General Structure of Wave Mechanics (Ch. 5)

1. General Structure of Wave Mechanics (Ch. 5) • Sections 5-1 to 5-3 review items covered previously • use Hermitian operators to represent observables (H,p,x) • eigenvalues of Hermitian operators are real and give the expectation values • eigenvectors for different eigenvalues are orthogonal and form a complete set of states • any function in the space can be formed from a linear series of the eigenfunctions • some variables are conjugate (position, momentum) and one can transform from one to the other and solve the problem in either’s “space” P460 - math concepts

2. Notation • there is a very compact format (Dirac notation) that is often used • |i> = |ui> = eigenfunction • <c|f> is a dot product between 2 function • |i><j| is an “outer” product (a matrix). For example a rotation between two different basis • if an index is repeated there is an implied sum P460 - math concepts

3. Degeneracy (Ch. 5-4) • If two different eigenfunctions have the same eigenvalue they are degenerate (related to density of states) • any linear combination will have the same eigenvalue • usually pick two linear combinations which are orthogonal • can be other operators which have only some specific linear combinations being eigenfunctions. Choice may depend on this (or on what may break the degeneracy) • example from V=0 P460 - math concepts

4. Degeneracy (Ch. 5-4) • Parity and momentum operators do not commute • and so can’t have the same eigenfunction • two different choices then depend on whether you want an eigenfunction of Parity or of momentum P460 - math concepts

5. Uncertainty Relations (Supplement 5-A) • If two operators do not commute then their uncertainty product is greater then 0 • if they do commute  0 • start from definition of rms and allow shift so the functions have <U>=0 • define a function with 2 Hermitian operators A and B U and V and l real • because it is positive definite • can calculate I in terms of U and V and [U,V] P460 - math concepts

6. Uncertainty Relations (Supplement 5-A) • rearrange • But just the expectation values • can ask what is the minimum of this quantity • use this  “uncertainty” relationship from operators alone P460 - math concepts

7. Uncertainty Relations -- Example • take momentum and position operators • in position space • that x and p don’t commute, and the value of the commutator, tells us directly the uncertainty on their expectation values P460 - math concepts

8. Time Dependence of Operators • the Hamiltonian tells us how the expectation value for an operator changes with time • but know Scrod. Eq. • and the H is Hermitian • and so can rewrite the expectation value P460 - math concepts

9. Time Dependence of Operators II • so in some sense just by looking at the operators (and not necessarily solving S.Eq.) we can see how the expectation values changes. • if A doesn’t depend on t and [H,A]=0  <A> doesn’t change and its observable is a constant of the motion • homework has H(t); let’s first look at H without t-dependence • and look at the t-dependence of the x expectation value P460 - math concepts

10. Time Dependence of Operators III • and look at the t-dependence of the p expectation value • rearrange giving • like you would see in classical physics P460 - math concepts