estquantica8_1s2011
TRANSCRIPT
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N N
= (q1,...qN)
qi
N
xp h
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3He
4He
(q1,...,qi,...,qj,...,qN) = (q1,...,qj ,....,qi,...,qN)
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nj = 0 1
nj = 0, 1, 2,...
1, 2
A, B
1 2
1(rA)1(rB)
2(rA)2(rB)
12[1(rA)2(rB) + 1(rB)2(rA)]
A B
1 2
12[1(rA)2(rB) 1(rB)2(rA)]
A B
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1 2
1 2
A, B
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1 2 3 4 1 2 3 4 1 2 3 4
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m L 0 < x < L
n(x)
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Hn(x) =nn(x)
H= p2
2m
n(x) =Ceikx
n=
2k2
2m
k
V(x) = 0, 0< x < L
= , x
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n(0) = n(L) = 0
n(x) =
2L
sin
nxL
, n= 1, 2,...
n =
22n2
8mL2
(x) = (x+ L)
exp(ikL) = 1
k = 2L
n, n= 0, 1, 2,...
L
ST P
k 2/L
L
k
f(k)
k
2/Lf(k) =
L
2
kf(k)
(r) =
8
V exp
ik r
k =
2
L1n1x1+
2
L2n2x2+
2
L3n3x3
ni = 0, 1, 2,...
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k =
2k2
2m
k
f(k) V(2)3
3kf(k)
{n1, n2,...,nj,...} {nj}
nj
N N
{nj
}
E{nj} =j
jnj
N=N{nj} =j
nj
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Z(T , V , N ) =
{nj},(P
jnj=N)
exp
j
jnj
{nj} N
Z(T , V , ) =
N=0
exp {N} Z(T , V , N )
=
N=0
exp {N}
{nj},(P
jnj=N)
exp {1n1 2n2 ...}
=
N=0
{nj},(
Pjnj=N)
exp{
(1
)n1
(2
)n2
...}
n1, n2,...
N=n1+ n2+ ... N.
Z=
n1,n2,...
exp {(1 )n1 (2 )n2 ...}
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Z =
N=0
{nj},(Pnj=N)...
=
N=0
n1,n2;n1+n2=N
...
={nj},0
... +{nj},1
... +{nj},2
... +{nj},3
... + ...
=
0,0
...
+
0,1
... +1,0
...
+
0,2
... +1,1
... +0,2
...
+
0,3
... +1,2
... +2,1
... +
= 0,0
... +0,1
... +0,2
... + ...+1,0
... +1,1
... +1,2
... + ...+2,0
... +2,1
... + ..
=
0,n2=0
...
+
1,n2=0
...
+
2,n2=0
...
+ ...
=
n1=0
n2=0
...
=n1,n2
...
Z=
n1
exp {(1 )n1}
n2
exp {(2 )n2}
...
Z= Z(T , V , ) =
jn
exp {(j )n}
nj
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nj = 1
jln Z
n 0
Z(T , V , ) =j
n=0
exp {(j )n}
n=0
exp {(j )n} = [1 exp {(j )}]1
exp {(j )} < 1 j j 0 exp {} < 1 < 0
0
Z(T , V , ) =j
[1 exp {(j )}]1
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lnZ(T , V , ) = j
ln[1 exp {(j )}]
nj = 1exp {(j )} 1
exp {(j )}
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lnZFD,BE(T , V , ) = j
ln[1 exp {(j )}]
njFD,BE= 1
exp {(j )} 1
F D BE
nj (j )
nj 1 j < nj 0 j > T 0 nj
nj 0 nj 1 j T 0 nj
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(j )
j T
j T
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nj 1 j
nj 1
exp {(j )} 1
j
z= exp {} 1
exp {}
ln ZFD,BE=j
exp {(j )} 1
2j
exp {2(j )} + ...
njFD,BE= exp {(j )} [1 exp {(j )} + ...]
ln Zsc=j
exp {(j }
njsc= exp {(j )}
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j k, = 2k2
2m
ln Zsc=k,
exp
2k2
2m
V
ln Zsc = V
(2)3
3k exp
2k2
2m
= V
(2)3exp ()
2kTm
2
3/2
= V exp() 2kTmh2
3/2
sc= V 2mkT
h2 3/2
(kT)5/2 exp
kT
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MBN = 1N!
distN
N
ZMBN = 1
N!
ZdistN
1 2 3
ZMBN =ZN1
N!N=2 ZMB2 =
Z212!
ZMB2 = 1
2!
e1 + e2 + e3
2=
1
2e21 +
1
2e22 +
1
2e23
+e(1+2) + e(2+3) + e(1+3)
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ZBE2 =e21 + e22 + e23 + e(1+2) + e(2+3) + e(1+3)
ZFD2 =e(1+2) + e(2+3) + e(1+3)
nk
x= (k)
x x
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nk
x= (k )
lnZsc= V z
2kTm
h2
3/2
j
nj
=z
zln Zsc= V z
2kTm
h2
3/2 N
N
exp() =N
V
1
h2
2kTm3/2
N
V
1
h2
2kTm
3/2 1
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N
V
1/3=a
T = h2mkT
(3/2)kT v =
3kT/m
=3kT
2h
T = v = 2h3kT m
a T
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lnZsc= zj
exp(j)
N
N=z
zln Zsc( , z , V ) =zj
exp(j)
njsc= zexp(j) = Nexp(j)
jexp (j)
njscN
= exp(j)
jexp (j)
njscN
p0(v) v=V
(2)3exp2k2
2m
3k
V(2)3
exp
2k2
2m
3k
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p= mv = k
p0(v) = 42kTm
3/2
v2 expmv2
2kT
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