Задачи по квантовой механике. Часть 2. Корнев А.С. - 56 стр.

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c
1s
(p) =
2i}
πp
µ
Z
2}a
0
3
/
2
(
·
Z
a
0
+
i
}
p
¸
−2
−
·
Z
a
0
−
i
}
p
¸
−2
)
=
= −
4}
πp
µ
Z
2}a
0
3
/
2
Im
µ
Z
a
0
+
i
}
p
−2
=
8
π
³
a
0
2}Z
´
3
/
2
·
1 +
³
a
0
p
}Z
´
2
¸
−2
.
w
1s
(p) = |c
1s
(p)|
2
=
8
π
2
³
a
0
}Z
´
3
·
1 +
³
a
0
p
}Z
´
2
¸
−4
.
R
w
1s
(p)d
3
p = 1
¤
Φ(ϕ) =
1
√
19π
(3 + cos ϕ)
I |c(E
m
)|
2
Φ(ϕ)
Ψ
m
(ϕ) =
e
imϕ
√
2π
m = 0, ±1, . . . E
m
=
}
2
m
2
2I
Φ(ϕ) = 3
r
2
19
Ψ
0
(ϕ) +
1
2
r
2
19
{Ψ
1
(ϕ) + Ψ
−1
(ϕ)},
c(E
0
) = 3
r
2
19
c(E
±1
) =
1
2
r
2
19
|m| > 1
c(E
m
) = 0 E
|c(E
m
)|
2
w
m
Ψ
m
(ϕ) Ψ(ϕ)
E
0
= 0 E
±1
w
0
= |c(E
0
)|
2
=
18
19
;
w
±1
= 2|c(E
±1
)|
2
=
1
19
P
m
w
m
= 1
¤
                                                             9?
                            µ     ¶ 3/2 (·                     ¸−2    ·                  ¸−2 )
                      2i}   Z            Z     i         Z    i
   c1s (p) =                                + p      −      − p         =
                      πp  2}a0           a0    }         a0  }
                       µ     ¶ 3/2    µ          ¶−2                ·    ³ a p ´2 ¸−2
              4}         Z              Z    i         8 ³ a0 ´ 3/2         0
       =−                          Im      + p       =                1+              .
              πp        2}a0            a0   }         π 2}Z               }Z
U  !#;#- P; ;*-.%! _9<;O-%!#.  2  !# ?  ! 2 EF6T 
 *
                                                       ·    ³ a p ´2 ¸−4
                                             8 ³ a0 ´3
                                                  2            0
                       w1s (p) = |c1s (p)| = 2           1+              .
                                            π }Z              }Z
M(#-+#*>!%!# 2 -. N#.%! 2  EF;- P !8- 2 "
R
   w (p)d3 p = 1
                          aQH  2  QU6T () * P;/-.%! O7                 ¤
f+(*,+ V G V G                                                                             
     1s

                           I4 < E F G 4I<[Pj5 B€5fF2>V\ < E < ZDGWK BDF25 Pj4`5 G 5 F G P E BDB G DB€5 T 4?V\ G \
  4 G 6lP R9EDT Z g GMG R_K B E  i Φ(ϕ) = √ 1 (3 + cos ϕ) n  > 4$d E > R&H >fF^B=:=> 5 g G \ R > K 5fF2 E B=B GX <[kWB€5f8 Q619π  G 5fg GWHJG R€G F2 E F G \ B E > R€c BDB I B G P_5 E B=F ]j57\ T 5V6 4I>V69BD:QB E |c(E )|2 n
x(y{z|y{}~€yI U )-/L &(%"X    \c       Φ(ϕ)
                                                            ;N%) !#E $&(%"X 2 ';O-%!#"%+ 3
                                                                                            m


6 Ψ (ϕ) = e√ 9 m = 0, ±1, . . . B!#; " !#" EFQBA
                                imϕ
                                                                                                            }2 m 2 ?       7
 M(  2O2 &(*-O2πl QO- ^ ;-%6* A
               m                                                                                     Em   =
                                                                                                             2I

                                      r                         r
                                          2                  1 2
                       Φ(ϕ) = 3               Ψ0 (ϕ) +                  {Ψ1 (ϕ) + Ψ−1 (ϕ)},
                                          19                 2 19
 /7 7 c(E ) = 3 2 ^ c(E ) = 1 2 7 I  !#66-.E !8- 2 9 |m| > 1?
                          r                               r
                                                                                     8               8
            0
               7  a  
                       %
                         + 
                             19  8
                                  ! 
                                    - 
                                       %
                                           ^ -2 19 2 &(%"X 2  E  ;#S!#66- \) N
                                          ±1

6  ! 2 N*!8" E  !#-NX  7 |c(E )|2  >  2 %!#. w N%
c(Em ) = 0
T  2 !#%!8 2  2 Ψ (ϕ) a!# !# 2  Ψ(ϕ)                                           6
                                                                          7 I !#%!# ^   Tm Q`)*/ *
                                                                                   )(8
                                                                    m

;a"O6 % )* a] + aN ; -.%! 2
                                 m
                                                                                         ) Q
 ]  +  A E = 0 9<%!8T!8%!# 2  ? - E 9<;  ) N \!#
  !# 2  ? 7RI 0  2  !#aQ8 N%T  2 ±1


                                              18
                       w0 = |c(E0 )|2 =          ;
                                              19
                                                      1 9 %"PEF/
                                                                                  ? 7
                       w±1 = 2|c(E±1 )|2 =
                                                      19
C,">4!8-#%6-3/O . ^ EF;- 2  ! 2  !8-("                P
                                                                                                             wm = 1
                                                                                                                      7
                                                                                                         m
                                                                                                                      ¤

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