Understanding Commutators in Quantum Mechanics: [AB] for Hermitian Operators

In quantum mechanics, the behavior and interaction of operators are crucial for understanding the dynamics and properties of quantum systems. One of the fundamental tools in this regard is the commutator. A commutator of two operators A and B, denoted as [A, B], provides a measure of how much the two operators fail to commute. Specifically, the commutator is defined as:

Definition of Commutator

The commutator of two operators A and B is given by:

[A, B] AB - BA

Here, A and B are Hermitian operators, which means they satisfy the property:

A? A and B? B

The dagger (?) denotes the adjoint or conjugate transpose of an operator. The commutator [A, B] measures the extent to which these operators do not commute. A commutator of zero, [A, B] 0, indicates that the two operators can share a common set of eigenstates and thus can be simultaneously diagonalized. On the other hand, a non-zero commutator, [A, B] ≠ 0, implies that the measurement outcomes of one operator affect the outcomes of the other.

Properties of the Commutator

The commutator exhibits several important algebraic properties:

Anti-symmetry

The anti-symmetry property of the commutator states that:

[A, B] -[B, A]

Jacobi Identity

The Jacobi identity is given by:

[A, [B, C]] [B, [C, A]] [C, [A, B]] 0

for any operators A, B, and C.

Example: Position and Momentum in Quantum Mechanics

One of the most significant examples of non-commuting operators in quantum mechanics is the position (x) and momentum (p) operators. Their commutator is given by:

[x, p] xp - px ih

The constant h is Planck's constant (or h/?), which is often denoted as hbar. This non-zero commutator is indicative of the Heisenberg uncertainty principle, which states that the uncertainty in position and momentum cannot both be arbitrarily small at the same time.

Further Insight

Non-commuting Hermitian operators in quantum mechanics are essential in representing physical properties that cannot co-exist simultaneously, such as position and momentum. The communication between these observables is the foundation for the Heisenberg uncertainty principle. This principle can be derived from the non-zero commutator using the Cauchy-Schwartz inequality, as demonstrated in the linked answer.

If you have specific operators in mind or need further clarification, feel free to ask!