The Laser Interferometer Gravitational-wave Observatory (LIGO) is an interferometer that detected the first gravitational waves in 2015, leading to a Nobel prize for Kip Thorne, Rai Weiss, and Barry Barish. There are mirrors at the end of the interferometer arms. Gravitational waves cause the positions of these mirrors to shift. Therefore, measuring the precise positions of the mirrors enables the detection of gravitational waves. The mirrors have masses of 40 kg.
(a) If the mirror positions are to be measured with a precision of ∆x < 10^(-18) m, what is the minimal perturbation to the mirror velocity according to the Heisenberg uncertainty principle?
(b) With this perturbation to the mirror velocity, approximately how long would it take for the mirror to move by more than 10^(-18) m because of quantum noise alone, making quantum noise the dominant source of positional uncertainty?
(c) What is the most precise measurement of position that can be made so that the mirror displacement due to the quantum noise accumulated over the period of a 100 Hz gravitational wave does not exceed the measurement precision?
(d) Gravitational waves travel at the speed of light. Two gravitational-wave detectors separated by a distance D ~ 3,000 km combine to create an Earth-scale instrument. (Hint: think of a telescope with an aperture of D ~ 3,000 km.) What is the diffraction-limited angular resolution of these detectors to 100 Hz gravitational waves?