Trotter theorem 669∣
∣
∣
∣
∣
∣
[(
PˆhQˆh)n
−Rˆnh]
ψ∣
∣
∣
∣
∣
∣=
∣
∣
∣
∣
∣
∣
(
PˆhQˆh−Rˆh)
Rˆ(n−1)hψ+PˆhQˆh(
PˆhQˆh−Rˆh)
Rˆ(n−2)hψ+···+
(
PˆhQˆh)n− 1 (
PˆhQˆh−Rˆh)
ψ∣
∣
∣
∣
∣
∣
∣
∣. (C.15)
Recall that for ordinary vectorsa,b, andc, such thata=b+c, the triangle inequality
|a|≤|b|+|c|holds. Similarly, we have
∣
∣
∣∣
∣
∣
[(
PˆhQˆh)n
−Rˆnh]
ψ∣
∣
∣
∣
∣
∣≤
∣
∣
∣
∣
∣
∣
(
PˆhQˆh−Rˆh)
Rˆ(n−1)hψ∣
∣
∣
∣
∣
∣
+
∣
∣
∣
∣
∣
∣PˆhQˆh(
PˆhQˆh−Rˆh)
Rˆ(n−2)hψ∣
∣
∣
∣
∣
∣+···. (C.16)
On the right, there arenterms, all of orderO(h). Thus, the right side varies as
nO(h) =nO(t/n). Asn→∞,nO(t/n)→0. Hence,
lim
n→∞∣
∣
∣
∣
∣
∣
[(
PˆhQˆh)n
−Rˆnh]
ψ∣
∣
∣
∣
∣
∣→ 0 , (C.17)
which implies eqn. (C.1).