# Heiseinberg's Microscope - Trigonometric and Interpretation questions

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1. Jun 8, 2012

### s3a

1. The problem statement, all variables and given/known data
The problem along with its solution is attached as ProblemSolution.jpg.

2. Relevant equations
Δx = λ/sinθ (Eq. 1)
Δp_x = (h/λ)(sinε) (Eq. 2)

3. The attempt at a solution
In Wikipedia, I found this article: http://en.wikipedia.org/wiki/Heisenberg's_microscope which uses (Eq. 2) from the “Relevant equations” section above but, comparing it to my attached jpg file, ε = 2θ so I feel that the equation should become Δp_x = (h/λ)(sin(2θ)) but my jpg file states that it is Δp_x ≈ 2(h/λ)(sin(θ)) instead. Why is this the case? I even confirmed with Wolfram Alpha that sin(2θ) ≠ 2sin(θ). sin(2θ) = 2sinθcosθ but I don't see any cosθ anywhere.

Also, in the solution in the jpg file, there is the Δp_x ≈ 2h(v/c)sinθ equation; is it this equation that “[shows] that if we minimize Δx by reducing λ, this will result in a loss of information about the x-component of the elctron momentum?” If so, then what's the purpose of the Δx Δp_x product? Is the value of Δx Δp_x ≈ 4πħ = 2h supposed to be an approximation to the Heisenber Uncertainty Principle equation: Δx Δp_x ≥ ħ/2? Lastly, the last part of the solution says “We can attempt to overcome this difficulty by [ . . .].” What's the difficulty we are attempting to overcome? Is it the difficulty of measuring both the position and momentum of an electron simultaneously? Could someone please explain, confirm and/or deny these things to me even if it seems obvious to you?

Any input would be GREATLY appreciated!

#### Attached Files:

• ###### ProblemSolution.jpg
File size:
111.7 KB
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Last edited: Jun 8, 2012
2. Jun 8, 2012

### vela

Staff Emeritus
You didn't attach the file.

3. Jun 8, 2012

### s3a

Sorry lol but, I just did now. :)

4. Jun 9, 2012

### harts

All of these equations, on your pdf and on wikipedia, are all approximations. Maybe they all roughly explain the same thing?

Last edited: Jun 9, 2012
5. Jun 9, 2012

### harts

http://spiff.rit.edu/classes/phys314/lectures/heis/heis.html

This lecture thing I found when I googled heisenberg's microscope kinda defends my angle approximation idea.

Now that I look at this lecture, I'm pretty sure they are getting that 2 from the fact that the uncertainty in p ranges from -hθ/λ to +hθ/λ?

so Δpx=2hθ/λ (or sinθ if youd like)

As far as I can tell, wikipedia and your pdf are giving different but adequate explanations of the same thing.

Last edited: Jun 9, 2012
6. Jun 28, 2012

### s3a

Sorry for the late response and thank you! Those notes were also aesthetically pleasing to my eyes. :)