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# ps05 - Quantum Mechanics Problem Sheet 5(Solutions are to...

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Quantum Mechanics Problem Sheet 5 (Solutions are to be handed in for marking before the lecture on Monday, 21 Feb 2005) 1. φ ( x ) is any of the odd-parity solutions to the stationary Schr¨ odinger equation for a Fnite square well V ( x ) = - V 0 for - L 2 < x < L 2 , 0 for | x | > L 2 . (a) Show that the solutions for the bound states ( E < 0) can be written as φ ( x ) = F e - κx for x > L 2 B sin( kx ) for 0 x L 2 - φ ( - x ) for x < 0 , where F and B are constants, κ - 2 mE/ ¯ h , and k p 2 m ( E + V 0 ) / ¯ h . (b) Sketch the form of the lowest-energy odd-parity wave function φ ( x ). (c) Using the continuity conditions on φ ( x ) and d φ/ d x , show that y = - z cot z y 2 + z 2 = R 2 , where z kL/ 2, y κL/ 2, and R 2 mV 0 L/ (2¯ h ). (Because of the symmetry of the wave function you need to consider the continuity conditions only at x = - L/ 2 or at x = L/ 2 , but not both. If you consider them at both points you just get twice the

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ps05 - Quantum Mechanics Problem Sheet 5(Solutions are to...

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