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Suppose one has a flux of electrons with speed not exceeding 10 m/sec falling on a plate with an aperture of 100 nm. If the electrons can pass one would have a statistics of many electrons passed through the slit but their speed can not exceed 10 m/sec because the aperture is a passive medium unable to accelerate the electrons. HUP (Heisenberg Uncertainty Principle) says that the momentum in x direction should be from 0 to 7 km/sec and more. {dp.dx>h --- dv=h/mv ---dv=7.10(^-34)/10(-30).10(-7)= 7.10(^3)= 7 km/sec}. So if electrons pass there is a contradiction with HUP?

Mercury
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1 Answers1

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de Broglie wavelength: $\lambda = \frac{h}{p_{z}} \rightarrow p_{z} = \frac{h}{\lambda}$

Slit width D: $\Delta x = D$

HUP: $h < \Delta p_{x} \Delta x = \Delta p_{x} D \rightarrow \Delta p_{x} > \frac{h}{D}$

Angle of diffraction: $\Delta \theta = \frac{\Delta p_{x}}{p_{z}} > \frac{h}{D}\frac{\lambda}{h} = \frac{\lambda}{D}$

By comparing the behavior of electrons to the behavior of light, you can see that you will get a wavelike interference pattern with an angle of diffraction, with distribution of angles with uncertainty $\Delta \theta = \frac{\lambda}{D}$, as though the electrons were behaving as light with wavelength = de Broglie wavelength. This $\Delta \theta$ reflects the uncertainty in the momentum in the x-direction due to the width of 100nm in the x-direction, without changing the uncertainty in the momentum in the z-direction.

The reason behind the HUP ultimately comes from the non-commutativity of x and $p_{x}$ so that quantum states cannot simultaneously be position and momentum eigenstates. But x and $p_{z}$ commute so when considering multiple dimensions, a slit narrow in the $x$ direction does not affect the $p_{z}$ momentum.

So, yes, electrons can pass through, and there is no problem with the Heisenberg uncertainty principle.

Alwin
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  • If some electrons pass then their initial absolute value of momentum (before aperture) would be less then their final absolute value of momentum (after the aperture). Is that not a problem? Is it true or not? – Mercury Nov 12 '21 at 03:02
  • Not a problem: the overall wavefunction's momentum is conserved. If you look at a single-slit or double-slit experiment you will notice that as many electrons deflect -x as +x so they balance out. – Alwin Nov 12 '21 at 04:13
  • I don't know what you mean by overall wavefunction's (wf) momentum. I suppose you mean the common wf of all particles in some moment of time. But in fact one can have a flux but with one particle at a time (following after great intervals - an experiment made many times). So the energy and momentum laws must apply for every single particle at least in this case. And just one case is a problem for HUP. – Mercury Nov 12 '21 at 15:42
  • https://iopscience.iop.org/article/10.1088/1742-6596/701/1/012007/pdf

    https://physics.stackexchange.com/questions/667515/do-we-not-know-that-momentum-is-certainly-not-conserved-at-single-electron-doubl

    To be fair, I am not entirely sure what the mechanism is but there is no problem for HUP as electron interference has been clearly measured. It is very possible for the total momentum of all particles in the system to be conserved, for example by a backreaction where each electron imparts momentum on the slit.

     – Alwin Nov 12 '21 at 20:27
  • Both connections are useful thank you. There can not be any mechanism because HUP is valid for single electrons. I am rather inclined to think that the e- can not pass as its wavelength is about 70 000 nm as the opening is just 100 nm. (only e- with speed above 7 km can pass -> WL = 70 nm). Its a pity I can not find anything in literature and no one is able to tell. Thank you anyway. – Mercury Nov 13 '21 at 15:17