PHY 711 Fall 2016 Lecture 17 1 PHY 7 11 Classical Mechanics and Mathematical Methods 111150 AM MWF Olin 107 Plan for Lecture 17 Read Chapter 7 amp Appendices AD Generalization of the one dimensional wave equation ID: 537932
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10/10/2016
PHY 711 Fall 2016 -- Lecture 17
1
PHY
7
11 Classical Mechanics and Mathematical Methods
11-11:50 AM MWF Olin 107
Plan for Lecture 17:
Read Chapter 7 & Appendices A-D
Generalization of the one dimensional wave equation
various mathematical problems and techniques including:
Sturm-
Liouville
equations
Orthogonal function expansions
Green’s functions methodsSlide2
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Eigenvalues and
eigenfunctions
of Sturm-
Liouville
equationsSlide4
In general, there are several techniques to determine the
eigenvalues l
n and eigenfunctions
f
n
(x)
. When it is not
possible to
find the ``exact'' functions, there are several powerful
approximation techniques
. For example, the lowest eigenvalue can be
approximated by
minimizing the function 10/10/2016PHY 711 Fall 2016 -- Lecture 17
4
Variation approximation to lowest eigenvalue
where
is
a variable function which satisfies the
correct boundary values. The ``proof'' of this inequality is
based on the notion that
can
in
principle
be expanded
in terms of the (unknown) exact
eigenfunctions fn(x): where the coefficients Cn can be assumed to be real.Slide5
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Estimation of the lowest eigenvalue – continued:
From the
eigenfunction
equation, we know that
It follows that:Slide6
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Rayleigh-Ritz method of estimating the lowest eigenvalueSlide7
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Rayleigh-Ritz method of estimating the lowest eigenvalueSlide8
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Comment on “completeness” of set of
eigenfunctions
It can be shown that for any reasonable function
h(x)
,
defined within the
interval
a < x <b
,
we can expand that function as a linear
combination of
the eigenfunctions fn(x)
These ideas lead to the notion that the set of
eigenfunctions
fn
(x)
form
a ``complete'' set in the sense of ``spanning'' the space
of all
functions in the
interval
a
< x <b, as summarized by the statement:Slide9
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Green’s function solution methods
Recall:Slide10
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Solution to inhomogeneous problem by using Green’s functions
Solution to homogeneous problemSlide11
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Example Sturm-
Liouville
problem:Slide12
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(after some algebra)Slide16
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General method of constructing Green’s functions using homogeneous solutionSlide17
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