Skript / lecture notes - Universität Paderborn

Prof. Dr. Wolf Gero Schmidt
Universität Paderborn, Lehrstuhl für Theoretische Physik
the time-dependent solution to the hydrogen molecule ion, for simplification we assume
in the following vanishining overlap, i.e., S = 0
i ˙ψ = Ĥψ (1.115)
wobei where |ψ s (t)〉 = e − i E 1
S·t √2 (|φ A 〉 + |φ B 〉) (1.116)
|ψ a (t)〉 = e − i Ea·t 1 √2 (|φ A 〉 − |φ B 〉) (1.117)
damit lautet die allgemeine Lösung the general solution is given as
(1.118)
|ψ(t)〉 = c s |ψ s (t)〉 + c a |ψ a (t) (1.119)
= e − i Est 1
(
) Ea−Es
√2
−i t
c s + c a e |φ A 〉 (1.120)
⎛
⎞
+ e − i Est 1
Ea−Es
√2
−i t
⎝c s − c a e ⎠
}{{}
|φ B 〉 (1.121)
ω T
präparieren spezielle Anfangsbedingung prepare special initial values
d.h. e − am Kern A i.e., electron is at core A
|ψ(t = 0)〉 = |φ A 〉 (1.122)
Einsetzen in allg. Lösung liefert Insertion in general solution leads to
⎫
1
√ (c s + c a ) = 1⎪⎬
2
c
1
s = c a = √ 1 . (1.123)
√ (c s − c a ) = 0⎪⎭
2
2
daraus folgt it follows
|ψ(t)〉 = e − i Est 1 2
{(
1 + e
−iω T t ) |φ A 〉 + ( 1 − e −iω T t ) |φ B 〉 } . (1.124)
Wahrscheinlichkeit das Elektron am Kern A zu finden Probability to find the electron at
core A
|c A (t)| 2 = |e − i Est 1 (
1 + e
−iω T t ) | 2 (1.125)
2
ω
=
∣ e− i Est e −i ω T
2 t ei T t 2 + e −i ω T
2 t
2
(1.126)
2 ∣
= cos 2 ω T
2
t. (1.127)
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