**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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