Wave Function Weighting Coefficient

The coefficients multiplying eigenfunctions that add up to the wave function. May vary for continuous spectra

Energy

$|c_n{|}^2$

• Represents the probability of measuring an energy $E_n$ for $\Psi (\vec{r},t)=\sum _{n=1}^{\infty }{c}_n{\psi }_n(\vec{r})e^{-i{E}_{n}t/\hslash }$

$\Psi (x,0)=\sum _{n=1}^{\infty }{c}_n{\psi }_n(x)$

• Wave function at a specific point in time is a linear combination of different position component eigenstates

$c_n=\int {\psi }_n^{\ast }(x)\Psi (x,0)\mathrm{d}x$

• From Fourier analysis
• Equivalent to $a_i={\hat{e}}_i\cdot \vec{A}$

$\sum _n|c_n{|}^2=1$

• Normalization condition

$⟨E⟩=⟨H⟩=\sum _n|c_n{|}^2{E}_n$

• Expected value of energy, where $|c_n{|}^2$ is the PDF