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# 1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement ...

1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement on a particle prepared in the ground state of harmonic oscillator. The formula will depend on h, m andw (b) Estimate order of magnitude of the standard deviation in (a) for the LIGO mirror of mass 10 kg and w 1 Hz. (c) A coherent state lo) is defined to be the eigenstate of the lowering operator with eigenvalue a, i.e. à lo)a) Write la) as where In) are the energy kets of the harmonic oscillator hamiltonian, or of the number operator if you prefer. Use the definition of coherent state to show that the distribution of Ifn12 with respect to n s of the Poisson form. d) Calculate the expectation value of the number operator (N) in a coherent state la) (e) What are the position and momentum expectation values for a coherent state? (f) Prove that Δ2Ap-h/2 for a coherent state, ie, they saturate the uncertainty relation

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