4thQuiz - PHY4221 Quantum Mechanics I(Fall 2004 Quiz...

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PHY4221 Quantum Mechanics I (Fall 2004) Quiz 4 (Answers) Since n φ is the normalized n th eigen-function of this system, so we have that: ( 29 2 cos 1,3,5, 2 2 sin 2,4,6, 0 2 n n x n L L L x n x x n L L L x π π φ = ××× ÷ = = ××× ÷ (1) Thus we know that at time t =0, the wave function of this particle is: ( 29 ( 29 ( 29 1 3 2 3 ,0 3 3 cos cos x x x x x L L L π π ψ φ φ = + = + ÷ (2) Then, with the propagator we could construct the wave function at any time: ( 29 ( 29 ( 29 , , ; ,0 ,0 x t dx K x t x x ψ ψ -∞ = (3) In equation (3), we have the definition of the propagator as following: ( 29 ( 29 ( 29 , ; ,0 exp n n n n E t K x t x x x i φ φ = - ÷ h (4) So finally we get: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 /2 / 2 /2 /2 /2 1 3 /2 1 , exp ,0 exp ,0 exp ,0 exp 3 exp n n n n L n n n L n L n n n L n L n n n L n E t x t dx x x i x E t i dx x x x E t i x dx x x E t i x dx x x x E t i ψ φ φ ψ φ φ ψ φ
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