Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 41A / b / i

Table of contents
- Solution i

Solution

The matrix exponential is the absolutely convergent series

e^{X} = \sum_{j = 0}^{\infty} \frac{X ^{j}}{j !} .

Multiplying the two expansions gives

e^{k B} e^{k C} = I + k (B + C) + k^{2} (\frac{1}{2} B^{2} + BC + \frac{1}{2} C^{2}) + O (k^{3}) .

On the other hand,

e^{k (B + C)} = I + k (B + C) + \frac{k ^{2}}{2} (B^{2} + BC + CB + C^{2}) + O (k^{3}) .

Subtracting yields

e^{k B} e^{k C} = e^{k (B + C)} + \frac{k ^{2}}{2} (BC - CB) + O (k^{3}) .

This is the leading Lie-Trotter splitting commutator error.

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