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Unformatted text preview: Quantum Mechanics Problem Sheet 3 (Solutions are to be handed in for marking before the lecture on Monday, 7 Feb 2005) 1. Consider a particle that is confined in a one-dimensional box, ie in a potential V ( x ) = for 0 x L for x < 0 and x > L . The particle is initially distributed uniformly inside the box, so that its wave function at time t = 0 is ( x, 0) = N for 0 x L for x < 0 and x > L . (a) Determine the normalization constant N . (b) Write the initial wave function as a superposition of the stationary states n ( x ) in the box, ( x, 0) = X n =1 c n n ( x ) , and calculate the expansion coefficients c n . Check your result by sketching ( x, 0), 1 ( x ) , and 2 ( x ) and considering the symmetry of these functions around the middle of the box. (If you have understood how one gets the stationary states n ( x ) , you dont need to re-derive them just quote the result from your lecture notes or from the previous problem sheet.) (c) Use the result of (b) to determine the time-dependent wave function...
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