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HW5 - PHY4221 Quantum Mechanics(Fall Term Of 2004 Problem...

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Unformatted text preview: PHY4221 Quantum Mechanics (Fall Term Of 2004) Problem Set 5 (Answers) Question 1: As following: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 , , , , , , 2 , , , , , , , , 2 , , , , 2 2 , 2 J x t x t x t x t x t x t t x t mi x x x x t x t x t x t t t x t x t x t x t mi x x x t V x t x t V x t mi x mi x x t mi ρ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ∂ ∂ ∂ ∂ ∂ ∂ + = +- ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂ +- ∂ ∂ - ∂ ∂ =- + +- + ∂ ∂ ∂ + h h h h h ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 , , , , , , , 2 2 , , , , 2 x t x t x t x x x t x t x t x t mi x mi x x t x t x t x t mi x x ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ∂- ∂ ∂ ∂ ∂ =- ∂ ∂ ∂ ∂ +- ∂ ∂ = h h h (1) In above the definition of time-dependent Schrodinger Equation is used and so is the assumption that the potential V is a real function of x and t . Question 2: It is obvious that in the region of zero potential the wave function could be written as: ( 29 ikx ikx x Ae Be ψ- = + (1) The wave function in the region of potential V is given by: ( 29 i x i x V x Ce De κ κ ψ- = + (2) In above the k and κ are defined as following: ( 29 2 2 , m E V mE k κ- = = h h (3) In case (b), the zero-potential region should only have rightward-going wave since there is no...
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HW5 - PHY4221 Quantum Mechanics(Fall Term Of 2004 Problem...

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