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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

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general structure of wave mechanics ch 5
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

notation
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

degeneracy ch 5 4
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

degeneracy ch 5 41
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

uncertainty relations supplement 5 a
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

uncertainty relations supplement 5 a1
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

uncertainty relations example
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

time dependence of operators
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

time dependence of operators ii
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

time dependence of operators iii
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