In this section we are going to relate the quantum mechanical operators with the corresponding physical properties of the system. To do this we must take into account that each operator has its own set of eigenfunctions and eigenvalues.
Let us consider that the operator \(\hat{A}\) is associated to the physical property A (Hamiltonian operator to energy, position operator to the distance of the particle from the origin, kinetic energy operator to the kinetic energy of the particle...) . Let us represent by \(f_i\) the set of eigenfunctions of the operator \(\hat{A}\) and by \(a_i\) the respective eigenvalues.
In Quantum Mechanics it is postulated:
Postulate IV. Regardless of what the state function of a system is, the only values that can result from a measurement of the physical observable A are the eigenvalues \(a_i\) of the equation \(\hat{A}f_i =a_i f_i\) . Being \(f_i\) well-behaved functions.
For example, the only values that can be obtained when measuring the energy of a quantum system are the eigenvalues of the equation \(\hat{H}\psi_i =E_i \psi_i\).
The only values that can be obtained when measuring the momentum of a particle are the eigenvalues of the equation: \(\hat{p}\psi_i =p_i \psi_i\)
It is necessary to make two clarifications about this postulate:
- If the state function of the system $\psi$ is an eigenfunction of the operator $\hat{A}$ with eigenvalue a, a measure of the magnitude A associated with the operator \(\hat{A}\) will give as result the value to
- If the function is not an eigenfunction of the operator \(\hat{A}\), it is impossible to predict exactly which of the eigenvalues of $\hat{A}$ we will obtain when measuring that property.