , 3, 0
0
fo r x
n xx D w ith n fo r x a
a
fo r x a
(4)
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D .
November 13, 2012 University Physics, Chapter 37 20
0
2 2 2 2
0
2
0
2 22
0
2
1
1 0 s in 0
1 s in2
2,
a
a
a
a
x d x x d x x d x x d x
n xD d x
a
n x aD d x D
a
th u s Da
(5)
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n .
n . (1(x) , n=1)
November 13, 2012 University Physics, Chapter 37 21
0 0
2s in 1, 2 , 3, 0
0
fo r x
n xx w ith n fo r x a
a a
fo r x a
(6)
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(x)
n=1,2,3,4 . .
November 13, 2012 22 University Physics, Chapter 37
(7)
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()
. .
, , .
. [0,a] ( ) .
.
November 13, 2012 University Physics, Chapter 37 23
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0 a
.
x dx dt .
November 13, 2012 University Physics, Chapter 37 24
2 /v E m
: (1)
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dt v(x) . dt=dx/v(x). x x+dx .
November 13, 2012 University Physics, Chapter 37 25
1
cx d x
v x
.
.
: (2)
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(1)
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0, 0. .
.
n .
November 13, 2012 University Physics, Chapter 37 26
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K
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November 13, 2012 University Physics, Chapter 37 27
22 2 2
2 2
2
2 22 2
2s in
2 2
2c o s
2
2s in
2 2
n
n n n
n n
n n n
d x d n xE x x
m d x m d x a a
n d n xE x
m a d x a a
n n x nE x x
m a a a m a
K
2 2
2
21, 2 , 3 ,
2n
E n fo r nm a
(2)
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November 13, 2012 University Physics, Chapter 37 28
() , .
.
(3)
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37.1 (1)
:
2.00 ?
:
9.10910-31 kg 2.00 10-10 m. n=1 .
November 13, 2012 University Physics, Chapter 37 29
23 4 22 2
2
1 223 1 1 0
1 8
1
1 .0 5 4 6 1 01
2 2 9 .1 0 9 1 0 2 .0 0 1 0
1 .5 1 1 0 9 .4 3
J sE
m a k g m
E J e V
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10-10 m . 10eV .
:
? 1.0010-14 m.
:
1.67310-27 kg .
5MeV 50MeV . .
November 13, 2012 University Physics, Chapter 37 30
23 4 22 2
2 1 3
1 222 7 1 4
1 .0 5 4 6 1 01 3 .2 8 1 0 2 .0 5
2 2 1 .6 7 3 1 0 1 .0 0 1 0
J sE J M e V
m a k g m
37.1 (2)
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(1)
1 xy 2 , 3 . 1 .
, 1 2 ? x y, U(x,y) . (x,y) .
.
November 13, 2012 University Physics, Chapter 37 31
22
2 2
yxpp
Km m
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x y .
1 .
.
November 13, 2012 University Physics, Chapter 37 32
, ,
, ,
x
y
x y i x yx
x y i x yy
p
p
(2)
2 2
2 2 2 2
2 2
1 1, , ,
2 2
, , ,2 2
x yx y x y x y
m m
x y x y x ym x m y
K p p
K
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2 .
, . x y ,
,
.
November 13, 2012 University Physics, Chapter 37 33
2 22 2
2 2
, ,( , ) , ,
2 2
x y x yU x y x y E x y
m x m y
(3)
)()(),(21
yUxUyxU
)()(),(21
xxyx
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2
x y 1 .
November 13, 2012 University Physics, Chapter 37 34
1 2
1
2
,
0 0
2s in 1, 2 , 3, 0
0
0 0
2s in 1, 2 , 3, 0
0
x
x
y
y
x y x y
fo r x
n xx w ith n fo r x a
a a
fo r x a
fo r x
n yx w ith n fo r y b
b b
fo r x b
(4)
1 2
1 2
( , ) ( ) ( )
0 0
: ( ) 0 0 , ( ) 0 0
0 0
U x y U x U y
fo r x fo r y
w ith U x fo r x a U y fo r y b
fo r x fo r y
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November 13, 2012 University Physics, Chapter 37 35
nx, ny .
(5)
yxnn ,
yxnn ,
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2 1 n .
a=b() . . (nx=1, ny=2 ) (nx=2, ny=1 ) .
, . .
November 13, 2012 University Physics, Chapter 37 36
(6)
2 2 2 2
2 2
, 2 22 2
x yn n x y
E n nm a m b
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, .
1 .
November 13, 2012 University Physics, Chapter 37 37
1
0
0 0
fo r x
U x fo r x a
U fo r x a
(1)
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, .
x
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x>a . .
.
E , .
November 13, 2012 University Physics, Chapter 37 39
22
12
2
1
2 2
2
2
d xU x E x
m d x
d x m U Ex
d x
(3)
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(1)
E>U1, 0 , 0 a U1=0 .
? [0,a] .
x>a E>U1 .
November 13, 2012 University Physics, Chapter 37 40
1c o s s inx F x G x fo r x a a n d E U
2 2 2 2 2
22 2
n
nE
m a m
2 2
1
2K E U
m
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U1 .
D, F, G . x=a .
x=a .
1 .
November 13, 2012 University Physics, Chapter 37 41
1
2
2 m U
(2)
0 0
s in 0
c o s s in
fo r x
x D x fo r x a
F x G x fo r x a
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?
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November 13, 2012 University Physics, Chapter 37 42
. .
(3)
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, E
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E
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D F . x=a .
x=a .
1 .
.
November 13, 2012 University Physics, Chapter 37 45
2 21 1 12 2 2 2
2 2 2 2m U E m U m E m U
(3)
0 0
s in 0
x
fo r x
x D x fo r x a
F e fo r x a
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?
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x .
x
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37.2 (1)
:
. a=1.30nm, n=2 20% , ?
:
.
, , n=2
November 13, 2012 University Physics, Chapter 37 47
2
2 2
2
a aa
n
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20%
, .
37.20 .
November 13, 2012 University Physics, Chapter 37 48
2 21 .2 1 .2 a
2
2
2 2
1 .2 a
37.2 (2)
2 2
0 0
s in 0
x
fo r x
x D x fo r x a
F e fo r x a
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x=a .
x=a 1 .
November 13, 2012 University Physics, Chapter 37 49
2
2
s in
s in
a
a
D a F e
Fa e
D
2 2
2
2
c o s
c o s
a
a
D a F e
Fa e
D
37.2 (3)
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.
2 U1 .
November 13, 2012 University Physics, Chapter 37 50
2
2 2
2 2
s in c o s
c o t
a a
a
2 2 2 21
2 2 22
2 2
2 22
1 2 2 2
2c o t ( )
1 c o t ( ) 1 c o t ( )2
m Ua
U a E am
37.2 (4)
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. .
4/3
November 13, 2012 University Physics, Chapter 37 51
2 2 1 4
2 3 31 c o t ( ) 1 c o t ( 2 / 1 .2 ) 1a
222 2 2
2
2 2
23 4
2 23 1 9
2 0
2
4
1 23
2 / 1 .2
2 2 2 .8 8
6 .6 2 6 1 0
2 .8 8 9 .1 0 9 1 0 1 .3 0 1 0
9 .9 0 1 0 0 .6 1 8
0 .8 2 3
a hE
m m m a
J sE
k g m
E J e V
th u s U E e V
37.2 (5)
-
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37.3 (1)
:
U1=0.823eV, a=1.30nm , ?
:
37.2 =2/(1.2a) . .
November 13, 2012 University Physics, Chapter 37 52
21
2
c o t
2
a
a n d
m U
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.
, , .
November 13, 2012 University Physics, Chapter 37 53
21
2
22
1
2
2c o t
2c o t
m Ua
a m Ua a a
37.3 (2)
23 6 .5 co t ,y y y w ith y a
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a=1.30nm 2a2mU1/2=36.5
.
.
y1=2.68 y2= 5.23 .
y2 2=2/1.2 , 1=2.6/a .
221.
November 13, 2012 University Physics, Chapter 37 54
37.3 (3)
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(1)
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November 13, 2012 University Physics, Chapter 37 55
x/a .
x/a=1 , 1 .
.
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( ).
1 3.1%, 2 18.6% .
November 13, 2012 University Physics, Chapter 37 56
(2)
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(1)
, 37.15a ?
November 13, 2012 University Physics, Chapter 37 57
.
1
0
0 0
0
fo r x
fo r x aU x
U fo r a x b
fo r x b
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0 .
November 13, 2012 University Physics, Chapter 37 58
0 0
s in 0
s in
x
fo r x
D x fo r x a
xF e fo r a x b
G x fo r x b
(2)
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0 b . 0 a , a b .
x=b . 0 a x>b .
. x>b (b ) x=a x=b 1 .
http://www.youtube.com/watch?v=W9OKOw7gDSY&NR=1
November 13, 2012 University Physics, Chapter 37 59
(3)
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November 13, 2012 University Physics, Chapter 37 60
(4)
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0
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?
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37.15a .
. ( ) .
November 13, 2012 University Physics, Chapter 37 62
(6)
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37.4 (1)
:
22.4MeV 36.2MeV, 8.4fm ?
:
.
November 13, 2012 University Physics, Chapter 37 63
12
2 m U E
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.
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November 13, 2012 University Physics, Chapter 37 64
2
1 9 7 .3 4 /
9 3 9 .5 7 /n
M e V fm c
m M e V c
2
1
2
2 9 3 9 .5 7 / 3 6 .2 2 2 .40 .8 1 6
1 9 7 .3 4 /
M e V c M e V M e Vfm
M e V fm c
1
2 0 .8 1 6 8 .42 61 .1 1 1 0
fm fmb a
tT e e
37.4 (2)
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(1)
1981 , .
. 1986 .
.
November 13, 2012 University Physics, Chapter 37 65
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, x0 .
x1 x0 .
November 13, 2012 University Physics, Chapter 37 66
(2)
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. .
(STM)
.
November 13, 2012 University Physics, Chapter 37 67
(3)
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November 13, 2012 University Physics, Chapter 37 68
http://www.youtube.com/watch?v=kSsf-7VhUsI&feature=channel
. ( )
(4)
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(1)
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1/r
. ?
?
, k , 0 .
November 13, 2012 University Physics, Chapter 37 69
2 2 21 1
02 2U x kx m x
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, K(x)=E-U(x) . , .
November 13, 2012 University Physics, Chapter 37 70
( ) 0 .
. E
( ) .
(2)
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. .
, .
November 13, 2012 University Physics, Chapter 37 71
22
2 21
022
2
2 21
022 2
2 2 2
20
2 2 2
2
2( )
2
d xm x x E x
m d x
d x mE m x x
d x
d x m E mx x
d x
1 02 1, 2 ,nE n n
(3)
-
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. n .
.
November 13, 2012 University Physics, Chapter 37 72
2 2
2 2
2 2
2 2
2 2
2
0 1 4
2
1 1 4
2
2
2 21 4
3
2
3 31 4
4 2
2
4 4 21 4
1
1 12
2
1 14 2
8
1 18 1 2
4 8
1 11 6 4 8 1 2
3 8 4
x
x
x
x
x
x e
xx e
xx e
x xx e
x xx e
0m
(4)
-
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, .
.
Hn n . x n , .
November 13, 2012 University Physics, Chapter 37 73
2 2
2
1 4
1 1
!2
x
n nn
x H x e
n
2 2
1
nn x x
n n
dH x e e
d x
(5)
-
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1(x) (1)
. 37.26 37.24 .
n=1 .
1(x) x .
November 13, 2012 University Physics, Chapter 37 74
2 2 2 2
2 2
1 1 4
1 12
2
x xxx e A x e
2 2 2 2
2 2 2 2
2
2 2
1 2
2 3
2 2
12 2 4
3
x x
x x
d xx A e A e
d x
d x xx A e A e
d x
-
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x x3 0 .
November 13, 2012 University Physics, Chapter 37 75
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2
20
2 2 2
3 2 2
2 2 2 20
2 4 2 2
3 2 2
2 2 3 20
2 4 2 2
2
3 2
3 2
x x x
x x x
d x m E mx x
d x
x x m E mA e A e x A x e
x x m E mA e A e x x A e
2 2
2 2
2 30
2 2 4 2
2 3 10
x m E mA e x x
1(x) (2)
-
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x3 .
x .
1(x) .
November 13, 2012 University Physics, Chapter 37 76
2 2
0
4 2
0
10
m
m
2
3
22 2 2
2 30
m EE
m
1(x) (3)
2 2
3 3 3
02 2 22
0
Em m m
31 0 02 21E
-
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1 0 . .
.
. n .
November 13, 2012 University Physics, Chapter 37 77
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(1)
. , , , ?
.
, * .
November 13, 2012 University Physics, Chapter 37 78
2
*1x d x x x d x
-
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.
< > , .
.
November 13, 2012 University Physics, Chapter 37 79
* * d
p x x d x i x x d xd x
p
(2)
2 2
2 * 2 *
2
1 1
2 2 2
dK p x x d x x x d x
m m m d x
p
*
x x x x d x
-
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37.5 (1)
:
0=1.001016 s-1 n=0 ?
:
.
November 13, 2012 University Physics, Chapter 37 80
2 2 2 2
2 2
0 1 4
1 x xx e A e
-
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, .
*(x)= (x) .
November 13, 2012 University Physics, Chapter 37 81
2 2 2 2
2 2
*
2 2
2 2
0
x x
x
x x x x d x
x A e x A e d x
x A x e d x
x
37.5 (2)
-
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.
n=0 .
0 .
November 13, 2012 University Physics, Chapter 37 82
2 2 2 2
2 2
*
2
2 2
2 2
2
2
2
x x
dK x x d x
m d x
dK A e A e d x
m d x
2 2 2 2
2 2
2 2
2 4 2
1x xd xA e A e
d x
37.5 (3)
-
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.
.
November 13, 2012 University Physics, Chapter 37 83
2 2 2 2
2 2
2 2 2 2
2 2
2 2
4 2
2 2 2
4 2
2 2 2 2
2
4 2
2 2 2 2
3
4 2
2
2
1
2
1
2
2 2
1
2 2 2
4
x x
x
x x
xK A e A e d x
m
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F
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F
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F F
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November 13, 2012 University Physics, Chapter 37 110
1 1 1 2 1
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F
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n n n n
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x x xx x xn
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2 3
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i
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4
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,,
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,
r t
r tr t
r t
r t
2 2 2 2 2 2
2 2 2, ,
2 2 2r t i r t
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2
3 3
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4 4
0 0 w wm c p c
w wm c p cE
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() 1932,
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November 13, 2012 University Physics, Chapter 37 127
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mcE
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37.7 (2)
2hrs7500J/s102.1
J1000.9
9
12