Why cant we find exact position of an electron-Heisenberg Uncertainty principle?

Question

I am using the book Classical Dynamics of Particles and Systems by STEPHEN T. THORNTON, JERRY B. MARION, page 88, and they say:

"We can conceivably measure the position of an electron by scattering a light photon from the electron. The wave character of the photon precludes an $\textit{exact}$ measurement, and we can determine the position of the electron only within some uncertainty $\Delta x$ related to the extent (i.e. the wavelength) of the photon."

$\textbf{Question 1}$

What exactly do they mean about the $\textit{wave character of the photon}$? What characteristic are they referring to?

$\textbf{Question 2}$

I guess Q1 would shed light on to this question: Why is the uncertainty related to the wavelength of the photon that is being used to measure it?

I know about the $\textbf{Heisenberg Uncertainty principle}$ and I know about $\Delta p$ part of the principle and that photons add momentum to the electron per hit. I am wanting to find photons cant give an accurate position of a $\textit{still}$ electron at the place of where it got hit by the photon, what happens afterwards is not important.

I would really appreciate any answer that would shed some light onto my problem.

Answer 1

The HUP does not in itself say that an electron cannot have an exact position. What it reflects is the fact that the probability of finding an electron at a given place is related to the amplitude of its wave function at that place, so for an electron to have a well defined position it must have a wave-function whose amplitude drops off to zero over a very small distance, and that means that it must have a wide spread in momentum space. The result is that the more you constrain the electron's position the more you increase the uncertainty about its momentum.

The reason why we can't pinpoint the position of an electron exactly should be obvious if you consider the fact that the only options we have to measure the position is to observe the results of interactions between the electron and other microscopic particles, and those particles themselves have uncertain positions. If an electron scatters a photon, we can't know exactly where the scattering took place, because it could have been anywhere in the volume of space where the electron's wave function overlapped with the photon's. That volume gets bigger as the photon's associated frequency gets lower. An ultra-low frequency radio wave might have a wavelength of many kilometres, and the wave function associated with a photon of that frequency would have a larger spatial spread than the wave function of a highly energetic gamma ray, say.

The other problem, of course, is that we can only know what the approximate position of the electron was at the point the scattering occurred, since the electron itself will have recoiled from the scattering event and will now be somewhere else.

Answer 2

Imagine a standing wave and a particle (1D case). The wave exerts influence on the electron at its peaks but at its nodes the particle is not acted on. If you were to judge the location of the particle by the force (acceleration) acted on it then in the presence of measurement errors you can only be more certain about where it is not likely to be rather than where it is. That is because around the peak effect the curve is flat (the sinusoid has zero or small derivatives) and the particle can be with the same likelihood in that broad flat region almost anywhere.

In contrast, at the nodes the detection curve is steep (the sinusoid has the largest derivative), so to decide the location more precisely we can have shorter wavelength and by that we can learn the particle location within a wavelength that can be arbitrary small. This will improve accuracy but with a multiplicity of possibilities, the buzzword is "ambiguity", and each possibility has improved SNR and good discrimination against other possibilities but uniqueness is lost.

Answer 3

You cannot discover the exact position of an electron for exactly the same reason you cannot discover an electron's favorite movie. Electrons do not have favorite movies.

Answer 4

The author is not making clear the subject by:

The wave character of the photon

Light,classical electromagnetic radiation, described by Maxwell's equations, has a wave character. In contrast the photon is an elementary point particle in the standard model of particle physics and thus has no wavelength. Its energy $hν$ is connected with the frequency $ν$ of the light wave that a huge number of photons build up, but the concept of wavelength is only in the wavefunction of the photon , i.e. the probability of finding the photon at (x,y,z).

The wavefunction of the photon is given by a solution of a quantized mMxwell equation, thus the connection with the classical light wave frequency.

This link may help in acquiring an intuition of how single photons add up to the classical light.

Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

In the context of the quote, it is the probability of the photon interacting with the electron that gives the HUP uncertainty. To get a probability distribution one has to have a fair number of same energy photons interact with the electron which sits at (x,y,z) and the distribution in space of the reflected photon will have the uncertainty of the probability wave nature. One could not define the (x,y,z) of the electron with an accuracy better than the wave length of the probability distribution.