4. Schroedinger picture (a) Consider a harmonic oscillator of mass m and angular

Question

4. Schroedinger picture (a) Consider a harmonic oscillator of mass m and angular frequency w. At time 0, the state of the system is given by where In) is the energy eigenstate with En hw(n 1/2). What is the proba- bility that a measurement of the energy of the oscillator performed at t 0 will yield the result (3/2)hw? Answer the same question for an arbitrary t 0. (b) From now on, assume that only co and cu are nonzero. Write the normalization condition for (0) s, and the expression for (H) in terms of co and c1. Show that (H) is time independent. Suppose that (H) 3hw/4. Calculate lcol and (c) The overall phase of the state vector I (0) s has no physical significance, but the relative phase of co and ci does. Let us fix the overall phase by requiring co to be real and positive, and define the relative phase by writing c1 as ci lcileie As before, assume that (H) 3hu/4. Now suppose that I tell you that at time Calculate at t 0. Give all possible solutions d) Parts (b) and (c) together have specified l (0)) s. Write lve(t))s for t 0. Define (t) by writing ly (t) s in the form (t))s (lool 0) lcile"("l1))x (time dependent overall phase) Calculate d(t). Calculate (r) as a function of time. Calculate p as a function of time, using Ehrenfest's theorem if you choose.

Explanation / Answer

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/MANY_PARTICLE_PHYSICS/BROUWER=theory_of_many_particle_systems.pdf

http://uncw.edu/phy/documents/Shafer_09.pdf

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