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A real linear operator is one whose matrix elements are real. For example, given an orthonormal basis $\{|\psi_i\rangle\}$,and an operator $\hat O$, suitably defined on a Hilbert space; when we calculate the expectation value of the operator: $\langle \psi_i|\hat O|\psi_j\rangle$ we always get real numbers. Because we have assumed a particular basis, the realty of an operator is not guaranteed to be a property that holds for any arbitrary basis used to span the Hilbert space. Every Hermitian operator is also a real operator, however, not all real operators are Hermitian. So Hermiticity is the crucial requirement needed to do quantum mechanics, it does no good for an operator to be real in this sense, without having Hermiticity.

A real linear operator is one whose matrix elements are real. For example, given an orthonormal basis $\{|\psi_k\rangle\}$,and an operator $\hat O$, suitably defined on a Hilbert space $\mathcal H$; when we calculate the expectation value of the operator: $\langle \psi_i|\hat O|\psi_j\rangle, \;\forall |\psi_i\rangle,|\psi_j\rangle\in \{|\psi_k\rangle\}$; we always get real numbers. Because we have assumed a particular basis, the realty of an operator is not guaranteed to be a property that holds for any arbitrary basis used to span the Hilbert space. Every Hermitian operator is also a real operator, however, not all real operators are Hermitian. Although the realty of an operator is a basis dependent concept, Hermiticity is intrinsic to the operator and thus holds in any basis, i.e. $\langle\hat O\psi |\psi\rangle=\langle\psi |\hat O\psi\rangle, \;\forall |\psi\rangle\in \mathcal H$. So Hermiticity is the crucial requirement needed to do quantum mechanics, it does no good for an operator to be real in this sense, without having Hermiticity.

A real linear operator is one whose matrix elements are real. Operators like the momentum operator are real, for example, when we calculate its expectation value $\langle \psi|\hat p|\psi\rangle$ we always get real numbers. Every Hermitian operator is also a real operator, however, not all real operators are Hermitian. So Hermiticity is the crucial aspect needed in quantum mechanics, that is if they represent observables; it does not good for an operator to be real in this sense without having hermiticity.

A real linear operator is one whose matrix elements are real. For example, given an orthonormal basis $\{|\psi_i\rangle\}$,and an operator $\hat O$, suitably defined on a Hilbert space; when we calculate the expectation value of the operator: $\langle \psi_i|\hat O|\psi_j\rangle$ we always get real numbers. Because we have assumed a particular basis, the realty of an operator is not guaranteed to be a property that holds for any arbitrary basis used to span the Hilbert space. Every Hermitian operator is also a real operator, however, not all real operators are Hermitian. So Hermiticity is the crucial requirement needed to do quantum mechanics, it does no good for an operator to be real in this sense, without having Hermiticity.

One must keep in mind that whether or not an operator is an observable, i.e. that it is Hermitian, depends as much on the definition of its domain than on the form of the operator itself, however, given that there are no problems in this regard; it is true that real operators are also Hermitian operators. In all of the correct cases, a real linear operator just means that the operator is Hermitian.

A real linear operator is one whose matrix elements are real. Operators like the momentum operator are real, for example, when we calculate its expectation value $\langle \psi|\hat p|\psi\rangle$ we always get real numbers. Every Hermitian operator is also a real operator, however, not all real operators are Hermitian. So Hermiticity is the crucial aspect needed in quantum mechanics, that is if they represent observables; it does not good for an operator to be real in this sense without having hermiticity.