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?
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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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