Квантовые вычисления. Ожигов Ю.С. - 76 стр.

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МГУ | Москва

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ψ
j
s
s = 1, 2, . . ., n
ψ
k
k ∈ {1, 2, . . . , J} j
s
ψ
ψ
k
|¯n
Ψ
i = |n
1
, n
2
, . . . , n
J
i n
k
k
¯n
P
¯n
λ
¯n
|¯ni λ
1/2
Ψ =
1
p
(2n)!
ψ
j
1
(r
1
)α
1
ψ
j
1
(r
2
)α
1
. . . ψ
j
1
(r
2n
)α
1
ψ
j
1
(r
1
)β
1
ψ
j
1
(r
2
)β
1
. . . ψ
j
1
(r
2n
)β
1
ψ
j
n
(r
1
)β
n
ψ
j
n
(r
2
)β
n
. . . ψ
j
n
(r
n
)β
n
,
α
j
β
j
j
1/2 j −1/2 j
k
Ψ
R
hΨ| H |Ψid¯r
α
∗
j
α
j
= β
∗
j
β
j
= 1, [α
j
α
k
] = [β
j
β
k
] = [α
j
, β
k
] = 0
j 6= k α
j
α
j
= β
j
β
j
= 0 j
a
j
j a
+
j
a
j
|n
1
, . . . , n
J
i = δ
1,n
j
(−1)
σ
j
|n
1
, . . . , n
j−1
, n
j
− 1, n
j+1
, . . . , n
J
i σ
j
=
n
1
+ . . . + n
j
a
+
j
a
k
+ a
k
a
+
j
= δ
j,k
a
j
a
k
+ a
k
a
j
= a
+
j
a
+
k
+ a
+
k
a
+
j
= 0
H = H
one
+ H
two
V
1
(r)
V
2
(r, r
0
)
H
one
=
P
k,l
H
k,l
a
+
k
a
l
H
two
=
P
k,l,m,n
H
k,l,m,n
a
+
l
a
+
k
a
m
a
n
H
k,l
= hψ
k
| H
one
|ψ
l
i =
R
ψ
∗
k
(r)V
1
(r)ψ
l
(r)dr,
H
k,l,m,n
= hψ
l
, ψ
k
|H
two
| ψ
m
ψ
n
i =
R
ψ
∗
k
(r)ψ
∗
l
(r
0
)V
2
(r, r
0
)ψ
m
(r)ψ
n
(r
0
)drdr
0
.
ψ
i
                                                                                  
                  &        '
    &   $    ψ                           
                                                                       s = 1, 2, . . . , n    &
  ) $      ψ
                                 k
                                              
                                      k ∈ {1, 2, . . . , J}  
                                                                  js
                                                                            j     ψ 
                                                                                   s
      $    !         #
   #      #   #                  ψ       $
  $      $ $  &  
                                                                   k
                                                                                                  '      
         |n̄ i = |n , n , . . . , n i  n    &  k $      $
                       % 
                                Ψ      1   2           J        k

         #                       n̄       
    %   ) $         P λ |n̄i    λ 
                                                                                     n̄
                 $   n̄    '  

                                        
  $  #                    
              '  # %                                  
                                                                                           '  
    1/2                         

                                              ψj1 (r1 )α1    ψj1 (r2 )α1     . . . ψj1 (r2n )α1
                                   1          ψj1 (r1 )β1    ψj1 (r2 )β1     . . . ψj1 (r2n )β1
                         Ψ= p                                                                 ,                  
                             (2n)!                                                     
                                          ψjn (r1 )βn ψjn (r2 )βn . . . ψjn (rn )βn
 α          βj                             j  &    


       j
1/2 j          &        −1/2           jk  '          '   #  
           %          #   
  $ %&  '                                               
 #       Ψ %&     '  $  R hΨ| H |Ψidr̄ 
   '        α∗ α = β ∗ β = 1, [α α ] = [β β ] = [α , β ] = 0 

                                                                             
                                                           j j      j j          j k         j k           j   k
j 6= k  αj αj = βj βj = 0  # j   '     &   
                               #                           
          '         aj    &             j                  '  $         a+
                                                                                                                        j
       a |n , . . . , n i = δ                 σj                                               σ =
                                   j 1         J      1,nj (−1) |n1 , . . . , nj−1 , nj − 1, nj+1 , . . . , nJ i     j
n1 + . . . + nj      &  )   a+                                               +
                                                                                                j ak + ak aj = δj,k
                                                                                                                          
                    + +    + +
aj ak + ak aj = aj ak + ak aj = 0  
     '       $                            #  & 
     $                '   '      #
  #   $ $  H = Hone + Htwo       &  V1 (r)
             ('$   #'         ' 
V2 (r, r0 ) P                                                                                  '                 
                    +             P               + +         
Hone = Hk,l ak al Htwo =                Hk,l,m,n al ak am an
         k,l                        k,l,m,n
                                       R
             Hk,l = hψk | Hone |ψl i = ψk∗ (r)V1 (r)ψ
                                                   R l (r)dr,
             Hk,l,m,n = hψl , ψk |Htwo | ψm ψn i = ψk∗ (r)ψl∗ (r0 )V2 (r, r0 )ψm (r)ψn (r0 )drdr0 .
     &   #   $ $                        ψ    ' 
                                                                                              i
   &  $ #                   #   '                   '     
       

' %!$
             " %
                               $$ %"
                                                 
                                                                 %&
                                                                          
                                                                             %%
                                                                                             "%    " 
                                                                                                  ")
    

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