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# qmhw7 - | = I where | α = re iθ i is a coherent state and...

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PHY4221: Homework 7 Due Oct. 30, 2008 1. A harmonic oscillator of mass m and frequency ω is prepared in the state, | ψ i = 1 N N X n =0 | 2 n i (a) Calculate the expectation values of position x and momentum p . (b) Calculate the expectation values of variances h x ) 2 i and h p ) 2 i . 2. The Hamiltonian of a system is given by H = i ~ g ( a 2 - a 2 ) where g is a real constant. Write down and solve the Heisenberg equation of motion for a ( t ) and a ( t ). Then compare your results with the example given on p. 16 in the Lecture Notes. 3. (a) Show that |h α 1 | α 2 i| 2 = e -| α 1 - α 2 | 2 where | α 1 i and | α 2 i are coherent states. (b) Show the completeness of coherent states: 1 π Z rdrdθ | α i h
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Unformatted text preview: | = I where | α = re iθ i is a coherent state, and the integral is taken over the whole complex plane of α . (c) Using the result (b), express the number state | n i in terms of a superposition of coherent states. 4. Show that the time derivative of a coherent state deﬁned by a time-varying complex number α ( t ) satisﬁes: d dt | α i = dα dt a † | α i -1 2 • dα dt α * + dα * dt α ‚ | α i If the above equation is a Schr¨ odinger equation after multiplying both sides by i ~ , what is the Hamiltonian? 1...
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