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 Posted: Wed Jul 19, 2006 5:56 am
I have a question regarding something that I found in the Feynman Lectures in Physics. When they get to explaining why there is uncertainty in the momentum when you measure position they talk about the angle of spread. They explain that the angle of spread is from one minimum of the wave interferance pattern. I don't think the explaination is well made, esp since I'm still not sure what it is, how do you masure it? Is it in the triangle formed from the middle of the slit to the minimum wave? If so is it in a tan/cos/sin ratio already? How can the angle of spread be such:
Δ(theta)=λ/B
if angles cannot be obtained by simple ratios.
Also, why does the angle of spread increase with a decreasing of the width of the slit? Is it because the angle of spread represents the particle's momentum? ANd thus the better we know the place it goes through (the position) the less we know the momentum?
The angle of spread is the ratio between the vertical and horizontal momentii. thus Py = P0*Δ(theta).
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Posted: Wed Jul 19, 2006 10:04 pm
Yeah, the angle of spread is the angle formed at the slit by rays going to two adjacent minimums.
One way (classical) to explain the increase of the spread when the width of the slit decreases is to look at why we get minimums to start with:
Consider two rays of light starting from either side of a slit of width d and going to the first minimum on the detection screen. They make cancel and make a minimum because sin(theta)d = lambda/2.
If you decrease d, but keep lambda the same, then sin(theta) must necessarily be bigger, and thus theta must be bigger.
The quantum-mechanical explanation is basically what you gave, or you could choose to look at it as the interference pattern is basically a "fourier transform", to use the term oddly, if not loosely, of the slit. There's some justification for this but I've since forgotten it.
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Posted: Fri Jul 21, 2006 9:30 am
Layra-chan Yeah, the angle of spread is the angle formed at the slit by rays going to two adjacent minimums. One way (classical) to explain the increase of the spread when the width of the slit decreases is to look at why we get minimums to start with: Consider two rays of light starting from either side of a slit of width d and going to the first minimum on the detection screen. They make cancel and make a minimum because sin(theta)d = lambda/2. If you decrease d, but keep lambda the same, then sin(theta) must necessarily be bigger, and thus theta must be bigger. The quantum-mechanical explanation is basically what you gave, or you could choose to look at it as the interference pattern is basically a "fourier transform", to use the term oddly, if not loosely, of the slit. There's some justification for this but I've since forgotten it. I'm sorry layra-chan, but I am having trouble picturing this, is it possible you make a diagram? I don't see how sing(theta)d = lambda/2 I've been trying to draw it myself, but isn't the width of the slit parallel to lambda? If we analyze the diffraction pattern on the opposing side.oh and also I forgot to state that I defined B as the width. d's fine as a variable though.you see, you use sing(theta) they just use dealta(theta). why.
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Posted: Fri Jul 21, 2006 5:12 pm
poweroutage Layra-chan Yeah, the angle of spread is the angle formed at the slit by rays going to two adjacent minimums. One way (classical) to explain the increase of the spread when the width of the slit decreases is to look at why we get minimums to start with: Consider two rays of light starting from either side of a slit of width d and going to the first minimum on the detection screen. They make cancel and make a minimum because sin(theta)d = lambda/2. If you decrease d, but keep lambda the same, then sin(theta) must necessarily be bigger, and thus theta must be bigger. The quantum-mechanical explanation is basically what you gave, or you could choose to look at it as the interference pattern is basically a "fourier transform", to use the term oddly, if not loosely, of the slit. There's some justification for this but I've since forgotten it. I'm sorry layra-chan, but I am having trouble picturing this, is it possible you make a diagram? I don't see how sing(theta)d = lambda/2 I've been trying to draw it myself, but isn't the width of the slit parallel to lambda? If we analyze the diffraction pattern on the opposing side.oh and also I forgot to state that I defined B as the width. d's fine as a variable though.you see, you use sing(theta) they just use dealta(theta). why.  Here lambda is the wavelength of the incoming light, and m is an integer not equal to 0. I goofed a bit on the formula, but the idea is still there. I think what you were taking as lambda was the displacement on the interferece detection screen. Taking m = 1 gives you the first minimum, and thus the spread.
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Posted: Tue Aug 01, 2006 5:27 pm
Thanks very much layra chan, sorry for the long response, that pretty much explains it, and the pic really helps. biggrin
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