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fourier_series_and_fourier_transforms [2021/03/08 22:02] – adminfourier_series_and_fourier_transforms [2021/03/10 22:05] (current) – [2.v.4 Dirac $\delta$-functions] admin
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 and then we will have and then we will have
 \[\boxed{\delta(\vec{r}-\vec{r}_0) = \frac{1}{(2\pi\hbar)^{3/2}} \int_{-\infty}^{+\infty} e^{i\vec{p}\cdot(\vec{r}-\vec{r}_0)/\hbar}\,\D^3 \vec{p}.}\] \[\boxed{\delta(\vec{r}-\vec{r}_0) = \frac{1}{(2\pi\hbar)^{3/2}} \int_{-\infty}^{+\infty} e^{i\vec{p}\cdot(\vec{r}-\vec{r}_0)/\hbar}\,\D^3 \vec{p}.}\]
 +
 +{{:question-mark.png?direct&50|}}
 +====== In Class Activity ======
 +
 +  - Prove that
 +  \[\braket{\psi_n}{\psi_m} = \int_{-\frac{L}{2}}^{+\frac{L}{2}} \psi^*_n(x)\psi_m(x)\,\D x = 0,\]
 +  for $n\neq m$, where $\psi_n(x) = e^{i2\pi nx/L}$.