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Physics/De broglie wavelength

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Question
Neutrons traveling at 0.400 m/s are directed through a pair of slits separated by 1.00 mm. An array of detectors is placed 10.0 m from the slits.
a) What is the de Broglie wavelength of the neutrons?
b) How far from the axis is the first zero-intensity point on the detector array?
c) When the neutron reaches a detector, can we say which slit the neutron passed through?

Answer
The deBroglie wavelength of a neutron is given by:
lambda=h/p=6.63x10^-34/(1.67x10^27kg*0.4m/s)=9.93x10^-7m=993nm
The angle at which the first minimum occurs will be given by:
n*lamba=d*sin(theta)  where n=1/2
Therefore, the angle at which the first minimum occurs will be:
theta=invsin(n*lambda/d)=invsin(1/2*9.93x10^-7/(1.0x10^-3)=0.0284degrees
Since this angle clearly falls within the small angle approximation sin(theta) can be replaced by theta which can in turn be replaced by x/L where L is the distance to the screen and x is the distance from the central antinode and the first minimum.
Therefore:
n*lamba=d*sin(theta)=d*x/L
Solving for x:
x=n*lambda*L/d=1/2*9.93x10-7*10/0.001=4.97x10^-3m
c) This cannot be determined since the neutron wave did not go through either slit! As a wave it went through both slits. I you position a detector in an attempt to determine whether or not the neutron went through a particular slit the interference patter goes away!

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James J. Kovalcin

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I am teaching or have taught AP physics B and C [calculus based mechanics & electricity and magnetism] as well as Lab Physics for college bound students. I have a BS in Physics from the University of Pittsburgh and a Master of Arts in Teaching from same. I have been teaching physics for 34 years. I am constantly updating my skills and have a particular interest in modern physics topics.

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