385lec9 - PHYS 385 Lecture 9 - Time-independent Schrdinger...

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PHYS 385 Lecture 9 - Time-independent Schrödinger equation 9 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 9 - Time-independent Schrödinger equation What's important : time-independent Schrödinger equation eigenfunctions and eigenvalues particle in a 1D box Text : Gasiorowicz, Chap. 4 Time-independent Schrödinger equation Let's apply the ideas of eigenfunctions to the time-dependent Schrödinger equation to extract the time-independent Schrödinger equation. In this, and the next several lectures, we continue to work in one dimension. Recall the explicit representation of the Schrödinger equation: i h ( x , t ) t = - h 2 2 m 2 ( x , t ) x 2 + V ( x ) ( x , t ), (1) where we have made the potential energy a function only of position. If an equation can be segregated into parts which depend on only one variable, then a fruitful approach to solving it is the so-called separation of variables approach. As implemented here, that means we should try to write the wavefunction ( x , t ) as a product of two functions, one containing all the position dependence and one containing all the time dependence: ( x , t ) = T ( t )• u ( x ). (2) Aside: not everyone uses this notation; frequently one sees ( x , t ) for the complete wavefunction, and ( x ) for the position dependent part. Substituting (2) into (1), and shuffling some functions to show what derivatives need attention i h u ( x ) dT ( t ) dt = - h 2 2 m T ( t ) d 2 u ( x ) dx 2 + T ( t ) V ( x ) u ( x ). Note how has become d . Dividing by T ( t )• u ( x ) yields i h dT ( t )/ dt T ( t ) = - ( h 2 /2 m )( d 2 u ( x )/ dx 2 ) + V ( x ) u ( x ) u ( x ) (3) Now, the time dependence is completely isolated on the left hand side, and the position dependence is on the right hand side. Since the LHS cannot depend on x , and the RHS cannot depend on t , then NEITHER side can depend on
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385lec9 - PHYS 385 Lecture 9 - Time-independent Schrdinger...

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