General Structure of Wave Mechanics (Ch. 5)

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# General Structure of Wave Mechanics (Ch. 5) - PowerPoint PPT Presentation

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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Presentation Transcript
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”

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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

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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

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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

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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]

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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

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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

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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

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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

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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

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