Uniqueness of eigenvector representation in a complete set of compatible observables

Sakurai states that if we have a complete, maximal set of compatible observables, say A,B,C… Then, an eigenvector represented by |a,b,c….>, where a,b,c… are respective eigenvalues, is unique. Why is it so? Why can’t there be two eigenvectors with same eigenvalues for each observable? Does maximality of the set has some role to play in … Read more

Eigenstates of momentum and energy of a free particle

Given the momentum operator ˆP:=ℏiddx, as I understand, the eigenvalue equations are ˆPfp(x)=ℏiddxfp(x)=pfp(x) and the eigenfunctions which correspond to this are fp(x)=Aeipxℏ. Given then that the square of this operator ˆP2 commutes with ˆP, so [ˆP,ˆP2]=0, it follows that the two operators share a common set of eigenstates which form a basis (apparently in something … Read more

Why does Quantum Mechanics use Linear Algebra? [closed]

Closed. This question needs to be more focused. It is not currently accepting answers. Want to improve this question? Update the question so it focuses on one problem only by editing this post. Closed 1 year ago. Improve this question I am currently doing Linear Algebra in hopes of one day tackling QM, and I … Read more

Proof of the set of compatible observable have a bound

Sakurai states that We can obviously generalize our considerations to a situation where there are several mutually compatible observables, namely, [A,B]=[B,C]=[A,C]=⋯=0 Assume that we have found a maximal set of commuting observables, that is; we can’t add any more observables to our list without violating the above. I don’t understand, How are you sure that … Read more

Can incompatible observables share an eigenvector?

I was recently introduced to the concept of compatible and incompatible observables and specifically the generalized uncertainty principle, which is written in my textbook as: σ2Aσ2B≥(12i⟨[A,B]⟩)2 where A and B are some observables. If A and B do not commute then they cannot have a complete set of common eigenfunctions and this is given as … Read more

Vector spaces without a basis and observables in quantum theory

In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), there exist models in which not all vector spaces have a basis. Suppose V is a Hilbert space (over the complex numbers), and assume that V does not have a basis. Then an observable U (Hermitian operator) cannot have an orthogonal eigenbase. It still has … Read more

What exactly implies the need of quantum mechanics for self-adjoint and not only symmetric operators? [duplicate]

This question already has answers here: Why is quantum mechancis is not content with symmetric operators, but wants self-adjoint operators? (2 answers) Closed 7 years ago. We know that quantum mechanics requires self-adjoint operators, not only symmetric. Can we say that this follows ONLY from the two following axioms of quantum mechanics, namely that each … Read more

The location of an object is gauge dependent. Therefore, it’s not measurable?

The location of an object x depends on how we choose our coordinate system. If we move the zero point, x also changes. However, since we have translational invariance, we can always do such shifts without changing anything. Now, in quantum theories, there is usually a lot of emphasis on quantities that are gauge independent. … Read more

Fundamental Understanding of Hamiltonians

First of all, my major is CS for several months I have been exploring the area Quantum Computing, therefore my background in Quantum Mechanics is a bit lacking. I know that a Hamiltonian is a self-adjoint operator which describes the total energy of a system, and its eigenvalues refer to the possible energy levels of … Read more

Is it obvious that the Hamiltonian observable in Quantum Mechanics should also be the Energy observable?

In Quantum Mechanics, the Hamiltonian observable is defined as the generator of time translations. It’s easy to show that if we take this to be the definition of the Hamiltonian, then it is of the form – ˆH=iℏ∂t where ˆH is Hermitian. The Unitary evolution map is then the exponential of this Hermitian operator given … Read more