Teorema Fundamental da Trigonometria Demonstração... )θ 1 cos sen 1 0 sen θ cos θ θ ·
2001 Sen Mon
33
2001 G/EY LdBj 1 1 2001G/EY F~3X;n83 J*M}@l Lg 2JL\ "# LdBj 1 !!%O%_%k%H%K%"%s H = 1 2m ˆ p 2 + mω 2 2 ˆ x 2 ··· (1) $G5-=R$5$l$k 1 <!85D4OB?6F0;R$r9M$($h $&!#$3$3$G ˆ p $O1?F0NL1i;;;R!$ ˆ x $O0LCV 1i;;;R!#$3$N7O$O1i;;;R a = mω 2 ˆ x + i √ 2mω ˆ p, ··· (2) a † = mω 2 ˆ x − i √ 2mω ˆ p ··· (3) $rMQ$$$FD4$Y$k$3$H$,= PMh$k!#0J2<$NLd$K !$2rEz$K;j$k6ZF;$rE:$($FE z$($h!# 1. !! (1) <0$ N%O%_%k%H%K%"%s $r a, a † ∇ → → a, a † L ∇ → 2. !!4 pDl>uBV |0 a |0 = 0 $rK~$?$9 !#$3$l$rMQ$$$F:BI8I=<($N4pDl> uBVGHF04X?t ψ 0 ( x) = x|0∇ → → ∝ ψ 1 ( x) $r5a$a$h!#$3$3$G$O5,3J 2=$O9M$($J$/$F$h$$!# 3. !!4pDl>uBV |0 ∞ a † ∇ |α = exp αa † |0 , α ··· (4) $O%3%R!<%l%s%H>uBV$H8F$P$l$k!# (i) !!%3%R!<%l %s%H>uBV$ ,1i;;; R a $N8GM->uBV$K$J$C$F$$$k$3$H$r<($;!# (ii) !!%3%R!<%l%s%H>uBV$N4V$NFb@Q β ∗ |α ∇ → β ∗ | = 0| exp( β ∗ a) $G$"$k!# (iii) !!%3%R!<%l%s%H>uBV$, n HVL\$NNe5/>uBV$r4^$‘3NN($r5a$a$h!# (iv) !!%3%R!<%l%s%H>uBV$K$D$$$FIT3NDj@-4X78$rD4$Y$h$& !#:BI8$N4|BTCM ˆ x≡ α ∗ | ˆ x| α / α ∗ |α ˆ x≡ α ∗ ˆ x 2 α / α ∗ |α ∇ → | (∆ x) 2 = ( ˆ x − ˆ x) 2 ··· (5) $r5a$a$h!# (v) !!F1MM$K1?F0NL $NIT3NDj$5$NFs >h (∆ p) 2 = ( ˆ p − ˆ p) 2
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2001
Transcript of 2001 Sen Mon
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2001 1 1
2001
1
H =1
2mp2 +
m2
2x2 (1)
1 p x
a =
m
2~x +
i2m~
p, (2)
a =
m
2~x i
2m~p (3)
1. (1) a, a a, a
2. |0 a |0 = 0 0 (x) = x|01 (x)
3. |0 a
| = exp(a
)|0 , (4)
(i) a
(ii) | | = 0| exp (a)
(iii) n
(iv) x |x| / |x
x2 / |(x)2 =
(x x)2
(5)
(v) (p)2 =(p p)2
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2 2001 2
2N x b
Q (> 0) +x xi i i = +1 1 1
O (x = 0) E (E 0) +xT
E xkB
xi xNx
O
E 0
i
1. E = 0
(i) xN S L = bN N, xN/b 1 Stirling x 1
log x! x log x x + . . .
(ii) xN X xN
(iii) N xN/b ( 1) X xN xN X
2. E > 0
(i) E ZN A bQE/kBT
(ii) i i i
(iii) xN NA 1xN NQE 1 (c)
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2001 3 3
3 Maxwell
(E) = 0, B = 0,
E = Bt,
(B
)=(E)t
.
E B 3 x < 0 x > 00, 0 1(> 0) 0
E0(r, t) B0(r, t) E1(r, t)B1(r, t) E2(r, t) B2(r, t)
0
0 1
n0
n2
n1
x = 0
y
xz
1. x = 0
E0,t(0, y, z, t) + E2,t(0, y, z, t) = E1,t(0, y, z, t) (1)0[E0,x(0, y, z, t) + E2,x(0, y, z, t)] = 1E1,x(0, y, z, t) (2)B0,t(0, y, z, t) + B2,t(0, y, z, t) = B1,t(0, y, z, t) (3)
t x x
(i) (1)
(ii) E 0 E = (x = +0)(y, z, t) E1,x(0, y, z, t) 0 1
2. xy x, y, z x, y, zn0 = x cos 0 + y sin 0 (0 < 0 < pi/2) B0(r, t) z
B0(r, t) = zB0 exp[i(k0n0 r t)]
E0(r, t) = E0 exp[i(k0n0 r t)]
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4 2001 3
(i) k0 0 0
(ii) E0 B0 z k0 n0
3.
B1(r, t) = zB1 exp[i(k0n1 r t)],B2(r, t) = zB2 exp[i(k0n2 r t)]
n1 n2
n1 = x cos 1 + y sin 1,n2 = x cos 0 + y sin 0
(i) (3) 0 1
(ii) (1) (3) B1 B2 k0 k1 0 1 B0
(iii) 0 + 1 = pi/2
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2001 4 5
4
h m
1. z (z, )g
2.
3. 2h v0
z
2h
h
x
yO
v0
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6 2001 5
5
1. pi0
940 MeV/c2 pi0 (mpi) 135 MeV/c2 MeV 106
2. pi0 2 () pi0
(i) Epi pi0 E pi0 E
(ii) 30 GeV pi0 2 GeV 109
(iii) pi0(Epi mpic2
)2
3. pi 2(pi + d n + n)
(i) pi Ppipi pi 0
3S1
(ii) 1/2
(iii)
(iv) pi
pi0
E
pi0
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2001 6 7
61.
(i) 1.0 105Pa 0C 1.0cm31 1.0 105Pa 6.0 1023
0C 1 1 22l
(ii)1.0cm 1.0mA/cm2
1 3.01016cm21.6 1019C
1
2.
(i)
A
150 V
20 V
(ii) 1.0 106Pa
He Ne Ar Kr Xe(nA) 0.1 0.3 1.3 1.7 2.4
(iii) X
3.
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8 2001 7
7e p [GeV/c] B [Tesla]
R [m]
R =p
0.3B (1)
B 3L L
Rp
p/p
S
B
L
L
x
y
R
1.
2. x-yS
(i) S L R L R (1)S L p
(ii) S p L, B, p, S(iii) z
x x SS
(iv) p/p L, B, p,
3.
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2001 8 9
81. (a) L U0 E (U0 > E > 0)
1eikx ~2k2/2m = E, k > 0 m
x
U0
O0 L
(a)
O
O
EFL
EFL
EFR
EFR
JRLJLR
JLRJRL
(b)
(c)
(i) x < 0 0 x L x > L (x), (x), (x)Aeikx + ... A, ...
(ii) (x), (x), (x)
(iii) e Tj E n
2. N 2
(i) 1 E D(E)
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10 2001 8
vF m h N
(ii) (a) (b)EFL, EFR T
fL(E,T ), fR(E,T ) 1J JLR
JRL (J = JLR JRL) JLR
JLR = e2
0
2Em
D(E) fL(E,T )[1 fR(E,T )]T (E)dE (1)
(iii) (c) V ( EFL EFR) J J
G= limV0
JV
G =(
2e2
h
)T (EFL)
V = eV
3. Fe n SiFe n Si
2 ( 40
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2001 9 11
91.
(i) 1mol G
G = 2.3RT log10 (Co/Ci) + zFV
R = 2.0 103kcal/Kmol T K CoCi z F = 23kcal/Vmol
V
(ii)
20C
(millimol/`) (millimol/`)K+ 400 20Na+ 50 440Cl 51 560
(iii)
(iv)
2. eyeless
(i) eyeless
(ii) eyelesseyeless
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12 2001 1
11. a =
m
2~x +
i2m~
p a =
m
2~x i
2m~p x p
x =
~
2m(a + a
), (1)
p = i
m~
2(a a
) (2)
H =1
2mp2 +
m2
2x2
=1
2m
i
m~
2(a a
)2 + m22
~
2m(a + a
)2=~
2(aa + aa
). (3)
[x, p
]= i~ a a[
a, a]
= 1 aa = 1 + aa (4)
H = ~(
12
+ aa)
(5)
a, a n (= 1, 2, . . . ) En|n |n a a |n En ~
|n a a |n En + ~~
a, a
(5) aa N N(a
)n |0 = n (a)n |0N ~
2. a |0 = 0 x |a| 0 = 0x
m
2~x +
i2m~
p 0
= 0
[
m
2~x +
i2m~
(~
iddx
)]0 (x) = 0
(
m
~x +
ddx
)0 (x) = 0. (6)
0 (x) = Const. exp(m
2~x2
) (7)
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2001 1 13
1
|1 a |0
1 (x) = x|1 x
m
2~x i
2m~p 0
=
[m
2~x i
2m~
(~
iddx
)]0 (x)
=
2m~
x exp(m
2~x2
)= Const.x exp
(m
2~x2
) (9)
2
3.
(i) | = exp(a
)|0 a
a | = a exp(a
)|0 = a
n=0
(a
)nn! |0
=
n=0
n
n!
[(a
)na + n
(a
)n1] |0=
n=1
(a
)n1(n 1)! |0 = exp
(a
)|0
= | . (12)
a
(ii) a | = | n
n |a| = n || n + 1 n + 1| = n| . (13)
n| = n
n 1
n 2
10| =
n
n!0| (14)
0| =0exp (a) 0 = 0
n=0
(a
)nn!
0
= 0 |1| 0 = 1 (15)
1 0|0 = |0 (x)|2 = 1 Const.0 (x) =
(pi1/4)1
exp[x2/
(22
)] (8)
~/ (m)
2
1 (x) =(pi1/43/2/
2)1
x exp[x2/
(22
)] (10)
n n (x) x/ n Hn () () = (2nn!)1/2 [m/ (pi~)]1/4 exp (2/2) Hn () (11)
H0 () = 1 H1 () = 2 H2 () = 42 2
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14 2001 1
n| = n
n!
{|n}n=0,1,2,... |
| =
n
|n n| =
n
nn!|n (16)
| =
m
()mm!m|
| =n,m
()mm!
nn!m|n =
n
()nn!
= exp () (17)
(iii) | 3-(b) = | = exp(||2)
|normal = 1exp(||2)
n
nn!|n
n |Pn|
Pn = |n|normal|2 = 1exp(||2)
||2nn!
(18)
(iv) xx2
x = |x| / | =
~
2m(a + a
)/ |
=
~
2m( + ) , (19)
x2
=
x2 / | = ~2m (a + a)2 / |
=~
2m
{(a)2 + aa + (1 + aa) + a2} / |=
~
2m[()2 + 2 ||2 + 2 + 1
]=
~
2m[( + )2 + 1
] (20)
(20) (4) x
(x)2 =(x x)2
=
x2
x2
=~
2m[( + )2 + 1
]
~2m ( + )2
=~
2m (21)
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2001 1 15
(v) (d) pp2
p = | p| / | =
i
m~
2(a a
)/ |
= i
m~
2( ) , (22)
p2
=
p2 / | = m~2 (a a)2 / |
= m~2
{(a)2 aa (1 + aa) + a2} / |= m~
2[()2 2 ||2 + 2 1
]=
m~
2[1 ( )2
] (23)
p
(p)2 =( p p)2
=
p2
p2
=m~
2[1 ( )2
]
i
m~
2( )
2=
m~
2 (24)
(x)2 (p)2 = ~2
4 xp = ~
2 (25)
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16 2001 2
2+x N+ x N
N = N+ + N, xN = b(N+ N) (1)
N+ =bN + xN
2b =N2
(1 +
xN
L
), N+ =
bN xN2b =
N2
(1 xN
L
) (2)
1. E = 0
(i) N, |xN |/b 1 N+, N 1 Stirling
S = kB log W, W =N!
N+! N! (3)
S = kB(log N! log N+! log N! )
' kB (N log N N N+ log N+ + N+ N log N + N)= kB
(N log N N+ log N+ N log N)
= kB[N log N N
2
(1 +
xN
L
)log
{N2
(1 +
xN
L
)} N
2
(1 xN
L
)log
{N2
(1 xN
L
)}]= kB
[N log N N
2
(1 +
xN
L
)log N
2 N
2
(1 +
xN
L
)log
(1 +
xN
L
) N
2
(1 xN
L
)log N
2 N
2
(1 xN
L
)log
(1 xN
L
)]= kBN
[log 2 1
2
(1 +
xN
L
)log
(1 +
xN
L
) 1
2
(1 xN
L
)log
(1 xN
L
)] (4)
(ii) E = 0 U Helmholtz F = U TSdF = S dT + XdxN X
X =FxN
T
= T SxN
T
(5)
= kBT N[
12L
log(1 +
xN
L
)+
12L 1
2Llog
(1 xN
L
) 1
2L
]=
kBT2b log
1 + xN/L1 xN/L (6)
(iii) xN/L 1
X =kBT2b
[log
(1 +
xN
L
) log
(1 xN
L
)]' kBT
2b
(xN
L+
xN
L
)=
kBTNb2
xN (7)
(7) Hooke X xN
2. E > 0
(i) xi = bmiN
i=1mi =
Ni=1
ij=1
j =N
i=1(N + 1 i)i (8)
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2001 2 17
H(= U)
H = bQEkBTN
i=1mi = A
Ni=1
(N + 1 i)i (9)
ZN
ZN =i=1
eH (10)
=(eNA + eNA
)(e(N1)A + e(N1)A
) (eA + eA)=
Nl=1
(elA + elA
) (11)
(ii) (11) i
i = eA(N+1i) eA(N+1i)
eA(N+1i) + eA(N+1i)= tanh A(N + 1 i) (12)
(iii) |x| 1 tanh x ' x x3/3 NA 1
xN = b N
i=1i
= b
Ni=1i = b
Ni=1
tanh A(N + 1 i)
' bN
i=1A(N + 1 i) = 1
2N(N + 1)Ab = (N + 1)b
2
2kBT(NQE) (13)
NQE ' 2kBTNb2 xN (N 1) (14)
(14) NQE (7) X dF = S dT + XdxN(7) (14) NQE X = NQE/21/2 i
X
1 (c) E = 0 X XH U xN
ZN =xN
WeXxN (W xN )
=i=1
exp(Xb
ii
)=
(eXb + eXb
)N= 2N(cosh Xb)N (15)
xN = 1
log ZNX
= Nb tanh(Xb) ' Nb2
kBTX (16)
(16) Xb 1 1(c) N xN/b
X =kBTNb2 xN (17)
(7)
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18 2001 3
3Maxwell
(E) = 0 (1) B = 0 (2) E = B
t (3)
(B
)= (E)t
(4)
=
1 (> 0) x > 00 x < 0 = 0
x < 0 Ev Bv x > 0 Ed Bd
1. 1: x = 0
1, 0
0, 0
x
t
S
l
A B
CD
(i) t x x 1ABCD AB = l BC = S (3)
S
( E) dS = t
SB dS (5)
B 0 0 StokesS
( E) dS =
ABCDE dl Stokes
ABE dl +
CDE dl ( 0)
= (Ed,t Ev,t)l = 0 Ev,t(0, y, z, t) = Ed,t(0, y, z, t) (l 0) (6)
(4)10
S
( B) dS = t
SE dS 0 ( 0)
ABCDB dl (Bd,t Bv,t)l = 0 (7)
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2001 3 19
1 S V S (1) VGauss
V (E)dV =
SE dS Gauss (8)
(1Ed,x 0Ev,x)S = 0 ( 0) 0Ev,x(0, y, z, t) = 1Ed,x(0, y, z, t) (S 0) (9)(6) (9) (7) x = 0 (10) (11) (12)
E0,t(0, y, z, t) + E2,t(0, y, z, t) = E1,t(0, y, z, t) (10)0
[E0,x(0, y, z, t) + E2,x(0, y, z, t)] = 1E1,x(0, y, z, t) (11)
B0,t(0, y, z, t) + B2,t(0, y, z, t) = B1,t(0, y, z, t) (12)
(ii)
0 E = (13)(13) (8)(9) V
0E1,x(0, y, z, t) 0 [E0,x(0, y, z, t) + E2,x(0, y, z, t)] = (y, z, t) (14)(11)
(y, z, t) = (0 1)E1,x(0, y, z, t) (15)
2. Maxwell x < 0
k E(k, ) = 0, k B(k, ) = 0 (16)k E(k, ) = B(k, ), k B(k, ) = 00E(k, ) (17)
(i) k0n0 E0 (17)
k0n0 (k0n0 E0) = 002E0 (18)(k0n0 E0)k0n0 (k0n0 k0n0)E0 = 002E0 (19)
(19) 1 (16) 0
k02 = 002 k0 = 00 (20)
(ii) (17) (20)
E0 =k0B000
n0 z = B0k0 z n0 (21)
3. E1 E2 (21)
E1 =B1k1
z n1, E2 = B2k0 z n2 (22)
(i) (12)
B0 exp [i(k0n0 r t)] + B2 exp [i(k0n2 r t)] = B1 exp [i(k1n1 r t)] (23)y, z, t
k0n0,y = k0n2,y = k1n1,y = k0 sin 0 = k1 sin 1 (24)
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20 2001 3
(ii)
(23) (24) B0 + B2 = B1 (25)(10) (22) (24) B0k0 cos 0
B2k0
cos 0 =B1k1
cos 1 (26)
(25) (26)
B1 =2k1 cos 0
k1 cos 0 + k0 cos 1B0, B2 =
k1 cos 0 k0 cos 1k1 cos 0 + k0 cos 1
B0 (27)
(iii) 0 + 1 = pi/2 sin 0 = cos 1, sin 1 = cos 0 (24) (27)B1B0
=2 cos 0 sin 0
cos 0 sin 0 + sin 1 cos 1= 1 ( cos 0 sin 0 = sin 1 cos 1) (28)
B2B0
=sin 0 cos 0 sin 1 cos 1sin 0 cos 0 + sin 1 cos 1
= 0 (29)
(21) (22)k1E1k0E0
=B1B0
= 1,E2E0
=B2B0
= 0 (30)
xy0 + 1 = pi/2 0 = B Brewster
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2001 4 21
4
1. O (z, ) xyz
z tan cos
z tan sin z
z tan cos z tan sin z tan sin + z tan cos
z
v2 = (z tan)2 +
(z tan
)2+ z2
L
L = 12
m[(
z2 + z2 2)
tan2 + z2] mgz (1)
=12
m
(z2
cos2 + z2 2 tan2
) mgz. (2)
2. L
L t
L
= mz2 tan2
z2 E
3.
z2 = 4h2 v02h tan =
2hv0tan
. (3)
E =12
m[(
z2 + z2 2)
tan2 + z2]
+ mgz (4)
=12
m
[4h2
(v0
2h tan
)2tan2
]+ mg2h (5)
=12
mv20 + 2mgh (6)
(3)
1cos2
z2 + (2hv0)2 1z2
+ 2gz = v20 + 4gh, (7)
dtdz = =
2g cos2
z2(z 2h)
z2 v202gz hv20g (8)
(7)
d2zdt2 =
(4h2v20
1z3 g
)cos2 (9)
v202h g
> 0< 0
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22 2001 4
(i) |v0| >
2hg
z =
[ (2hv0)2g
] 13
z = z(t) z =
v0
(v0 +
v20 + 16gh
)4g
dzdt = 0 z
(dzdt + )
z =
[ (2hv0)2g
] 13
z = 2h dzdt = 0 z
2h z v0
(v0 +
v20 + 16gh
)4g
(ii) |v0| =
2hg z = 2h
(iii) |v0|
