Significance of using Eigenvalues / Vectors in QM?

Question

A fundamental idea in Quantum Mechanics is that observable quantities are represented by linear, hermitian operators. Why is it that we represent distinguishable states as the eigenvectors of operators, and corresponding eigenvalues are the results of measurements of those states?

I am reading "Quantum Mechanics The Theoretical Minimum" by Leonard Susskind and Art Friedman, in which this is said:

"If the system is in the eigenstate $$|\lambda_i \rangle$$ the result of a measurement is guaranteed to be $$\lambda_i$$"

I understand the geometrical meaning of eigenvectors and eigenvalues in linear algebra, but why are they used and so important in Quantum Mechanics?

Answer 1

According to the postulates of QM,

1) observables are raised to the status of linear self-adjoint operators

2) the only measurable values of an obeservable are its eigen-values.

Now, self-adjoint(for finite dimensions, they are same as Hermitian)operators are guaranteed to have real eigenvalues. Hence the measurable quantities are the eigenvalues of the observable which are real(as they are obtained from experiment).

This is how eigenvalues stem in QM. Take the postulates as your guiding principle! Hope this helps.

Answer 2

As far as I know, this comes from the structure of the Schroedinger equation, the equation that rules a particle in non relativistic quantum mechanics (I have not read that book, but I think that this is its subject).

The time dependent Schroedinger equation: $$ i \hbar \frac{\partial }{\partial t} \Psi = \hat{H} \Psi $$ with $\hat{H} $ hamoltonian operator of the problem and $\Psi$ time dependent solution of this equation. From this formulation one can show that it is possible to separate the variables of the problem, writing an equation for the temporal part and another one for the three spatial coordinates. The latter can be easily written as: $$\hat{H} \psi = E\psi$$ with $\psi$ solution of the time independent equation and $E$ energy of the particle. From a mathematical point of view this problem is the same as looking for the spectrum of eigenvalues (the energies) and eigenvectors (the solutions of this equation, projected on a certain basis, also called eigenfunctions) of the hamiltonian operator, that is by definition the operator whose eigenvalues are the allowed energies.

Therefore, this kind of language is a direct consequence of the mathematical formulation of the problem.

(Note that in my answer I have skipped a lot of mathematic, that would have make my answer considerably longer. You can find it in most books of introductive quantum mechanics, such as Griffiths or Eisberg - Resnick ones. You may also find this Wikipedia's link quite interesting.)