6_postulates2-2

Set orthonormal sets are the orthonormal basis set

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Unformatted text preview: avefunction solutions to our model problems coefficients function Use orthonormality of basis set to get coefficients ∗ left multiply function by ∗ and integrate substitute top relation for function ∗ pull constants outside of integral ∗ orthonormality , Recipe for coefficients ∗ ∗ If Postulate 4 is an eigenfunction of , then If . Then , ∗ ∗ ∗ ∑ is not an eigenfunction of , then . Then Any function can be written as a sum of coefficients multiplying orthonormal set. ∗ ∗ ∑ and ∗ ∗ , ∗ ∗ , More complicated, find coefficients Look at this from another perspective If and is an eigenfunction of , then ∗ ∗ , . Then ∗ ∗ ∗ , 0 If is not an eigenfunction of , then ∑ ∑∗ . Then ∗ no variation ∑ and ∑ , ∗ ∗ ∑ , and not zero generally, variation 9 9/16/2013 Postulate 5 The wave function, or state function, of a system evolves in time according to the Ψ, time-dependent Schrödinger equation Ψ , . energy operator is the Hamiltonian operator + kinetic potential energy energy When is independent of time, Ψ ,, / Stationary states. problem reduces to This is an eigenvalue problem. Time Independent Schrödinger Equation Degeneracy 3D Particle-in-Box abcL a cube , , 8 , , 1,2, … 6-fold degenerate (2,1,1) (1,2,1) - za20 za21 za22 zb20 zb21 zb22 za20 za21 za22 zb20 zb21 zb22 (1,1,2) - - za20 za21 za22 zb20 zb21 zb22 Degeneracy and H atom 10 9/16/2013 Degeneracy and H atom 3d typical representations our solutions 3p Any linear combination of degenerate wavefunction solutions is also a legitimate representation of the system. Quantum Mechanics homework: “It's all fun and games until someone loses an i ” Quantum Mechanics homework: “They're just about finished, they just have to dot the i's and cross the h's” 11...
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