- #1
spaghetti3451
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I was wondering what the difference is between the two. Would be nice if someone could explain the difference in simple terms, because it appears to be essential to my quantum mechanics course.
Self-adjoint operators and Hermitian operators are very similar, but the main difference lies in the context in which they are used. Self-adjoint operators are defined on a finite-dimensional vector space, while Hermitian operators are defined on an infinite-dimensional vector space. Additionally, self-adjoint operators have real eigenvalues, while Hermitian operators have complex eigenvalues.
An operator is self-adjoint if it is equal to its adjoint, or conjugate transpose. In other words, if A is a self-adjoint operator, then A* = A. An operator is Hermitian if it satisfies the same condition, but with an additional requirement that its eigenvalues are all real.
In quantum mechanics, self-adjoint and Hermitian operators are used to represent physical observables, such as position, momentum, and energy. The eigenvalues of these operators correspond to the possible values that can be measured for each observable, and the eigenvectors represent the states in which those measurements are certain.
No, a non-square matrix cannot be self-adjoint or Hermitian. These properties only apply to square matrices, as they involve the transpose and conjugate transpose of the matrix, which can only be taken for square matrices.
Yes, all self-adjoint operators are also Hermitian. This is because the requirement for a Hermitian operator is that its eigenvalues are real, which is already satisfied by self-adjoint operators. However, the reverse is not necessarily true, as there are Hermitian operators that are not self-adjoint because they are defined on an infinite-dimensional vector space.