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Unformatted text preview: Welcome back!! Today: Brief review of Schrdinger Equation and: Nanotechnology: When is small small? Review: Schrdinger equation t t x i t x t x V x t x m = + - ) , ( ) , ( ) , ( ) , ( 2 2 2 2 h h General form in 1D: Solving this equation gives us the matter wave function ( x,t ) for a particle in a potential V ( x,t ) . Many physical situations (e.g. H atom): no time dependence in V! ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x x m = + - h Time independent Schrdinger equation in 1D: With ( x,t ) = ( x ) ( t ), and ( t ) =exp(- iEt/ ) 1. Figure out what V(x) is, for situation given. 2. Guess or look up functional form of solution (x). 3. Plug in to check if s, and all xs drop out, leaving equation involving only bunch of constants; showing that trial solution time independent eq. ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x x m = + - h Review: Recipe to solve Schr. eqn. involving only bunch of constants; showing that trial solution is correct functional form. 4. Figure out what boundary conditions must be to make sense physically. 5. Multiply (x) by time dependence (t)=e-iEt/ to have full solution: (x,t) STILL HAS TIME DEPENDENCE! 6. Figure out values of constants to meet boundary conditions and normalization | (x)| 2 dx =1- ) ( ) ( 2 2 2 2 x E x x m = - h ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x x m = + - h Example: simplest case, free space V(x) = const. Smart choice: constant V(x) V(x) 0! kx A x cos ) ( = E m k = 2 2 2 h Solution: with: kx B x sin ) ( = , or: No boundary conditions not quantized! kx A x cos ) ( = h 2 2 m k 2 = E k p h = So almost have solution, but remember still have to include time dependence: k, and therefore E, can take on any value....
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- Spring '08