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mwalmasri
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Why all operators in QM have a Hermitian Matrices ?
mwalmasri said:yes so all operators have a self-adjoint Matrices, which operator can be Represented as anti-Hermitian operators?
The reason for this is that Hermitian matrices represent observables in quantum mechanics. This means that the eigenvalues of the matrix correspond to the possible outcomes of a measurement of that observable, and the eigenvectors represent the states in which the observable is well-defined.
Hermitian matrices are important in quantum mechanics because they correspond to observables, which are physical quantities that can be measured. The Hermitian property ensures that the eigenvalues and eigenvectors are real, which is necessary for a physically meaningful measurement.
The uncertainty principle states that certain pairs of observables cannot be measured simultaneously with arbitrary precision. Hermitian matrices play a role in this principle by representing compatible observables, which can be measured with arbitrary precision at the same time. Non-compatible observables, on the other hand, do not have Hermitian matrices and cannot be measured simultaneously with arbitrary precision.
Yes, there are exceptions. Some operators, such as the time evolution operator, do not have Hermitian matrices. However, even in these cases, the operator can be written as a linear combination of Hermitian matrices.
No, two different Hermitian matrices cannot represent the same observable. This is because the eigenvalues and eigenvectors of a Hermitian matrix uniquely determine the observable it represents. Therefore, any two Hermitian matrices representing the same observable must be equal.