385lec10

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 = !...
View Full Document

## This note was uploaded on 09/07/2009 for the course PHYS 385 taught by Professor Davidboal during the Spring '09 term at Simon Fraser.

### Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online