The possible values that can result from measuring an observable $\hat{O}$ is the observables spectrum $\sigma(\hat{O})$. The spectrum of an operator is the set of its eigenvalues (this is not completely true, but let's just ignore that here).

The states that describe a physical system are vectors in a Hilbert space H: $|\Psi\rangle \in H$. You can always find a complete set of eigenvectors of any observable. This allows you to write any vector $|\Psi\rangle$ as a linear combination of eigenvectors

$|\Psi\rangle = \sum_n c_n |\Phi_n\rangle$, with $\hat{O}|\Phi_n\rangle = \lambda_n |\Phi_n\rangle$ and $c_n = \langle \Phi_n|\Psi\rangle$.

The $c_n$'s in the above sum have a physical meaning: When measuring $\hat{O}$, you will measure $\lambda_n$ with a probability of $|c_n|^2$ (this requires $\sum_n |c_n^2| = 1$).

So if the state of the system is described by an eigenvector of some observable $\hat{O}$ (e.g. $c_1 = 1$, and $c_n = 0\ \forall n\geq 2$), then you know with absolute certainty that the measurement of $\hat{O}$ will yield $\lambda_1$.

The possible values that can result from measuring an observable $\hat{O}$ is the observables spectrum $\sigma(\hat{O})$. The spectrum of an operator is the set of its eigenvalues (this is not completely true, but let's just ignore that here).

The states that describe a physical system are vectors in a Hilbert space H: $|\Psi\rangle \in H$. You can always find a complete set of eigenvectors of any observable. This allows you to write any vector $|\Psi\rangle \in H$ as a linear combination of eigenvectors

$|\Psi\rangle = \sum_n c_n |\Phi_n\rangle$, with $\hat{O}|\Phi_n\rangle = \lambda_n |\Phi_n\rangle$ and $c_n = \langle \Phi_n|\Psi\rangle$.

The $c_n$'s in the above sum have a physical meaning: When measuring $\hat{O}$, you will measure $\lambda_n$ with a probability of $|c_n|^2$ (this requires $\sum_n |c_n^2| = 1$).

So if the state of the system is described by an eigenvector of some observable $\hat{O}$ (e.g. $c_1 = 1$, and $c_n = 0\ \forall n\geq 2$), then you know with absolute certainty that the measurement of $\hat{O}$ will yield $\lambda_1$.

The possible values that can result from measuring an observable $\hat{O}$ is the observables spectrum $\sigma(\hat{O})$. The spectrum of an operator is the set of its eigenvalues (this is not completely true, but let's just ignore that here).

The states that describe a physical system are vectors in a Hilbert space H: $\Psi \in H$ (normally I would use braket notation but it doesn't seem to work here on stackexchange). You can always find a complete set of eigenvectors of any observable. This allows you to write any vector $\Psi$ as a linear combination of eigenvectors

$\Psi = \sum_n c_n \Phi_n$, with $\hat{O}\Phi_n = \lambda_n \Phi_n$ and $\sum_n |c_n|^2 = 1$.

The $c_n$'s in the above sum have a physical meaning: When measuring $\hat{O}$, you will measure $\lambda_n$ with a probability of $|c_n|^2$.

So if the state of the system is described by an eigenvector of some observable $\hat{O}$ (e.g. $c_1 = 1$, and $c_n = 0\ \forall n\geq 2$), then you know with absolute certainty that the measurement of $\hat{O}$ will yield $\lambda_1$.

The possible values that can result from measuring an observable $\hat{O}$ is the observables spectrum $\sigma(\hat{O})$. The spectrum of an operator is the set of its eigenvalues (this is not completely true, but let's just ignore that here).

The states that describe a physical system are vectors in a Hilbert space H: $|\Psi\rangle \in H$. You can always find a complete set of eigenvectors of any observable. This allows you to write any vector $|\Psi\rangle$ as a linear combination of eigenvectors

$|\Psi\rangle = \sum_n c_n |\Phi_n\rangle$, with $\hat{O}|\Phi_n\rangle = \lambda_n |\Phi_n\rangle$ and $c_n = \langle \Phi_n|\Psi\rangle$.

The $c_n$'s in the above sum have a physical meaning: When measuring $\hat{O}$, you will measure $\lambda_n$ with a probability of $|c_n|^2$ (this requires $\sum_n |c_n^2| = 1$).

So if the state of the system is described by an eigenvector of some observable $\hat{O}$ (e.g. $c_1 = 1$, and $c_n = 0\ \forall n\geq 2$), then you know with absolute certainty that the measurement of $\hat{O}$ will yield $\lambda_1$.