11.2 The Suzuki-Trotter Expansion
A particularly useful approximation of U(0,T) can be obtained using the Suzuki-Trotter expansion [303]. If A1,A2,…,Ak are operators that do not necessarily commute, then
![( ) [ ( A ) ( A ) (A )]m exp A1 + A2 + ...+ Ak = lim exp --1 exp -2- ⋅⋅⋅exp --k . m→ ∞ m m m](https://static.packt-cdn.com/products/9781836209614/graphics/media/file1235.svg)
|
Recall that two operators A and B are said to commute if AB = BA. Many operators introduced in previous chapters do not commute, for example, rotations around different axes do not, and the end result (the end quantum state) depends on how rotations are ordered.
As mentioned in Chapter 1, the expectation values of Hermitian operators are real and correspond to physical observables (e.g., the expectation of a Hermitian Hamiltonian is the physically observable energy). If operators commute, we can measure them in an arbitrary order and obtain the same answer. There is no uncertainty in the values of the corresponding physical observables.
The Suzuki-Trotter expansion, however, does not require operators to commute to remain valid. This has...
