Understanding Electron Diffraction: Heisenberg Uncertainty Principle

[bmrick]

So my teacher regularly makes mistakes on his powerpoints and this is leaving me feeling uncertain about something he has posted. We are studying the Heisenberg uncertainty principle and discussing a situation in which an electron beam is being fired in the x direction at a small slit (length d) and due to the wave nature of particles some electrons are bending as they leave the slit and have gained momentum in the y direction. so P will denote momentum. P(y) is momentum in the y direction and P(x) is momentum in the x direction.
He notes that P(y)/P(x) is equal to the tan of the angle between the vectors.
He then notes that because the angle is so small tan θ is roughly equal to the θ and so θ≈P(y)/P(x)
Combining this equation with one received earlier from the single slit experiment done by___________:
θ=λ/d
and so P(y)/P(x)=λ/d

So then, this is to represent the max momentum in the y direction of an electron that is diffracted? So the Y component of an electron's momentum ranges P(x)λ/d and -P(x)λ/d

Here's where i assume my teacher made a mistake, but i might just be misinterpreting the slide
He continuously, for like the next six slides, uses variations of the equation P(x)λ/d∠P(y), which implies the Y momentum is actually always greater than P(x)λ/d?

common sense tells me that he messed up on the powerpoint. But the fact that it's written in six different places and the greater than relationship is even expressed in words, has me wondering if I'm missing something. Thanks for the help yall!

 

[bmrick]

bump. Any chance someone out there can give me a hand?

 

[Jilang]

bump. Any chance someone out there can give me a hand?

The equation you first give is for the first minimum intensity fringe only. More generally py/px = nλ/d where n is an integer. The maximum value of py will therefore always be greater than pxλ/d. Note the word maximum. Most of the momenta may lie within the range you think, but there will be others that lie outside.

 

[bmrick]

can you define what you mean by FIRST minimum intensity FRINGE?
Alright so let me get this straight. the relationship between the Y momentum is always a quantized value of this function of X momentum (P(x)*wavelength/slit distance).
So then what is this function meant to tell us? It seems to me that you're saying it says the maximum Y momentum capable by an electron being fired is always greater than this quantity, and is a quantized value of it. And the minimum Y momentum is the quantity explicitly defined by the equation (P(x)*wavelength/slit distance).

 

[Jilang]

The condition for the first minimum intensity. See this diagram:
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/sinslit.html
The momentum in the y direction is not quantised. The pattern of intensity is caused by constructive and destructive interference from contributions over the whole width of the slit. If the slit was sufficiently small the intensity would be more even over the whole screen, like light passing though the hole of a pinhole camera.

 

[bmrick]

alright i see what you're saying there. The electrons experience interference just as photons do. And the equation posted above is meant to mathematically describe the y length between the centerline and the first minimum in intensity. Thanks a lot Jilang, you were a ton of help.

So it seems like the distinguishing feature between light and matter then is just the velocity at which they travel? Since both carry energy and momentum in quantized form, and they can both be understood as having wave properties that induce the same reactions.

What exactly does it mean, that particles have a wave nature. I mean, it seems that introductory physics classes expects us to accept this as fact based on the evidence presented in the form of experiments and the acceptance in the scientific community, but what does this really mean for matter?

 

[Jilang]

alright i see what you're saying there. The electrons experience interference just as photons do. And the equation posted above is meant to mathematically describe the y length between the centerline and the first minimum in intensity. Thanks a lot Jilang, you were a ton of help.

So it seems like the distinguishing feature between light and matter then is just the velocity at which they travel? Since both carry energy and momentum in quantized form, and they can both be understood as having wave properties that induce the same reactions.

What exactly does it mean, that particles have a wave nature. I mean, it seems that introductory physics classes expects us to accept this as fact based on the evidence presented in the form of experiments and the acceptance in the scientific community, but what does this really mean for matter?

What the underlying real meaning is still a question of debate and that is what makes Quantum Mechanics so fascinating! Quantum field theory describes all matter as fields for example and particles are just excitations of the fields. Other theories prefer to describe particles as real entities that behave the way they do for other reasons that may involve Many Worlds. This is all known as interpretation as nothing can be proved by an experiment one way or another. All we can really say at this stage is that the maths works!