Master Dissertation

Note that
〈Woutφ, ψ〉 = 〈 lim
t→∞ U(0, t)e −iH0t φ, ψ〉
= lim
t→∞ 〈U(0, t)e −iH0t φ, ψ〉
= lim
t→∞ 〈φ, e iH0t U(0, t) ∗ ψ〉
= 〈φ, lim
t→∞ e iH0t U(t, 0)ψ〉.
Hence W ∗ out = s- limt→∞ e iH0t U(t, 0). Further
W ∗ outWinψ = lim
t→∞ lim
s→−∞ eiH0t U(t, 0)U(0, s)e −iH0s ψ
This leads us to the following definition.
= lim
t→∞ lim
s→−∞ eiH0t U(t, s)e −iH0s ψ.
Definition 3.3. The scattering matrix is defined as follows
S = W ∗ outWin = s- lim
t→∞ s- lim
s→−∞ eiH0t U(t, s)e −iH0s .
The physical meaning of the scattering matrix is now apparent. A
normalized initial asymptotic state ψ considered at time t = 0, say, is first
transformed to s = −∞ by free dynamics, then it is evolved from −∞ to
t = ∞ by full interacting dynamics and finally it is transformed back from
∞ to t = 0 again by free dynamics. Thus Sψ is in fact the outgoing
scattering state transformed to t = 0 by free dynamics. The probability for
a transition form ψ to φ is given by
P (ψ → φ) = |〈φ, Sψ〉| 2 .
Now ψ(t) as given by equation (3.1) is the solution to the Schrödinger
equation
i d
dt ψ(t) = (H0 + V (t))ψ(t).
If we go over to the interaction picture 2 by substituting φ = e iH0t ψ, we get
i d d
φ(t) = i
dt dt (eiH0tψ(t)) = −H0e iH0t iH0t
ψ(t) + e (H0 + V (t))ψ(t)
= e iH0t −iH0t
V e φ(t)
= ˜ V (t)φ(t).
FIXME: Make sure you can differentiate like this.
2 See [8] page 318.
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