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Unformatted text preview: Quantum Mechanics Problem Sheet 3 These problems will be discussed in the practice session on Friday, 3 Feb 2006) 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 . Its initial wave function at time t = 0 is ψ ( x, 0) = N x- L 2 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 φ n ( x ) in the box, ψ ( x, 0) = ∞ X n =1 c n φ n ( x ) , and calculate the expansion coefficients c n . Check your result by sketching ψ ( x, 0), φ 1 ( x ) , and φ 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 φ n ( x ) , you don’t need to re-derive them — just quote the result.) (c) Use the result of (b) to determine the time-dependent wave function ψ ( x, t ) of the particle....
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- Spring '06
- mechanics, wave function