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P460 - Sch. wave eqn.1 Schrodinger Wave Equation Schrodinger equation is the first (and easiest) works for non-relativistic spin-less particles (spin added ad-hoc) guess at form: conserve energy, well-behaved, predictive, consistent with =h/p free particle waves

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P460 - Sch. wave eqn.2 Schrodinger Wave Equation kinetic + potential = “total” energy K + U = E with operator form for momentum and K gives Giving 1D time-dependent SE For 3D:

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P460 - Sch. wave eqn.3 Operators Operators transform one function to another. Some operators have eigenvalues and eigenfunctions In x-space or t-space let p or E be represented by the operator whose eigenvalues are p or E Only some functions are eigenfunctions. Only some values are eigenvalues

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P460 - Sch. wave eqn.4 Interpret wave function as probability amplitude for being in interval dx

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P460 - Sch. wave eqn.5 No forces. V=0 solve Schr. Eq Find average values Example

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P460 - Sch. wave eqn.6

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7 Solving Schrodinger Equation If V(x,t)=v(x) than can separate variables G is separation constant valid any x or t Gives 2 ordinary diff. Eqns.

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P460 - Sch. wave eqn.8 G=E if 2 energy states, interference/oscillation 1D time independent Scrod. Eqn. Solve: know U(x) and boundary conditions want mathematically well-behaved. No discontinuities. Usually except if V=0 or =0 in certain regions

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P460 - Sch. wave eqn.9 Solutions to Schrod Eqn Gives energy eigenvalues and eigenfunctions (wave functions). These are quantum states. Linear combinationsof eigenfunctions are also solutions

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P460 - Sch. wave eqn.10 Solutions to Schrod Eqn Linear combinationsof eigenfunctions are also solutions. Asuume two energies assume know wave function at t=0 at later times the state can oscillate between the two states

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P460 - Sch. wave eqn.11 The normalization of a wave function doesn’t change with time (unless decays). From Griffiths: Use S.E. to substitute for J(x,t) is the probability current. Tells rate at which probability is “flowing” past point x substitute into integral and evaluate The wave function must go to 0 at infinity and so this is equal 0

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