Eigenstate
Definition
An eigenstate for an operator \(\hat{A}\) is a state \(|a\rangle\) such that \(A|a\rangle = a|a\rangle\).
Eigenvalues and Eigenvectors
A state vector \(|\psi\rangle\) is an eigenvector of \(\hat{A}\) iff \(\hat{A}|\psi\rangle = \alpha|\psi\rangle\). \(\alpha\) is the eigenvalue of the operator.
Theorem
The eigenvalues of a Hermitian operator are real and eigenstates of \(\hat{A}\) correspond to different eigenvalues are orthogonal.
Proof.
\(A|\varphi_n\rangle = a_n|\varphi_n\rangle\).
(1) \(\langle\varphi_m|A|\varphi_n\rangle = a_n\langle\varphi_m|\varphi_n\rangle\).
\(\langle\varphi_m|A^\dagger = a_m^*\langle\varphi_m|\).
(2) \(\langle\varphi_m|A^\dagger|\varphi_n\rangle = a_m^*\langle\varphi_m|\varphi_n\rangle\).
(1) - (2): \(a_n\langle\varphi_m|\varphi_n\rangle - a_m^*\langle\varphi_m|\varphi_n\rangle = (a_n-a_m^*)\langle\varphi_m|\varphi_n\rangle = 0\)
For \(m=n\), \(a_n-a_m^*=0\), hence \(a_n=a_n^*\). Therefore, the eigenvalues are real.
For \(a_m\neq a_n\) (different eigenvalues), \(\langle\varphi_m|\varphi_n\rangle = 0\), thus they are orthogonal.
Finding Probabilities of Measurements
Measuring \(A\). \(\psi(\vec{r},t)=\sum_n c_n|\varphi_n\rangle\), \(a_n\to|c_n|^2\), \(c_n = \langle\varphi_n|\psi\rangle\).