? d?

1
? d?µ? t?? at?µ?? .
??. ?? s??????? at?µ??? p??t?p?.
?p? t?? ??µ????t? µ???? t? s??????? ?ßa?t???
?t?µ?.
?p?µ??e?a ???????? ??sµ?p?????2? G??
???????p????.
2
? ?????s? t?? ?ßa?t?µ??a?????. (1/2)

• ??? ? ?d?a p?? ??ete ??a t? ?t?µ? ap?d?deta? µe
  t?? e????a t?? ??e?t?????? p?? pe??st??f??ta?
  ???? ap? t?? p????a ??ete ??se? t? t?a??? t??
  ?ßa?t?µ??a????? pe??p?? 80 ?????a.

• ? ?ßa?t?µ??a???? ep?fe?e sa??t???? a??a??? t??
  ?de?? ??a t?? s?µpe??f??? t?? ???? se at?µ???
  ep?ped? ?a? a?a?????se t? ?e???t??? ??µe?a
  s?et??? µe t?? µ??f? t?? at?µ??, t?? t??p?
  a??pt???? desµ??, ta s??µata t?? µ?????, t???
  µ??a??sµ??? t?? a?t?d??se?? ?t?.

3
? ?????s? t?? ?ßa?t?µ??a?????. (2/2)

• ? ?????s? t?? ?ßa?t?µ??a????? ßas?st??e st??
  ?d?e? ?aµp??? e?e???t?? p?? ? s?µa?t???te?e?
  e??a?

• ? ?ß??t?s? t?? e????e?a?. (Max Planck 1900)
• ? ??µat?s?µat?d?a?? ?e???a. (Louis De Broglie
  1924)
• ? a??? t?? aßeßa??t?ta?. (Werner Heisenberg 1925)
• ? e??s?s? t?? Schrodinger. (Årwin Schrodinger
  1926)

4
? ??µat?s?µat?d?a?? ?e???a, (d?ad???t?ta t??
f?t?? ?a? t?? ????)
?p?? t? ??e?t??µa???t??? ??µa ??e? ?a?
s?µat?d?a?? f?s? (f?t????) , ?ts? ?a? ???e
?????µe?? s?µat?d?? µp??e? ?a ??e? ?a? ??µat???
?p?stas?.
Louis De Broglie 1924

• ?? µ???? ??µat?? t?? ?????µ???? s?µat?d??? e??a?

?µ???? ??µat??, p??µ?, mµ??a, ?ta??t?ta
5
? pe??aµat??? ap?de??? t?? ??µat?s?µat?d?a???
?e???a?.
?? 1927 d?ap?st????e ap? t??? Davisson ?a? Germer
?a? pe??aµat??? ?t? ta ?????µe?a ??e?t????a
s?µpe??f?????ta? ?a? ?? ??µata. ??d???te?a,
d?ap?st????e ?t? ?????µe?? d?sµ? ??e?t?????? ??e?
t?? ?d??t?ta ?a pe?????ta? (?a?a?t???st???
?d??t?ta t?? ??µ?t??) ?ta? pe??? ap? ???sta?????
p???µata, ?d??t?ta p?? ß???e efa?µ??? st??
?e?t?????a t?? ??e?t??????? µ????s??p???.
6
?p?????sµ?? t?? µ????? ??µat?? de
Broglie.(pa??de??µa ap? t? µ??????sµ?)
1? pa??de??µa ??a ??e?t????? p?? ??e?
ep?ta????e? ap? t?s?
V100volts ap??t? ta??t?ta
T? a?t?st???? µ???? ??µat?? De Broglie e??a?
7
?p?????sµ?? t?? µ????? ??µat?? de
Broglie.(pa??de??µa ap? t?? µa?????sµ?)
2? pa??de??µa ??a ?????µe?? µp??a
µe µ??a m0,15kg ?a?
ta??t?ta ?40m/sec
??e? µ???? ??µat?? De Broglie
?? µ???? ??µat?? a?t? ?p??e?peta? p??? a??µa ?a?
ap? t? d??µet?? t?? p????a p?? e??a? t?? t????
10-14m . Ge????te?a, st?? µa?????sµ? de? ??e?
???µa ? ??µat??? f?s? t?? ?????µ???? s?µ?t??
??at? e??a? ad??at?? ?a a????e??e? ? ??µat????
t??? ?a?a?t??a?.
8
?a at?µ??? ??e?t????a ?a? ? ??µat??? t???
s?µpe??f???.
? ? ????s? t?? ??e?t?????? st? ?t?µ? e??a?
pe?????sµ??? ???? ap? t?? p????a t?? at?µ??,
?ts? ?ste ?? ??µa ?a s??a?t? t?? ea?t? t?? ?a?
?a d?µ?????e?ta? st?s?µ? ??µa.
? ? d?µ??????a st?s?µ?? ??µat??, de?
ap?de????eta? ?e???t??? a??? e??a? µ?a eµpe?????
pa?ad??? p?? ???s?µ?p??e? ? ?ßa?t?µ??a???? ??at?
s?µf??e? µe pe??aµat???? µet??se?? sta
?a?a?t???st??? t?? ??e?t??????.
9
? a??? aßeßa??t?ta? ? ap??sd????st?a?. (?
?a?d?? t?? s???????? ?ßa?t?µ??a?????)
???a? ad??at?? ? ta?t??????? ?a????sµ?? t?? ??s??
?a? t?? ??µ?? t?? ??e?t??????.
?xsf??µa ?a????sµ?? ??s??. ?px sf??µa
?a????sµ?? ??µ??.
Werner Heisenberg 1925
? ??a ???ssa pe????af?? t?? at?µ?? ?a e??a?
???ssa p??a??t?t??.
10
? ??µat??? e??s?s? t?? Schrodinger .

• ? ??µat??? e??s?s? t?? Schrodinger e??a? ?
  ?eµe???d?? ??µ?? p?? eµpe????e? t?s? t??
  ??µat?s?µat?d?a?? ?e???a ?s? ?a? t?? a???
  ap??sd????st?a? ?a? p??sa?µ??eta? sta d??f??a
  s?st?µata t?? µ??????sµ??.

Årwin Schrodinger 1926

• ?t? e??a? ??a t?? µa?????sµ? ?? ??µ?? t?? ?e?t??a
  , e??a? ??a t?? µ??????sµ? ? ??µat??? e??s?s?
  Schrodinger .

• ? ??µat??? e??s?s? st?? ?ßa?t?µ??a???? e??a? ??a?
  eµpe?????? ßas???? ??µ?? p?? epa???e?eta?
  pe??aµat???.

• ?p? t?? ep???s? t??, p????pt??? ??
  ??µat?s??a?t?se?? p?? d????? t?? ??µat????
  s?µpe??f???? t?? s?µat?d??? t?? µ??????sµ??.

11
?a t????a??. (1/3)
? ? efa?µ??? t?? ??µat???? e??s?s?? Schrodinger
sta ?t?µa, ???eta? af?? ep??????µe t? st?s?µ?
??µa sa? p??t?p? t?? ??e?t??????? ?at?stas??.
? ? ep???s? t?? e??s?s?? Schrodinger e??a?
d??at?? ?a ???e? µ??? ??a t? ?t?µ? t?? ?d???????,
?? de ??se?? t?? e??s?s?? ??a t? ?t?µ? t??
?d??????? ???µ????ta? at?µ??? t????a?? (atomic
orbital AO) ?a? s?µß??????ta? µe t? e???????
???µµa ? .
? ?a ?? µp????? ?a ???s?µ?p??????? µe µe????
p??s????s? ?a? sta p?????e?t?????? ?t?µa.
12
?a t????a??. (2/3)
? ?a ?? e??a? st?? ??s?a s??a?t?se?? E?(x, y, z)
p?? s?s?et????? t?? e????e?a ? t?? ??e?t?????? µe
t?? s??teta?µ??e? x, y, z t?? ??se?? p?? µp??e?
?a ß?e?e? t? ??e?t????? t?? ?d???????.
? Se ???e ?? ?1 , ?2 , ?3 a?t?st???e? µ?a
µ??? e????e?a ?1 , ?2 , ?3 d??ad? ? e????e?a
t?? t????a??? e??a? ?ßa?t?sµ???.
? ?ta? d?? ? pe??ss?te?a at?µ??? t????a?? ?????
t?? ?d?a e????e?a t?te ???µ????ta? e?f???sµ??a
t????a??.
? ?a at?µ??? t????a?? ?p?????? d???t???.
13
?a t????a??. (3/3)
? ?????eta? ?t? ta at?µ??? t????a?? (?) de? ?????
f?s??? s?µas?a, e??a? µa??µat???? s??a?t?se??,
??se?? t?? e??s?s?? Schrodinger ??a t? ?t?µ? t??
?d??????? ?a? ????? ??a t?? d??f??e?
s??teta?µ??e? x, y, z t?? ??se?? t?? ??e?t??????
?et???? ? a???t???? t?µ??.
? ?? tet?????? t?? t????a??? (?2) ??e? f?s???
s?µas?a ?a? d??e? t?? p??a??t?ta p?? ??e? t?
??e?t????? ?a ß?e?e? se ??p??e? ??se?? ???? ap?
t?? p????a.
??? fa???µa? ?at' as?????st? t??p? saf?? se sa? ,
p??pe? ?a ??ete pa?a???se? t? e?pa."
(Alan Greenspan)
14
? a?apa??stas? t?? ??e?t??????? ?ef??µe
p????t?ta st??µ?t??
?? ??e?t?????? ??f?? ap?d?deta? µe t?? p????a
µ?sa se ??a ??f?? st??µ?t?? ?p??, ta st??µata
e??a? p???? st?? ??se?? p?? ??e? µe????
p??a??t?ta (µe???? t?µ? ?2) ?a ß?e?e? t?
??e?t?????.
µ???? p????t?ta
µe???? p????t?ta
1s ??e?t?????? ??f??
p????a?
??d???te?a t? -e?2 e?f???e? t?? ??e?t??????
p????t?ta t?? ?a???µe??? ??e?t??????? ??f???.
15
? a?apa??stas? t?? ??e?t??????? ?ef?? µe
p????t?ta ???µat??
To ??e?t?????? ??f?? ap?d?deta? ?a? µe p????t?ta
???µat?? p?? ?s?d??aµe? µe t?? ??e?t??????
p????t?ta
1s ??e?t?????? ??f??
16
? a?apa??stas? t?? ??e?t??????? ?ef?? µe
???a??? ?aµp??e?
?? ???a??? ?aµp??e? ap?d?d??? t? s??µa ?a? t?
s?et??? µ??e??? t?? ??e?t??????? ??f???.
90-99
1s ??e?t?????? ??f??
?????eta? ?t? ta ??e?t?????? ??f? de? ????? ???a
??at? ? p??a??t?ta ?a ß?e?e? t? ??e?t?????
µ?de???eta? se ?pe??? ap?stas? ap? t?? p????a.
?ts? ?? ???a??? ?aµp??e? ap?d?d??? t? ???? µe t?
µ???st? ??e?t?????? p????t?ta 90-99 .
17
? a?apa??stas? t?? ??e?t??????? ?ef??µe
??af???? pa?ast?se??.
?2
1s ??e?t?????? ??f??
?2f(r)
? ??af??? pa??stas? ep?de????e? p?? µetaß???eta?
? p??a??t?ta ?2 µe t?? ap?stas? r ap? t?? p????a.
r
18
?? t????a?? de? e??a? ??e?t?????? ??f??.
19
O? ?ßa?t???? a???µ?? ?a? ta t????a??.
ap? ??????? ??sµ?p????
20
?? ?ßa?t???? a???µ??.

• Se ???e AO a?t?st???e? µ?a t???da ?ßa?t????
  a???µ?? n, l, ml ?a? a?t?st??fa.

• ?? ?ßa?t???? a???µ?? de? e?s????ta? a??a??eta
• (?p?? ????e st? p??t?p? t?? Bohr), a???
  p????pt??? sa? apa?t?s? ???e pa?ade?t?? ??s??
  (t????a?o?), t?? e??s?s?? t?? Schrodinger ??a t?
  ?t?µ? t?? ?d???????.

• ?? ?ßa?t???? a???µ?? s?et????ta? µe s?µa?t???
  ?a?a?t???st??? t?? ??e?t?????? .

• ?? ?ßa?t???? a???µ?? µp????? ?a ???s?µ?p???????
  ?a? se ???a p?????e?t?????? ?t?µa ? ???ta.

21
1?? ? ?????? ?ßa?t???? a???µ?? (n).
?a???e? a???a?e? t?µ?? 1, 2, 3, . . . , n .
µe?a???e? ? e????e?a t?? t????a???.
?s? µe?a???e? ? ?????? ?ßa?t???? a???µ??
t?s?
µe?a???e? t? µ??e??? t?? t????a???.
µ???a??e? ? ???? ??e?t??????? ??f??? ?a? p????a.
22
2?? ? a??µ????a??? ?ßa?t???? a???µ?? ( l ).
?a???e? a???a?e? t?µ?? 0, 1, 2, . . . , n-1 .
? 2?? ?ßa?t???? a???µ?? (l) s?et??eta? µe t??
d???µe?? µeta?? t?? ??e?t??????? ?ef?? ?a? ??
a?t? ?a?????e? t?? µ??f? t?? ??e?t??????? ?ef??.
S?et??eta? µe t?? e????e?a t?? t????a??? µ???
sta p?????e?t?????? ?t?µa ? ???ta. ?s?
µe?a??te??? e??a? ? ?ßa?t???? a???µ?? (l) t?s?
µe?a???e? ? e????e?a t?? t????a???
23
S?µß???sµ?? t????a???.
G?a t?? d??f??e? t?µ?? t?? ?ßa?t???? a???µ?? (l)
s?µß??????µe ta t????a?? µe ???µµata ?? e???
t?µ? 2?? ?ßa?t???? (l) 0 1 2 3 4
s?µß???sµ?? t????a??? s p d f g
?? µp??st? ap? ta ???µµata s, p, d, ?p???e?
a???µ??, t?te a?t?? ?p?d????e? t?? 1? ?ßa?t???
a???µ? (n) t?? t????a??? . p. ?. µe t??
s?µß???sµ? 2s e?????µe t????a?? µe n2 ?a? l0,
µe t?? s?µß???sµ? 3d e?????µe t????a?? µe
n3 ?a? l2 ?.?.p.
24
3?? ? µa???t???? ?ßa?t???? a???µ?? ( ml ).
?a???e? a???a?e? t?µ?? l 0 l .
S?et??eta? µe t? µa???t??? ped?? ???? t??
pe??f???? t?? ??e?t??????.
?a?????e? t?? p??sa?at???sµ? t?? t????a???.
25
St?ß?de? ?a? ?p?st?ß?de?.
St?ß?da e??a? t? s????? t?? t????a??? p?? ?????
t?? ?d?? ????? ?ßa?t??? a???µ? (n) .
?p?st?ß?da e??a? t? s????? t?? t????a??? p??
????? t??? ?d???? ??????? ?ßa?t????? a???µ??? (n)
?a? (l).
26
4?? ? µa???t???? ?ßa?t???? a???µ?? ( ms ).
?e? ?a?a?t????e? t? t????a?? a??? t? ??e?t?????.
?a???e? t?µ?? ½ ? -½
S?et??eta? µe t? µa???t??? ped?? t?? ??e?t??????
???? t?? ?d??pe??st??f?? t??.
N
S
S
e
e
N
d?sµ? at?µ?? ?
N
S
27
?a s t????a??.
3s
2s
1s
?a s t????a?? ????? ??a sfa????? s?µµet??a.
G?af???? pa?ast?se?? t?? s????t?s?? ? ?a? t??
p??a??t?ta? ??s?? ?2 t?? ??e?t?????? ??a t? 1s.
G?af??? pa??stas? t?? p??a??t?ta? ?a ß?e?e? t?
??e?t????? 1s se ep?f??e?a sfa??a? a?t??a? r.
A?t??a t?????? Bohr.
28
?a p t????a??.
?a p t????a?? ????? ??a s??µa d?? ??ß?? . ? ??a?
??ß?? a?t?st???e? st?? ?et???? t?µ?? t??
??µat?s????t?s?? ? e?? ? ????? st?? a???t????.
2px
2py
2pz
?a p ??e?t?????? ??f? ????? t? ?d?? s??µa µe ta
t????a?? a??? e??a? pe??ss?te?? e?tetaµ??a ?at?
t?? d?e????s? t?? ????? t???.
29
?a p t????a?? - ??af???? pa?ast?se?? - ??µß???
ep?ped?.
?2
?2
?
?
x
x
Sta p t????a?? ?p???e?, µeta?? t?? ??ß??, ??a
ep?ped? µe µ?de???? ??e?t?????? p????t?ta p??
???µ??eta? ??µß??? ep?ped?. To ??µß??? ep?ped?
pa??e? µe???? ???? st? ??µ??? d?ast???t?ta ?a?
?d?a?te?a st? s??µat?sµ? ?a? t?? ?s?? t??
?µ???p?????? desµ??.
??µß??? ep?ped?
30
?a d t????a??.
?a d t????a?? de? ????? ??a t?? ?d?a µ??f?.
z
y
x
?a d t????a?? ????? p????? ??µß???? ep?f??e?e?.
31
?a f t????a??.
?a f t????a?? - ?p?? ta d t????a?? de? ????? ??a
t?? ?d?a µ??f?.
z
y
x
?a f t????a?? ????? ?a? a?t? p????? ??µß????
ep?f??e?e?.
32
??µß???? ep?f??e?e? sta 2s, 3s, t????a??.
3s
2s
1s
G?af???? pa?ast?se?? t?? ??µat?s????t?s?? ?
??a ta 1s, 2s, 3s t????a??.
??µß???? ep?f??e?e?
?? ??µß???? ep?f??e?e? e?????? ta s??µata t??
?ß??d???? t????a???.
33
??µß???? ep?f??e?e? sta 2s, 3s, t????a??.
1s
2s
3s
?
?
?
??µß???? ep?f??e?e?
G?af???? pa?ast?se?? t?? s????t?s?? ? ??a ta
1s, 2s, 3s t????a??.
4pr2?2
4pr2?2
4pr2?2
G?af???? pa?ast?se?? t?? s????t?s?? 4pr2?2 ??a
ta 1s, 2s, 3s t????a??.