Point source of monochromatic photons - self-contradictory in QM?

Question

Suppose we have a point source of photons located somewhere in space.

So when the photons are released their location is well known, $\Delta x \approx 0, \Delta y \approx 0, \Delta z \approx 0$

Heisenberg's uncertainty principle tells us: $\Delta p_x \Delta x \ge \dfrac{\hbar}{2}$, and same with y and z axes.

This tells me that $p_x,p_y,p_z$ can take on any values (since the uncertainties in position is 0).

So the debroglie relation $\lambda = \dfrac{h}{\sqrt{p_x^2+p_y^2+p_z^2}}$ leads to an infinite spectrum of wavelengths.

Is my analysis correct or wrong? Can we have a monochromatic point source of photons?

Answer 1

The point dipole is an approximation from classical physics - note that it also involves an infinite field strength in its center, where the field amplitude is not differentiable. I think such a source is not compatible with the common approach to quantum mechanics.

If you take such a very small, subwavelength source, it is true that the evanescent near field around it consists of many components. Some of with have a very high spatial frequency - but they are not radiated, they just reside around the source.

In the contrary, the radiated wave is restricted by the dispersion relations in the surrounding medium to have only a certain wavelength at a given frequency. If the source operates for long enough, you get a very narrow spectrum of frequencies in the radiated wave - which essentially means that even the wave of a point dipole will have a single well-defined wavelength λ=c/f.