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The matrix element ${\displaystyle \langle f|H'|i\rangle }$ is called that because it is the ${\displaystyle (f,i)}$ element of the matrix representation of ${\displaystyle H'}$. This is not a different definition from the mathematical matrix element. This page serves no purpose other than confusion, in my opinion. AndreasBM (talk) 10:01, 21 September 2021 (UTC)[reply]

I was confused by this article as I was studying some very preliminary lectures on quantum mechanics. As a result, I queried via ChatGPT the accuracy of the first paragraph, along with the statement “Not to be confused with the entries or elements of a matrix, nor with the matrix coefficient.” Although I don’t assume the ChatGPT is correct, it’s response seemed to clear up the confusion I had due to the inconsistencies with my lecture and other sources. Here is that ChatGPT response (with very minor edits) as food for thought for someone more knowledgeable than me on the subject:

The Wikipedia entry is confused and partially incorrect. Let's analyze it point by point. Wikipedia Quote Breakdown: 1. "Matrix element (physics) Not to be confused with the entries or elements of a matrix, nor with the matrix coefficient." Partially correct but misleading. · Matrix elements are in fact the entries of a matrix when a linear operator is represented in a basis. · The intention may have been to distinguish abstract matrix elements in quantum mechanics from generic matrix entries in linear algebra, but the phrasing is confusing. 2. "In physics, particularly in quantum perturbation theory, the matrix element refers to the linear operator of a modified Hamiltonian using Dirac notation." Incorrect. · Saying "the matrix element refers to the operator" is just wrong. The matrix element is not the operator - it is a complex number (a scalar), typically of the form ⟨j|H|i⟩, where H is a linear operator. The scalar matrix element describes how a linear operator acts between two basis states. · A linear operator (like the Hamiltonian) is the whole object - it acts on vectors in a vector space (e.g., quantum states). · Dirac notation (bra-ket notation) helps express both operators and their matrix elements, but the matrix element is not the operator itself. 3. "It is in fact referring to the matrix elements of a Hamiltonian operator which serves the purpose of calculating transition probabilities between different quantum states." This part is mostly correct. · Yes, matrix elements of the Hamiltonian (especially in time-dependent or perturbation theory) are used to compute transition amplitudes and probabilities. · The sentence could be clearer, but the core idea is right here. 4. "The matrix element considers the effect of the newly modified Hamiltonian (i.e. the linear superposition of the unperturbed Hamiltonian plus interaction potential) on the quantum state." Misleading wording again. · A matrix element doesn't consider anything - it quantifies the overlap: ⟨f|H^' |i⟩ where H^'=H_0+V is the modified Hamiltonian. · This expression tells you how the operator H^' maps the initial state into the component along . · So, it's better to say: "The matrix element of the modified Hamiltonian expresses how the operator connects or couples two quantum states." ►Here's a much better version of the concept, grounded in standard usage:

In physics, particularly in quantum mechanics and perturbation theory, a matrix element refers to the scalar quantity ⟨j|H|i⟩, which represents the action of a linear operator (such as a Hamiltonian) between two quantum states |i⟩ and |j⟩. Matrix elements are used to compute observable quantities such as transition amplitudes, energy shifts, and probabilities, and they are expressed in Dirac notation. When the Hamiltonian is modified (e.g., by adding a perturbation), the matrix elements of the modified operator reveal how the system's dynamics are affected. 002Frank (talk) 00:11, 24 July 2025 (UTC)[reply]