The Exercise

Let a Hamiltonian be a sum of terms, $H = \sum_k H_k$ , every two of which commute, $H_j H_k = H_k H_j$ . The exercise is to show that the evolution under $H$ for a time $t$ is then the product of the evolutions under the separate terms for the same time: for every $t$ , the operator $e^{-iHt}$ is the product of the operators $e^{-iH_k t}$ .^✦

The Claim

The library states the identity for elements of any normed algebra over the complex numbers, with the terms indexed by a finite set and the exponential of an element given by its power series.

Theorem 1. Let $\mathcal{A}$ be a normed algebra over $\mathbb{C}$ , let $S$ be a finite set, and let $H_k \in \mathcal{A}$ for each $k \in S$ , every two of them commuting: $H_j H_k = H_k H_j$ for all distinct $j, k \in S$ . Let $H = \sum_{k \in S} H_k$ . Then for every real $t$ $e^{-iHt} = \prod_{k \in S} e^{-iH_k t}.$ (1)^✦

With $\mathcal{A}$ the operators on a system’s state space and the $H_k$ the terms of its Hamiltonian, this is the exercise.

What Is Checked

The claim is stated over a finite family of pairwise commuting elements of a normed algebra over $\mathbb{C}$ , a real time $t$ , the exponential, and a product over a finite set taken with no order fixed among its factors.

About these constructions the library proves that $e^{-iH_j t}$ and $e^{-iH_k t}$ commute whenever $H_j$ and $H_k$ do, so that the product in (1) has the same value in every order of its factors.