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EGN123

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## Homework Statement

In-phase light from a laser with an effective power of 2x10

^{5}J and a wavelength of 1064nm is sent down perpendicular 4km arms of the LIGO detector.

(i) Determine the number of photons traveling in the interferometer arms.

(ii) Assuming the detector is sensitive enough to detect single photons at the initial position, estimate the precision with which a change in the length of one of the arms can be detected.

## Homework Equations

c=ƒλ

E=hƒ

ρ=E/c

ΔpΔx ≥ ħ/2

## The Attempt at a Solution

(i) Photon frequency = c/λ = (3x10

^{8})/(1064x10

^{-9}) = 2.82x10

^{14}Hz

Photon Energy = hƒ = (6.626x10

^{-34})(2.82x10

^{14}) = 1.87x10

^{-19}J

Time for light to travel the length of the arms twice (i.e. return to initial position)= 2 x (4000/c) = 2.66x10

^{-5}s

Laser releases 2x10

^{5}J per second, which is 1.07x10

^{24}photons per second

1.07x10

^{24}photons per second x 2.66x10

^{-5}s = 2.85 x 10

^{19}photons

(ii) I was very unsure how to do this part. My initial thought was the uncertainty principle so I tried that but I don't think it is correct.

Sensitive to single photons, ΔE = hf = 1.87x10

^{-19}J

Δρ = ΔE/c = 6.23x10

^{-28}Ns

ΔpΔx ≥ ħ/2

Δx = ħ/2Δp = 8.46 x 10

^{-8}m

I don't think my answers are correct, especially (ii). Am I using the wrong methods? Or correct methods but have made a mistake?