385lec10 - PHYS 385 Lecture 10 Potential wells in one...

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Unformatted text preview: PHYS 385 Lecture 10 - Potential wells in one dimension 10 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 10 - Potential wells in one dimension What's important : • step potential in 1D • square well in 1 D Text : Gasiorowicz, Chap. 5 Step potential Next to the particle-in-a-box, with infinitely hard walls, the simplest potential is the step potential: V ( x ) V o x Here, the time-independent Shrödinger equation in one dimension ! h 2 2 m d 2 u ( x ) dx 2 + V ( x ) u ( x ) = E u ( x ) or d 2 u ( x ) dx 2 + 2 m h 2 [ E ! V ( x )] u ( x ) = (1) has a particularly simple form: For x < 0, d 2 u ( x ) dx 2 + k 2 u ( x ) = with k 2 = 2 mE / h 2 , (2) so that u ( x ) ~ exp(± ikx ) For x > 0, d 2 u ( x ) dx 2 + q 2 u ( x ) = with q 2 = 2 m ( E- V o ) / h 2 , (3) so that u ( x ) ~ exp(± iqx ). To understand what solution corresponds to what, recall our discussion of the probability current in Lec. 6. PHYS 385 Lecture 10 - Potential wells in one dimension 10 - 2 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. j = ! i h 2 m " * # " # x ! " # " * # x $ % & ’ . (4) Because our potential is time independent, then the wavefunction ψ ( x , t ) decomposes into the product u ( x )exp( iEt / h ). Not subject to a derivative, the time part disappears from Eq. (4), leaving j = !...
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385lec10 - PHYS 385 Lecture 10 Potential wells in one...

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