In quantum mechanics, measurements are associated with observables, which are physical properties of a quantum system that can be measured. Each observable in quantum mechanics is associated with a corresponding mathematical operator. When you perform a measurement on a quantum system, the result you obtain is always one of the eigenvalues of the corresponding operator.
The eigenvalues of an operator represent the possible outcomes of a measurement for that observable. An eigenvalue is a scalar value that characterizes the state of the system with respect to the observable being measured. The corresponding eigenvector represents the state of the system that would yield that particular eigenvalue when measured.
The reason why measurements in quantum mechanics yield eigenvalues of the relevant operator can be understood through the mathematical formalism of quantum mechanics. According to the principles of quantum mechanics, the state of a quantum system is described by a mathematical object called a wavefunction. The wavefunction evolves in time according to the Schrödinger equation.
When a measurement is performed on a quantum system, the wavefunction "collapses" into one of the eigenstates of the operator corresponding to the observable being measured. This collapse is described by a mathematical operation called the projection postulate or the collapse postulate. The probability of obtaining a particular eigenvalue is given by the square of the coefficient of the corresponding eigenstate in the wavefunction.
Therefore, when a measurement is made, the wavefunction collapses to one of the eigenstates of the observable, and the corresponding eigenvalue is obtained as the measurement outcome. This is known as the eigenvalue-eigenstate link in quantum mechanics.
It's worth noting that the process of measurement in quantum mechanics can be probabilistic, meaning that the outcome of a measurement is not deterministically predicted. Instead, the probabilities of obtaining different eigenvalues are determined by the coefficients of the corresponding eigenstates in the wavefunction. This probabilistic nature of measurements is one of the distinct features of quantum mechanics compared to classical physics.