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Past exam of the mathematics course of the University of Cambridge
/
2024
/
ii
/
Paper 1
/
41A
/
b
/
i
/
Solution
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2024
ii
Paper 1
41A
b
i
Created
2026-09-24
Updated
2026-09-25
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The
matrix
exponential is the absolutely convergent
series
e
X
=
∑
j
=
0
∞
j
!
X
j
.
(1)
Multiplying the two expansions gives
e
k
B
e
k
C
=
I
+
k
(
B
+
C
)
+
k
2
(
2
1
B
2
+
BC
+
2
1
C
2
)
+
O
(
k
3
)
.
(2)
On the other hand,
e
k
(
B
+
C
)
=
I
+
k
(
B
+
C
)
+
2
k
2
(
B
2
+
BC
+
CB
+
C
2
)
+
O
(
k
3
)
.
(3)
Subtracting yields
e
k
B
e
k
C
=
e
k
(
B
+
C
)
+
2
k
2
(
BC
−
CB
)
+
O
(
k
3
)
.
(4)
This is the leading
Lie-Trotter splitting commutator error
.
Ancestors
(12)
i
b
41A
Paper 1
ii
2024
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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