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homework8 - 28 Consider a harmonically bound electron in 1D...

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28. Consider a harmonically bound electron in 1D with Hamiltonian H 0 = 1 2 ~ ω ( q 2 + p 2 ), initially in its ground state. A heavy particle passes through the region at high speed. This heavy particle interacts with the electron through a weak short-range interaction that can be approximated by the potential V ( t ) = V 0 δ ( q x ( t )) = V 0 δ ( q vt ) , where v is essentially the velocity of the heavy particle (in dimensionless units of the oscillator), and V 0 << ~ ω is a constant. Sketch the potential seen by the electron for times t < 0 , t = 0 , and t > 0 . Find the probability that the electron is left in the fi rst excited state as a result of this collision. Sketch this probability as a function of v . 29. Consider a particle of mass m and charge e constrained to move on the circumference of a circle of radius a lying in the xy -plane, initially in its ground state. A weak, time-varying, but spatially uniform electric fi eld “pulse” is applied of the form ~ E = E 0 e τ 2 / τ 2 ˆ x which has its peak at t = 0 . In fi
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