Wave Function Weighting Coefficient
#Physics
The coefficients multiplying eigenfunctions that add up to the wave function. May vary for continuous spectra
Energy
$\displaystyle |c_{n}|^{2}$
- Represents the probability of measuring an energy $\displaystyle E_{n}$ for $\displaystyle \Psi(\vec{r},t)=\sum_{n = 1}^{\infty}c_{n}\psi_{n}(\vec{r})e^{-iE_{n}t/\hbar}$
$\displaystyle \Psi(x,0)=\sum_{n = 1}^{\infty}c_{n}\psi_{n}(x)$
- Wave function at a specific point in time is a linear combination of different position component eigenstates
$\displaystyle c_{n}=\int \psi_{n}^{*}(x)\Psi(x,0) , \mathrm{d}x$
- From Fourier analysis
- Equivalent to $\displaystyle a_{i}=\hat{e}_{i}\cdot \vec{A}$
$\displaystyle \sum_{n}\lvert c_{n}\rvert^{2}=1$
- Normalization condition
$\displaystyle {\left\langle{E}\right\rangle}={\left\langle{H}\right\rangle}=\sum_{n}\lvert c_{n}\rvert^{2}E_{n}$
- Expected value of energy, where $\displaystyle \lvert c_{n}\rvert^{2}$ is the PDF