Why is it essential in quantum mechanics for the eigenvectors of an observable to compose a basis spanning the entire state space?

Question

After going through the questions and answers, I still have a question lingering in my mind.

So, an observable is defined as a Hermitian operator whose eigenvectors make up a basis for the state space.

But why is it so important that this basis covers the entire state space? Could it be because if it doesn't, we can't express $|\psi\rangle$ as a combination of all the basis elements? And then, we'd struggle to understand the quantum state of the system, right? I guess this also means we wouldn't be able to describe all the possible outcomes of observables and their probabilities accurately.

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Answer 1

So, an observable is defined as a Hermitian operator whose eigenvectors make up a basis for the state space.

No, simply any Hermitian (or rather self-adjoint) operator defines, in principle an observable.

Usually, physical observables have an additional property, that of being local in space. However sometimes this requirement (arising from the laws of physics) doesn't apply. For example in quantum computing sometimes one can try to build observables with long range interactions.

So the most general definition of observable, without any restriction, is simply that of a self-adjoint operator.

Consider a single particle in 3D space (such as an electron-proton system in it's center of mass, i.e. an hydrogen atom). The momentum $P$ has purely continuous spectrum. Therefore no eigenvalue and no eigenvector.

Something similar as what you say anyway, still holds. Indeed one can decompose any wavefunction via it's Fourier decomposition:

$$ \psi(x) = \int_{\mathbb{R^3}} dp \, \frac{e^{ipx}}{(2\pi)^3}\hat{\psi}(p) $$

In the above equation $p$ is the wannabe eigenvalues of $P$ and $\frac{e^{ipx}}{(2\pi)^3}$ is the wannabe eigenvector. You will notice though, that this function is not normalizable. So it doesn't belong to the Hilbert space. But one can still build legitimate wavefunctions with them (via the equation above). In this specific case one speaks, for example, of building wave packets.