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ps2 - Quantum Mechanics Problem Sheet 2(Individual help...

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Quantum Mechanics Problem Sheet 2 (Individual help with these problems is available in the workshop on Friday, 27 Jan 2006. Solutions should be handed in for marking at the start of the lecture on Tuesday, 31 Jan 2006.) 1. Consider a particle that is confined in a one-dimensional box, ie in a potential V ( x ) = 0 for 0 x L for x < 0 and x > L . (a) Determine the solutions φ n ( x ) of the stationary Schr¨ odinger equation for this problem. Make sure that you have normalized them correctly. (You can in principle copy the derivation from your lecture notes, but make sure you can reproduce this derivation for yourself. The particle in a 1D box is one of the simplest problems in quantum mechanics, and thus it is often built on in other problems and it is also often asked in exams!) (b) Calculate the energy eigenvalue E n corresponding to φ n ( x ). (c) Use the results of (a) and (b) and write down the complete time-dependent wave function ψ n ( x, t ) for the n th stationary state in this potential.
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