Commutator Calculator

Enter operator matrices

Matrix size 2 × 2 3 × 3

Rows separated by semicolons or new lines.

Tolerance for commute check

Use 0 for exact checking.

Display decimals

Affects tables and downloads.

Matrix A 1 0; 0 -1

Separate values with spaces or commas.

Matrix B 0 1; 1 0

Supports scientific notation, like 1e-3.

Compute expectation value using a real state vector ψ

State vector ψ (length equals matrix size)

Uses ⟨ψ|M|ψ⟩ / ⟨ψ|ψ⟩ for real vectors.

Calculate Reset

Formula used

The commutator of two operators A and B is defined as: [A,B] = AB − BA. For matrix operators, AB and BA are standard matrix products.

The anticommutator is: {A,B} = AB + BA. A and B commute when [A,B] equals the zero matrix.

This calculator also reports the Frobenius norm of [A,B]. It can test commutation using your chosen tolerance.

How to use this calculator

1. Select the matrix size (2×2 or 3×3).
2. Enter matrices A and B using rows separated by “;” or new lines.
3. Separate each row’s values with spaces or commas.
4. Set a tolerance to decide when values count as zero.
5. Optionally enable expectation value and enter a state vector.
6. Press Calculate to view the commutator above the form.
7. Use Download buttons to export CSV or PDF files.

Example data table

Commutator in physics: practical notes

1) What the commutator measures

The commutator is defined as [A,B] = AB − BA. It measures how strongly the order of two operations matters. When [A,B]=0, A and B commute and can be applied in either order without changing the outcome for matrix multiplication.

2) Operator order and quantum outcomes

In quantum mechanics, observables are represented by operators. A non‑zero commutator indicates incompatibility and links directly to measurement limits through the Robertson relation ΔA·ΔB ≥ ½|⟨[A,B]⟩|. A classic data point is [x,p]=iħ, which embeds Planck’s constant in the algebra.

3) Matrix method used here

This calculator treats A and B as square matrices of the same size n×n. It computes two products, AB and BA, then subtracts element‑wise to form [A,B]. The cost scales roughly as O(n³), because each matrix product uses n dot products per entry.

4) Worked numeric example

For Pauli‑like matrices A=[[1,0],[0,−1]] and B=[[0,1],[1,0]], the commutator is [A,B]=[[0,2],[−2,0]]. The non‑zero off‑diagonal values show that reversing the order flips signs and changes the resulting transformation.

5) Interpreting zero and non‑zero results

If the output matrix is all zeros, A and B commute for the chosen inputs. Many diagonal pairs commute, while mixing a diagonal matrix with a rotation‑like matrix often produces a structured, antisymmetric‑looking result. Small values close to zero can also appear from rounding; adjust decimals to check stability.

6) Expectation value option

The optional expectation calculation evaluates ⟨ψ|[A,B]|ψ⟩ / ⟨ψ|ψ⟩ for a real state vector ψ. This is useful when you want a single scalar that summarizes how the commutator “acts” on a particular state rather than inspecting every matrix element.

7) Units and scaling notes

The commutator inherits units from the product of the two inputs. For example, if A has units of energy and B has units of time, then [A,B] has energy‑time units. Rescaling A by a factor k scales the commutator by the same k.

8) Common use cases

Commutators appear in angular momentum algebra, ladder operators, quantum harmonic oscillators, and symmetry testing. In linear systems, they also act as a quick diagnostic for whether two transformations can be simultaneously diagonalized, which is valuable in eigen‑analysis and mode decoupling. It also appears in control models and filtering math.

FAQs

1) What inputs does this commutator calculator accept?

It accepts two square matrices A and B of the same size. Enter rows separated by semicolons or new lines, and values separated by spaces or commas.

2) What does it mean if the commutator is zero?

A zero commutator means AB equals BA for your inputs. In physics, that often implies compatible operations or observables that can share a common eigenbasis.

3) Can I use non‑integer or scientific notation values?

Yes. Decimals and scientific notation like 1e−3 are supported. Use the decimals option to control rounding in the displayed table and downloads.

4) Why do I sometimes see very small non‑zero numbers?

Floating‑point arithmetic can introduce tiny rounding differences, especially for larger matrices or mixed magnitudes. Increase displayed decimals to verify whether values are truly non‑zero.

5) What is the expectation value feature used for?

It computes a single scalar ⟨ψ|[A,B]|ψ⟩/⟨ψ|ψ⟩ using a real vector ψ. This helps connect the commutator to state‑dependent uncertainty or dynamics checks.

6) Are complex numbers supported?

This version is designed for real numbers. If you need complex matrices, you can still separate real and imaginary parts, or request an extended version with complex arithmetic.

7) How should I choose matrix size for fast results?

Start with 2×2 or 3×3 for conceptual work. Computation time grows roughly with n³, so very large matrices will be slower and harder to interpret.