# Is there a difference between a Hermitian operator and an observable? [duplicate]

My poorly written lecture notes say that any Hermitian operator does have a complete set of orthogonal eigenstates with real corresponding eigenvalues and is therefore an observable.

In the article Observables, it is said that in order for a Hermitian operator to be observable its eigenvectors must form a complete set.

• basically each H operator is an observable
– user65081
May 12, 2019 at 13:04
• – user65081
May 12, 2019 at 13:09
• @Wolphramjonny can you have a H operator without a complete set of orthogonal eigenstates? May 12, 2019 at 13:09
• I dont remember the proof, but I am 99% sure the answer is no
– user65081
May 12, 2019 at 13:10
• you should wait for someone with better knowledge to confirm
– user65081
May 12, 2019 at 13:12

According to the postulates of quantum mechanics, each observable $$p$$ quantity is associated with an operator $$\hat{p}$$ that acts on the wavefunction $$\psi$$.

The relationship is given by the eigenvalue equation: $$\hat{p}\psi = p\psi.$$

$$\hat{p}$$ is an operator, which means nothing on its own. $$p$$ is the eigenvalues, the observable which is a number.

For instance, if $$p$$ is the momentum:

• $$\hat{p} = \frac{\hbar}{i}\nabla$$, i.e. a functional operator so quite useless on its own;

• Acting of a plane wave $$\psi = e^{ikx}$$, $$\hat{p}\psi = \hbar k\,\psi$$. I.e. the observable momentum is $$p=\hbar k$$.

• What do you mean observable is a number? All observables are operators. You can measure an observable and get a real number. May 12, 2019 at 12:50
• All observables are represented by operators. The actual observable is something that you measure, so it has to be a number, i.e. not the function $\hat{p}$ but its eigenvalue $p$. May 12, 2019 at 12:53
• The Observable Wikipedia article says "an observable is a physical quantity that can be measured." It then goes on to say "In quantum physics, it is an operator". May 12, 2019 at 13:08
• Semantic differences. When you measure the physical quantity, you get a number. IN order to get a number from the wavefunction $\psi$, you need to act on it with an operator. You could say "observable=operator" then, I would prefer "observable $\leftrightarrow$ operator". May 12, 2019 at 13:10
• Then it depends on what you mean by “physical quantity”. I would say it’s the result of the measurement. May 12, 2019 at 13:19

An operator need not be hermitian. For instance, the harmonic oscillator creation operator $$\hat a^\dagger$$ is not hermitian, and neither is the angular momentum lowering operator $$\hat L_-$$. Yet both are perfectly legitimate (linear) operators, i.e. they act linearly on a state and produce a different state.

Setting aside subtle points about domains of operators and self-adjointness, observables must be hermitian (in the sense that their matrix representations are hermitian matrices) because eigenvalues of hermitian matrices are real, which is good since in a lab we measure real (rather than complex) quantities. Moreover, hermitian matrices have a complete set of eigenvectors that spans the entire space.

Note that it is important to realize that this doesn’t imply that non-hermitian operators cannot have eigenvalues or eigenvectors, just that there’s no guarantee the eigenvalues are real and the eigenvectors for a complete set. The usual example of this is the harmonic oscillator coherent state $$\vert \alpha\rangle$$ (where $$\alpha$$ is any complex number) which is an eigenvector of the annihilation operator $$\hat a$$, with complex eigenvalue $$\alpha$$. The eigenvalue need NOT be real since $$\alpha$$ can be complex, and the coherent states form an overcomplete set of vectors for the Hilbert space of the harmonic oscillator.

• All observables are Hermitian operators but are all Hermitian operators also observables? May 12, 2019 at 18:18
• It might be worth noting that $\hat{a}$ has an overcomplete basis of eigenfunctions because it is not only non-hermitian, but also non-normal ($[\hat{a},\hat{a}^\dagger]\ne 0$). A normal operator always has a nice orthonormal basis of eigenfunctions even if it is non-hermitian. May 12, 2019 at 18:37

First, a mathematical subtlety. In finite dimensional Hilbert spaces spaces, every self-adjoint operator can be represented by a Hermitian matrix, i.e. one that is equal to its conjugate transpose, and every Hermitian matrix corresponds to a self-adjoint operator. However, in infinite dimensional spaces, as it is often the case for the Hilbert spaces describing quantum systems, a symmetric operator $$A$$—which in finite dimensions is equivalent to a Hermitian matrix—is only self-adjoint if its domain is the same as the one of its adjoint $$A^\dagger$$, namely $$\mathrm{dom}\ A = \mathrm{dom}\ A^\dagger$$. A self-adjoint operator, though, is always symmetric. This being said, every observable corresponds to a self-adjoint operator.

Nevertheless, it is generally false to suppose the converse: not every self-adjoint operator is an observable, and a typical example of such is the density operator $$\hat{\rho}$$. More on the subject can be read here: not every self-adjoint operator is an observable.