P460 - math concepts1 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 eithers “space”

P460 - math concepts2 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 concepts3 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 concepts4 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 =0 define a function with 2 Hermitian operators A and B  U and V and real because it is positive definite can calculate I in terms of U and V and [U,V]

P460 - math concepts5 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 concepts6 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 concepts7 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 concepts8 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  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 concepts9 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