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Lecture_13 - PHYS PHYS342 Fall2011 Lecture 13: Schrdingers...

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HYS 342 PHYS 342 Fall 2011 ecture 13: Schrödinger’s Equation in 2 dimensions Lecture 13: Schrödinger s Equation in 2 dimensions Ron Reifenberger Birck Nanotechnology Center Purdue University Lecture 13 1
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m i l s tm xists ft i h t If we know the wavefunction, we know everything it is possible to know For every dynamical system, there existsaw a vefunction Ψ that is a continuous, square-integrable*, single-valued function of the coordinates of all the particles and of time. If you know Ψ , all possible predictions about the physical properties of the system can be obtained. “The coordinates of all the particles”? For a single particle moving in one dimension:   , x t   x For a single particle in one dimension: or a single particle moving in three dimensions: , t r For a single particle in a superposition of two quantum states, a and b:   ,( , ) ( , ) ab x tA x t B x t   For a single particle moving in three dimensions: For two particles moving in three dimensions:   12 ,, t rr    * Square-integrable means that the normalization integral is finite 2
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Real space vs. k-space () x real space x (m) 1 ikx kx e d x   1 ikx x ke dk   2  ) o r g k 2  pace k (m -1 ) ko k-space or wavevector space or eciprocal space 3 reciprocal space or inverse space
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A particle of mass m e in a rigid 2-D square box 2 2 2 ) e m xy E  y U(x,y)= m e   2 2 2 ( , 00 , ( , ) 0 Uxy E for x L and y L U x y x y   L ( ,y) U(x,y)=0 2 2 2 2 2 2 e we then have m E L x 22 (, ) ()() ( , ) x dXx Suppose x y XxY y then y    L x L square well () ( ( ) ( ) ) Yy xd x dY y X x m e 2 2 ( ) y dy 4
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2 2 2 e m E now becomes   2 2 2 2 2 2 2 2 2 ()() () e m EXxYy y dY y Xx dy x dXx Yy dx    2 2 2 2 2 2 11 e m E hi ti i f t d f
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This note was uploaded on 12/08/2011 for the course PHYS 342 taught by Professor Staff during the Fall '08 term at Purdue.

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Lecture_13 - PHYS PHYS342 Fall2011 Lecture 13: Schrdingers...

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