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class30 - Welcome back Today Brief review of Schrödinger...

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Unformatted text preview: Welcome back!! Today: Brief review of Schrödinger Equation and: Nanotechnology: When is small ‘small’? Review: Schrödinger equation t t x i t x t x V x t x m ∂ Ψ ∂ = Ψ + ∂ Ψ ∂- ) , ( ) , ( ) , ( ) , ( 2 2 2 2 h h General form in 1D: Solving this equation gives us the matter wave function Ψ ( 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 ψ ψ ψ = + ∂ ∂- h Time independent Schrödinger equation in 1D: With Ψ ( x,t ) = ψ ( x ) Φ ( t ), and Φ ( t ) =exp(- iEt/ ħ ) 1. Figure out what V(x) is, for situation given. 2. Guess or look up functional form of solution ψ (x). 3. Plug in to check if ψ ’s, and all x’s 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 ψ ψ ψ = + ∂ ∂- 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 ψ (x) by time dependence Φ (t)=e-iEt/ ħ to have full solution: Ψ (x,t) STILL HAS TIME DEPENDENCE! 6. Figure out values of constants to meet boundary conditions and normalization | Ψ (x)| 2 dx =1- ∞ ∞ ) ( ) ( 2 2 2 2 x E x x m ψ ψ = ∂ ∂- h ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x x m ψ ψ ψ = + ∂ ∂- h Example: simplest case, free space V(x) = const. Smart choice: constant V(x) V(x) ≡ 0! kx A x cos ) ( = ψ E m k = 2 2 2 h Solution: with: kx B x sin ) ( = ψ , or: No boundary conditions not quantized! kx A x cos ) ( = ψ 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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class30 - Welcome back Today Brief review of Schrödinger...

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