LightPhoton
- 42
- 3
Wikipedia says that the equation in the title is defined to be true.
But is it true always?
Working with the right-hand side,
$$\langle \mathbf r\vert\hat A\vert\Psi\rangle=\int\langle \mathbf r\vert\hat A\vert\mathbf r'\rangle\langle\mathbf r'\vert\Psi\rangle\ d\mathbf r'$$
If we assume that ##\langle \mathbf r\vert\hat A\vert\mathbf r'\rangle=\hat A(\mathbf r)\delta(\mathbf{r-r'})##, then above expression collapses to ## \hat A(\mathbf r)\Psi(\mathbf r)##. But how to show this in general?
But is it true always?
Working with the right-hand side,
$$\langle \mathbf r\vert\hat A\vert\Psi\rangle=\int\langle \mathbf r\vert\hat A\vert\mathbf r'\rangle\langle\mathbf r'\vert\Psi\rangle\ d\mathbf r'$$
If we assume that ##\langle \mathbf r\vert\hat A\vert\mathbf r'\rangle=\hat A(\mathbf r)\delta(\mathbf{r-r'})##, then above expression collapses to ## \hat A(\mathbf r)\Psi(\mathbf r)##. But how to show this in general?