Does an observable have to be represented by a self-adjoint operator?

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In summary, the conversation discusses the difference between self-adjoint and hermitian operators, and how they relate to observables in quantum mechanics. While some books use the terms synonymously, there is a distinction between the two in terms of the domains of their operators. An observable must be represented by a self-adjoint operator.
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StarsRuler
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¿ Is it the same self-adjoint operator that hermitian operator

If it is not the same, what is the difference? And an observable is an operator whose eigenvectors form basis in the Hilbert space, and it is hermitian, or self-adjoint?

I always considered both terms like sinonynms, in the textbook use both terms, but with the same definition, hermitian and self-adjoint ( the last term is obvious) : it is an operator that it is the same that his adjoint (transpose conjugate)
 
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The difference among the 2 terms is given by the difference between the books which use hand-waving mathematics versus (instead of) real functional analysis. I would advice for the use of <self-adjoint> in all possible (suitable) contexts.
 
  • #3
StarsRuler said:
¿ Is it the same self-adjoint operator that hermitian operator If it is not the same, what is the difference?
StarsRuler, I believe that some books use the two terms to disitnguish between the adjoint of an operator in Hilbert space and the matrix adjoint of a Dirac matrix.
 
  • #4
Ok. Then an observable in QM is represented by an self-adjoint operator which eigenvectors form basis in Hilbert Space ( therefore if we use function representation or bra and kets representation), is not it?
 
  • #6
At least in mathematical physics, a Hermitian or synonymously symmetric mean that the operator and it's adjoint have the same operational form (i.e. d/^2dx^2). However, for a symmetric operator to be self-adjoint, the (dense) domains of the two operators have to be the same. The later condition is non-trivial for unbounded operators which can't be defined on all the Hilbert space.
 
  • #7
StarsRuler said:
¿ Is it the same self-adjoint operator that hermitian operator
No.
StarsRuler said:
If it is not the same, what is the difference?
https://www.physicsforums.com/showpost.php?p=4401816&postcount=13
The difference is given on page 13, however you can read whole paper. It is interesting.
StarsRuler said:
And an observable is an operator whose eigenvectors form basis in the Hilbert space, and it is hermitian, or self-adjoint?
After studying it you will see that observable must be represented by self-adjoint operators.
 
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1. What is the difference between self-adjoint and Hermitian operators?

Self-adjoint and Hermitian operators are closely related concepts in linear algebra. A self-adjoint operator is defined as an operator that is equal to its own adjoint, or conjugate transpose. In other words, if A is a self-adjoint operator, then A* = A. On the other hand, a Hermitian operator is defined as an operator that is equal to its own conjugate transpose, or Hermitian adjoint. In other words, if A is a Hermitian operator, then A† = A. This means that self-adjoint and Hermitian operators are essentially the same, except that they use different notation for the adjoint operation.

2. How do you determine if an operator is self-adjoint or Hermitian?

To determine if an operator is self-adjoint or Hermitian, you can use the definition of these operators. For a self-adjoint operator, you can check if A* = A. For a Hermitian operator, you can check if A† = A. In both cases, you need to take the conjugate transpose of the operator and see if it is equal to the original operator. If it is, then the operator is self-adjoint or Hermitian. Another way to determine if an operator is self-adjoint is to check if all of its eigenvalues are real numbers. If this is true, then the operator is self-adjoint.

3. What are the applications of self-adjoint and Hermitian operators?

Self-adjoint and Hermitian operators have a wide range of applications in mathematics and physics. In mathematics, these operators are used in functional analysis, differential equations, and spectral theory. In physics, they are used in quantum mechanics to describe observable quantities, such as energy and momentum. They are also used in the study of quantum entanglement and quantum information processing.

4. Can a non-square matrix be self-adjoint or Hermitian?

No, a non-square matrix cannot be self-adjoint or Hermitian. These operators are defined for square matrices, where the number of rows is equal to the number of columns. This is because the adjoint operation involves taking the transpose of the matrix, which can only be done for square matrices. Therefore, a non-square matrix cannot be self-adjoint or Hermitian.

5. What is the relationship between self-adjoint and unitary operators?

Self-adjoint and unitary operators are related in that they are both special cases of normal operators. A normal operator is an operator that commutes with its adjoint, or A*A = AA*. A self-adjoint operator is a normal operator where A* = A, while a unitary operator is a normal operator where A*A = AA* = I, where I is the identity operator. This means that all self-adjoint operators are normal, but not all normal operators are self-adjoint. Similarly, all unitary operators are normal, but not all normal operators are unitary.

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