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PrelExam - PHY4211 Quantum Mechanics I Fall Term of 2005...

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PHY4211 Quantum Mechanics I Fall Term of 2005 Preliminary Examination (A Possible Set of Suggested Solution) Question 1: The time-independent Schrodinger equation is: () () 2 2 2 Vx E x m ψψ ⎡⎤ −∇ + = ⎢⎥ ⎣⎦ = , (1) and the potential is simply: () 0 2 else L x = . (2) Outside the well, the wave functions vanish; inside the well, they are given by: 2 2 2 0 mE xx ∇+ = = (3) Its solution is: 22 2 2 sin cos 0 sin cos 0s i n c o s 0 2 sin 0 2 cos 0 2 mE mE mE mE AL B L xA xB x Lm E m E L B L mE mE BL ψ ⎡⎤⎡⎤ += =+ ⎢⎥⎢⎥ ⎪⎣ ⎣⎦⎣⎦ ⎬⎨ ⎪⎪ ⎛⎞ = + = ⎜⎟ ⎝⎠ = = == = = , (4) Firstly it is possible that A=0 forever, then to obtain a nontrivial solution to the equation (1) we must require that B≠0 so that: () ( ) 2 2 cos 0 2 1 0,1, 2,3, 2 2 cos 2 1 mE L mE Ln n n x L π = ⇒= + =⋅ + = = , then the renormalized solution is easy to obtain: 2 cos 2 1 0,1, 2, xn x n LL = + = ⋅⋅⋅ . (5a) Similarly other solutions are: 2 sin 2 1,2,3, x n (5b) The corresponding eigen-states are respectively:
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() 2 22 2 12 0,1, 2, 2 En n mL π ⎡⎤ = + = ⋅⋅⋅ ⎢⎥ ⎣⎦ = ; (6a) 22 2 2 2 1, 2, n mL = = = . (6b) This problem could be also solved simply by shifting the range of the potential to, e.g. [0, 2L], and use the well known result from any introductory quantum mechanics textbook. This method will not be
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PrelExam - PHY4211 Quantum Mechanics I Fall Term of 2005...

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