1. In Depth Insights
The commutator between two linear operators andis defined as:
If , then , meaning that operatorcommutes with operator . Similarly, the anti commutator of two operators is defined as:
When , the operator is said to anti commute with . Many vital relations in quantum mechanics can be deduced directly using commutators and anti commutators.
Perhaps the most practically useful property of the commutator is its direct link to simultaneous diagonalization: two Hermitian operators andcommute if and only if they can be simultaneously diagonalized by a shared orthonormal basis. That is, there exists a complete set of orthonormal eigenvectors such that:
and
Simultaneous Diagonalization Theorem: Let and be Hermitian operators. Thenif and only if there exists an orthonormal basis in which both and are diagonal. In this case, we say that and are simultaneously diagonalizable.
This foundational result connects the commutator, which is typically straightforward to evaluate directly, with the capacity for simultaneous diagonalization, a property that is generally far harder to determine a priori.
For example, consider the Pauli matrices and :
Therefore, and do not commute.
Proof: One can easily prove that if and are diagonal in the same orthonormal basis, then. To prove the converse, let be an orthonormal basis for the eigenspace corresponding to the eigenvalue of operator , where indexresolves any degeneracy. Notice that:
Thus, the state is also an element of the eigenspace Let denote the projection operator onto the subspace and define the operator It is evident that the restriction of to the subspaceis Hermitian, thereby admitting a spectral decomposition in terms of an orthonormal set of eigenvectors spanning .
Let us denote these shared eigenvectors by, where indices and label the eigenvalues of and , respectively, and accounts for any residual degeneracy of . Because resides in , we have . Furthermore, since, we obtain:
Consequently, is an eigenvector of with eigenvalue . It follows that the statesform a complete orthonormal basis of eigenvectors for both and across the entire vector space, establishing that and are simultaneously diagonalizable.
2. Mathematical Formalism
Consider the time independent Schrödinger equation for a simple harmonic oscillator:
where represents the quantum momentum operator. The underlying strategy is to factor the Hamiltonian:
If these variables were ordinary numbers, the factorization would be elementary:
In quantum mechanics, however, this algebraic simplification is invalid because and are operators that generally do not commute; that is,is not equal to. Nonetheless, this inspires us to introduce the following ladder operators:
What algebraic structure do these operators possess? Let us calculate their product:
As anticipated, an additional cross term emerges involving. This expression is precisely the commutator of and , which quantifies the degree to which the two operators fail to commute. In commutator notation, the product reads:
To evaluate the commutator, we must exercise care: working with abstract differential operators directly can lead to algebraic slips. The proper rigorous procedure is to let the commutator act upon an arbitrary test function :
Dropping the test function because it holds for any arbitrary smooth function, we arrive at the canonical commutation relation:
This canonical commutation relation constitutes one of the most fundamental identities in quantum mechanics.
3. References
Nielsen, M. A., & Chuang, I. L. (2000). Quantum Computation and Quantum Information (Vol. 1). Cambridge University Press.