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08wave2 - Solving Schrodinger Equation If V(x,t)=v(x than...

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P460 - Sch. wave eqn. 1 Solving Schrodinger Equation If V(x,t)=v(x) than can separate variables [ ] G V i x t x V t x t x assume i x V dt d i mdx d dt d dx d m t x t x m = = + = + = Ψ = Ψ + Ψ Ψ φ φ ψ φ ψ ψ ψ ψ ψ φ φ φ ψ h h h h h h 1 ) ( ) ( ) ( ) ( ) ( ) , ( ) ( 2 2 2 2 2 2 2 2 2 2 2 ) , ( 2 G is separation constant valid any x or t Gives 2 ordinary diff. Eqns.
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P460 - Sch. wave eqn. 2 Solutions to Schrod Eqn Gives energy eigenvalues and eigenfunctions (wave functions). These are quantum states. Linear combinations of eigenfunctions are also solutions. For discrete solutions 1 ...... ) , ( 2 * / 2 2 1 1 = = Ψ Ψ Ψ = Ψ Ψ Ψ + Ψ = Ψ i ij j i i t iE i i n n c normalized dx orthogonal e each c c c t x i δ ψ h If H Hermitian
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P460 - Sch. wave eqn. 3 h h / ) ( iGt dt d i e t G = = φ φ φ G=E if 2 energy states, interference/oscillation h h / 2 2 2 ) ( ) , ( 2 iEt e x t x E V mdx d = Ψ = + ψ ψ ψ ψ 1D time independent Scrod. Eqn. Solve: know U(x) and boundary conditions want mathematically well-behaved. Do not want: = = = 2 2 ) ( x x x ψ ψ ψ No discontinuities. Usually except if V=0 or ψ =0 in certain regions
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P460 - Sch. wave eqn. 4 Linear Operators Operator converts one function into another an operator is linear if (to see, substitute in a function) linear suppositions of eigenfunctions also solution if operator is linear……use “Linear algebra” concepts. Often use linear algebra to solve non- linear functions…. dx x f d x Of x x f x Of ) ( ) ( ) ( ) ( 2 = + = dx d O ex linear x Of x Of x f x f O if = + = + : ) ( ) ( )] ( ) ( [ 2 1 2 1
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P460 - Sch. wave eqn. 5 Solutions to Schrod Eqn Depending on conditions, can have either
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