Question

I have a question about difference between physical observables and eigenstates in quantum mechanics

it is postulated in Quantum Mechanics that

physical observable in Classical Mechanics are represented by linear operators

the state of the particle was represented by a curve in phase space determined by generalized position and momentum

but in Quantum Mechanics it is represented by a vector in Hilbert space

and if we measure the physical observable of the particle, the previous state of the particle once superpositioned with eigenstates of the physical observables

with the corresponding weighting factor(square of it is probability of yielding the result when measure) collapses to a single eigenstate and gives out the corresponding eigenvalue

but I don't understand what is the difference between physical observable represented by linear operator and the eigenstate of the particle

Answer 1

In quantum mechanics (QM), physical observables are something that we measure. They are associated with linear operators in quantum numbers. A linear operator acts on a vector in Hilbert space and gives another vector of that space as output. For a particular linear operator, there will be certain vectors which are the same for input and output. These vectors are called eigenvectors. The operator of such operator may be represented as,

where,

• is linear operator
• is the eigenvector
• is the eigenvalue

The eigenvalue represents the observable represented by the linear operator .

It follows that during an experiment the state of a system jumps unpredictably to an eigenstate of the observable that was measured. This we know as the 'wave function collapse'. The eigenvalue associated to this eigenstate is known as observable. It must be noted that only the real numbers are represented as physical observables. The energy, position, momentum or spin are few physical observables.