Chem 3240 · Lecture 3.7
An operator acting on its eigenfunction returns the same function, scaled.
\[\hat{A}\psi_n = A_n\psi_n\]
Boundary conditions fix how many solutions exist: finite or infinite.
Numerically, operators become matrices, eigenfunctions become eigenvectors.
\[Av = \lambda v\]
Hermitian operators describe observables.
\[\langle \phi \mid \hat{H} \mid \psi \rangle = \langle \psi \mid \hat{H}\mid \phi \rangle^*\]
Three consequences: - Eigenvalues are real: \(\;E_n = E_n^*\) - Eigenfunctions are orthogonal: \(\;\langle \psi_n \mid \psi_m\rangle = \delta_{nm}\) - Eigenfunctions form a complete basis
Eigenfunctions of a Hermitian operator span the space, like unit vectors.
\[\mid f\rangle = \sum_i c_i \mid \psi_i \rangle\]
Any wavefunction can be expanded in the eigenfunctions of any observable.
A state is a linear superposition of the eigenfunctions of \(\hat{A}\).
\[\mid \psi \rangle = \sum_n c_n \mid \phi_n \rangle\]
The coefficients are projections onto each eigenfunction:
\[c_n = \langle n \mid \psi \rangle\]
The squared coefficients are probabilities.
\[p_n = \mid c_n \mid^2\]
\[\sum_n \mid c_n \mid^2 = \sum_n p_n = 1\]
Measuring \(\hat{A}\) yields eigenvalue \(A_n\) with probability \(p_n\).
The average is a probability-weighted sum of eigenvalues.
\[\langle E\rangle = \langle \psi \mid \hat{H}\mid \psi \rangle\]
For \(\mid \psi \rangle = c_1\mid 1\rangle + c_2\mid 2\rangle\):
\[\langle E\rangle = \mid c_1\mid^2 E_1 + \mid c_2\mid^2 E_2 = p_1 E_1 + p_2 E_2\]
In any single experiment you obtain one eigenvalue.
\[\mid \psi \rangle \rightarrow \mid \phi_n \rangle\]
The superposition collapses to a single eigenfunction.
Orthogonality means mutually exclusive outcomes:
\[\langle \phi_1 \mid \phi_2 \rangle = 0\]
A radioactive atom in superposition of decayed and not-decayed.
Measurement forces the system into one mutually exclusive state.
A quantum state is a superposition of eigenfunctions; measurement returns a single eigenvalue with probability \(\mid c_n\mid^2\), and the expectation value is the probability-weighted average of those eigenvalues.