Eq
Eq
1.23
2¥ '÷dj¥
:
1.24
24$
I.
:
.
[4*4
v4
=
.
Em "¥÷+Évy*
'
=
-
.de/=fIJ-+l4*Y.)dx=fI(FEht4iY;-)dx=fIl-Em
"¥÷÷I:/ 4.
=
=
-
Ent:( 5¥
"
-
Yi
+
:(¥mY¥
Y
-
Dax
:# lax
s÷÷.to
(e)
expectation
p
value of
'
"
=/ 4T¥ ¥ / Ydx
1¥.ca?-x4dx
titi fila
titi f- 21.2 tola xydx
4A't ( a'
¥11 !
lia
44¥ )ñ( ;-) 41¥ / (3)
(e)
'
-
✗
'
-
=
a
'
-
-
=
'
✗
=
a
=
=
(a)
/
-
-
f? A' la
1
=
2
I
=
2A
I
=/
141 'd ✗
y
'
-
'
1414×+1+9,yy×
+
+a
Y
'
141'd ✗
-0
✗
'
l 'd ✗
2A
=
'
[(
fix Esa
× ) /;
/ ¢5 Fast ¥1 ( ) )
>
-
2A
=
"
✗
'
a
"
-
za
a.
,
-0
'
'
=
✗ + ✗
-
-
o
2A as
=
=
¥
a
a
a.
-
(b) (
X
(c) ( P )
)
=
=
f.?
=
¥ A' f? Ca
=
use
0
=
f.
41-2×1 dx
✗
0
integrand
Odd
d¥
✗
'
141
'
because
only
we
dx
×
know
a>
'
2A
=
'
[
'
✗
ca
'
-
b) dx
fila
2A
( "¥ 20¥ ¥11 ;
a' J
(
2A
( ¥ 2¥ ¥)
a'
2A
f- ¥ E) 2A'a*( ¥;)
2A
"
'
'
'
"
✗ -2A ✗
a
+
4'd ✗
+ ✗
=
2A
'
+
✗
+
-
-
=
=
-
'
=
=
a
'
=
-
value of
"
=
'
F- f.? Ma -41 A¥a d×
'
dx
.
of
✗
of P
FEAT
=
=
EEF
1- FE
(odd integrand )
0
=
.
(d) Expectation
(x2 )
141 'd ✗
✗
f) 4*-3×4
F-
we can't
at t
Op
=
=
¥
(9) Uncertainty
-
F÷:
A-
=
Ydx
+
a
iEi=fÉ
=
Ox
>
-
5_z¥
(f) Uncertainty
Normalization
.
-
'
=
=Y¥as)aY¥,t=¥
'
-
F ¥)
+
(b)
Check the
0×0,
=
uncertainty
Principle
-4=0,59%-2 Iz
(g) (E. FE) JE
=
Equation
¥141
From
1.22
'
=
4*24
It
equation
,¥*y
1.23
¥=¥mY¥
-
w
,
(a)
→
vxy-fi.IT#+i-i i )Yyty--lYi--jh--mfY+-YI-.YI.ie/- ni-fYVY+vHiY- zih-m(Y*Y#-YY?-yf+ni=1yi(v*
÷Mi=4*¥++¥¥=*(¥m¥
-
.
-
y
-
lvotir votirl-o.ir
-
=¥m(*i¥ TEH ¥1M
-
-
→
÷%Mid×=÷±m(Y"¥ Max ¥1;1Y
d÷=¥a(Y"¥ ¥411 ¥1
-
-
.
"
.
-
-
÷
d÷ -2¥ p
CQED)
=
""
"
Tt
-
=
ZIP
=/ Edt
-
In
P
P
(f)
=
=
-
⇐
t
e- 2ft
+
C
|
"
"
=
÷
-
T=
-
"
"
¥
#
/
From
yµ
*
=
'
-
Nanni .ae
¥mY¥:¥v*yi
(b)
0
) PV
Nk ☐ T
=
Pd
?
k☐T
=
( t1
D=
→
1-
%
I
=
V
d
=
'
"
zh¥F¥j↳=¥?¥T
<
T
N
;
's
÷÷y
L
,
)
¥(÷|%p%
L
T
(
For Helium
°
P
I
=
=
4 Mp
1×105
=
6.8
=
Nlm
✗
1027kg )
'
!×(%÷÷÷→"ao,
'
T
atm
M
"
a
L
T
Under 2.8
÷E=
d L
,m!÷
<
T
o
(m
For electrons
"
T
(
T
C
16.6×10 )
D=
"
✗
1.3
✗
✗
105
T L
105k
nuclei
(
At
electrons in solid
are
,
m
=
28
Mp
=
28
quantum
mechanical
(1.7×10-27)=4.8×10"kg)
x¥÷;÷¥→y
.
T C 2.4
Below 2.4k
k
the nuclei are
quantum
mechanical
ideal gas
(m
=
are
Mp
quantum
-
=
1.7×10
mechanical
"
kg
)
x¥÷¥i¥
"
K
an
0.01m
1031kg )
3(g.¥y(1.y×1j2Ñ
Under 1.3
Silicon
9. I
=
atoms in
) For atomic Hydrogen
1- <
(a)
k
2.8 K
6.2×10
3 k
"
"
K
Hydrogen
are in the
classical
regime
.
.