Function Repository Resource:

BakerCampbellHausdorffTerms

Generate terms in the Baker–Campbell–Hausdorff expansion

Contributed by: Mohammad Bahrami

ResourceFunction["BakerCampbellHausdorffTerms"][{op₁,op₂,…,op_m},n]

generates the order-n term of the Baker-Campbell-Hausdorff expansion of operators {op₁,op₂,…,op_m}.

ResourceFunction["BakerCampbellHausdorffTerms"][{op₁,op₂,…,op_m},n,alg]

generates the order-n term of the Baker-Campbell-Hausdorff expansion of operators {op₁,op₂,…,op_m}, where alg can be a NonCommutativeAlgebra object, {Dot,n},Dot,Composition,TensorProduct or NonCommutativeMultiply.

Details and Options

The Baker-Campbell-Hausdorff (BCH) expansion is defined as

with alg as the underlying operation between the non-commutative operators op_i. The degree-n term in the BCH expansion is the part that involves n Lie algebra elements combined through n-1 nested commutators and coefficients.

The BCH formula solves e^Xe^Y=e^Z for noncommutative operators X and Y in a Lie algebra.

The BCH expansion is the sum of terms over n.

If the algebra argument is omitted, NonCommutativeAlgebra with the default property values is used.

ResourceFunction["BakerCampbellHausdorffTerms"] take the following option:

"CommutatorForm"

False

when True, it will hold the commutator form, otherwise it will compute the commutation.

For example, for two operators, the commutator form follows the formula where is iterative j commutation

ResourceFunction["BakerCampbellHausdorffTerms"] requires version 14.3 or higher of the Wolfram Language.

Examples

Basic Examples (7)

Generate the first-order term of the Baker–Campbell–Hausdorff formula for two operators:

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Generate the second-order term of the Baker–Campbell–Hausdorff formula for two operators:

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Show the formula in the commutator form:

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Release the commutator form and show the expanded formula:

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Generate the third-order term of the Baker–Campbell–Hausdorff formula for two operators:

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Show the formula in the commutator form:

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Verify explicitly that the two results above are identical:

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Show that the result is identical to :

In[8]:=

ResourceFunction["BakerCampbellHausdorffTerms"][{x, y}, 3] - 1/12 (Commutator[x, Commutator[x, y]] - Commutator[y, Commutator[x, y]]) // NonCommutativeExpand

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Generate the fourth-order Baker–Campbell–Hausdorff term for two operators:

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Show that the result is identical to :

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ResourceFunction["BakerCampbellHausdorffTerms"][{x, y}, 4] - (-(1/24) Commutator[x, Commutator[y, Commutator[x, y]]]) // NonCommutativeExpand

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Show that the result is identical to :

In[11]:=

ResourceFunction["BakerCampbellHausdorffTerms"][{x, y}, 4] - (-(1/24) Commutator[y, Commutator[x, Commutator[x, y]]]) // NonCommutativeExpand

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Compute the fourth-order Baker–Campbell–Hausdorff term for two operators in commutator form:

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Verify explicitly that the results above are identical:

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Compute the second-order Baker–Campbell–Hausdorff term for three symbolic matrices:

In[14]:=

$ResourceFunction["BakerCampbellHausdorffTerms"][ Table[MatrixSymbol["x" <> ToString[j], \[FormalN]], {j, 3}], 2, Dot] // NonCommutativeExpand[#, Dot] &$

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Compute the third-order Baker–Campbell–Hausdorff term for three operators with Composition as the action:

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Compute the third-order Baker–Campbell–Hausdorff term for two operators with NonCommutativeMultiply as the action between operators:

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Show the third-order Baker–Campbell–Hausdorff term for four operators with Dot as the action between operators:

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Scope (2)

Show a few terms of Baker-Campbell-Hausdorff formula 𝒵=Log(ⅇ^X₁ⅇ^X₂…ⅇ^X_n) with three non-commutative operators X_j:

In[18]:=

$With[{ops = Table[ToString[Subscript[x, j], StandardForm], {j, 3}]}, Grid[Table[{ToString[Subscript[\[ScriptCapitalZ], j], StandardForm], ResourceFunction["BakerCampbellHausdorffTerms"][ops, j, Dot] // NonCommutativeExpand[#, Dot] &}, {j, 4}], Frame -> All, Alignment -> Left]]$

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Show the commutator form of 𝒵=Log(ⅇ^X₁ⅇ^X₂):

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$Grid[Table[{ToString[Subscript[\[ScriptCapitalZ], j], StandardForm], ResourceFunction["BakerCampbellHausdorffTerms"][{x, y}, j, "CommutatorForm" -> True] // TraditionalForm}, {j, 5}], Frame -> All, Alignment -> Left]$

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Options (2)

Show the third-order Baker–Campbell–Hausdorff term for two operators while preserving the commutator form:

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Show the second-order Baker–Campbell–Hausdorff term for four symbolic matrices in commutator form:

In[21]:=

$ResourceFunction["BakerCampbellHausdorffTerms"][ Table[MatrixSymbol["x" <> ToString[j], \[FormalN]], {j, 4}], 2, Dot, "CommutatorForm" -> True] // TraditionalForm$

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Applications (19)

Annihilation and creation operators: (5)

The single-mode photon field satisfies the Bosonic commutation relation.

For the expression , compute the second-order term in the Baker–Campbell–Hausdorff series:

In[22]:=

$ResourceFunction[ "BakerCampbellHausdorffTerms"][{\[Mu] a, \[Nu] SuperDagger[a]}, 2]$

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Define a function that simplifies an noncommutative polynomial in a and a^†:

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$ClearAll[NCPolynomialReductionBosonic] NCPolynomialReductionBosonic[exp_, scalars_ : {}] := NonCommutativePolynomialReduce[ exp, {Commutator[a, SuperDagger[a]] - 1}, {a, SuperDagger[a]}, NonCommutativeAlgebra[<|"ScalarVariables" -> scalars|>]][[-1]]$

Reduce (simplify) it:

In[24]:=

$NCPolynomialReductionBosonic[%, {\[Mu], \[Nu]}]$

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Compute the third-order term of the Baker–Campbell–Hausdorff series:

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$ResourceFunction[ "BakerCampbellHausdorffTerms"][{\[Mu] a, \[Nu] SuperDagger[a]}, 3]$

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Reduce it:

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$NCPolynomialReductionBosonic[%, {\[Mu], \[Nu]}]$

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Likewise, the higher order terms disappear as well. Thus one finds: .

Canonical position and momentum (4)

Define canonical position and momentum operators and :

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Verify :

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Compute the second-order term of the Baker–Campbell–Hausdorff series for :

In[29]:=

$NCPolynomialReductionBosonic[ ResourceFunction[ "BakerCampbellHausdorffTerms"][{I \[Alpha] q, I \[Beta] p}, 2], {\[Alpha], \[Beta]}]$

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Show that the higher-order terms are zero:

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$Table[NCPolynomialReductionBosonic[ ResourceFunction[ "BakerCampbellHausdorffTerms"][{I \[Alpha] q, I \[Beta] p}, n], {\[Alpha], \[Beta]}], {n, 3, 6}]$

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Thus it means .

Displacement operator (3)

With the displacement operator as 𝒟(α)=ⅇ^{α a^†-α^*a}, show that .

Compute the second-order term of the Baker–Campbell–Hausdorff series:

In[31]:=

$NCPolynomialReductionBosonic[ ResourceFunction[ "BakerCampbellHausdorffTerms"][{\[Alpha] SuperDagger[a] - Conjugate[\[Alpha]] a, \[Beta] SuperDagger[a] - Conjugate[\[Beta]] a}, 2], {\[Alpha], \[Beta]}]$

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Show that the higher-order Baker–Campbell–Hausdorff terms vanish:

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$Table[NCPolynomialReductionBosonic[ ResourceFunction[ "BakerCampbellHausdorffTerms"][{\[Alpha] SuperDagger[a] - Conjugate[\[Alpha]] a, \[Beta] SuperDagger[a] - Conjugate[\[Beta]] a}, n], {\[Alpha], \[Beta]}], {n, 3, 6}]$

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Compute the second through fifth terms in the BCH series of 𝒟(α₁)𝒟(α₂)𝒟(α₃) and verify :

In[33]:=

$Table[NCPolynomialReductionBosonic[ ResourceFunction["BakerCampbellHausdorffTerms"][ Table[Subscript[\[Alpha], j] SuperDagger[a] - Conjugate[Subscript[\[Alpha], j]] a, {j, 3}], n], Table[Subscript[\[Alpha], j] , {j, 3}]], {n, 2, 5}]$

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SU(2) spin rotations (4)

Set up the variables and their SU(2) algebra[J_i,J_j]=ⅈϵ_ijkJ_k with ϵ the 3-dimensional Levi-Civita totally antisymmetric tensor. Note that in our relations we do not include quadratic identities such as J_j²=𝕀, because these are representation-dependent (e.g., for Pauli matrices) rather than algebraic identities.

Given the algebra and variables, define a function to reduce a polynomial in J_j (Cartesian basis):

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$ClearAll[NCPolynomialReductionSU2] NCPolynomialReductionSU2[exp_, vars_, scalars_ : {}, scale_ : 1] /; Length[vars] == 3 := NonCommutativePolynomialReduce[exp, DeleteCases[0]@ Flatten@(Outer[Commutator, vars, vars] - I scale TensorContract[ TensorProduct[LeviCivitaTensor[3], vars], {{3, 4}}]), {vars}, NonCommutativeAlgebra[<|"ScalarVariables" -> scalars|>]][[-1]]$

Consider the relation Log[ⅇ^{-ⅈ δt θ₁J₁}ⅇ^{-ⅈ δt θ₂J₂}]=-ⅈ δt H_eff. Our goal to find effective Hamiltonian describing the above rotations. We will use rotation angles θ_j and also time variable δt. We should set them as scalar variable in the algebra.

Define noncommutative variables and scalars:

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$vars = Table[Subscript[J, j], {j, 3}]; scalars = Append[Table[Subscript[\[Theta], j], {j, 3}], \[Delta]t];$

Compute the second-order Baker–Campbell–Hausdorff term for ⅇ^{-ⅈ δt θ₁J₁}ⅇ^{-ⅈ δt θ₂J₂}:

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$NCPolynomialReductionSU2[#, vars, scalars] &@ ResourceFunction[ "BakerCampbellHausdorffTerms"][{-I \[Delta]t Subscript[\[Theta], 1] Subscript[J, 1], -I \[Delta]t Subscript[\[Theta], 2] Subscript[J, 2]}, 2]$

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Compute H_eff up to the second order of δt:

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$Sum[NCPolynomialReductionSU2[ ResourceFunction[ "BakerCampbellHausdorffTerms"][{-I \[Delta]t Subscript[\[Theta], 1] Subscript[J, 1], -I \[Delta]t Subscript[\[Theta], 2] Subscript[J, 2]}, n], vars, scalars], {n, 3}]/(-I \[Delta]t) // Expand$

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Numerical example (3)

Consider ⅇ^xⅇ^y=ⅇ^A with , .

Compute the first 10 terms of the Baker–Campbell–Hausdorff series for A:

In[38]:=

$x = \[Alpha] PauliMatrix[3]; y = {{0, \[Beta]}, {0, 0}}; A = Sum[ResourceFunction["BakerCampbellHausdorffTerms"][{x, y}, n, Dot], {n, 11}]; A // MatrixForm$

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For A₁₂ element of the matrix, show that the Taylor series of the function below matches the result above:

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$Normal[\[Beta] Series[(2 \[Alpha])/( 1 - Exp[-2 \[Alpha]]), {\[Alpha], 0, 10}]] == A[[1, 2]]$

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Therefore, one gets

Publisher

Mads Bahrami

Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

1.2.0 – 17 November 2025
1.1.0 – 29 October 2025
1.0.0 – 23 July 2025

Source Metadata

Citation:
- Our implementation of Zassenhaus formula is based on the following paper:
  - A simple expression for the terms in the Baker-Campbell-Hausdorff series https://www.arxiv.org/pdf/math-ph/9905012

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

BakerCampbellHausdorffTerms

Details and Options

Examples

Basic Examples (7)

Scope (2)

Options (2)

Applications (19)

Annihilation and creation operators: (5)

Canonical position and momentum (4)

Displacement operator (3)

SU(2) spin rotations (4)

Numerical example (3)

Publisher

Requirements

Version History

Source Metadata

Related Resources

Related Symbols

License Information