PHY 711 Fall 2014 Lecture 20 1 PHY 7 11 Classical Mechanics and Mathematical Methods 101050 AM MWF Olin 103 Plan for Lecture 20 Summary of mathematical methods Sturm Liouville equations ID: 272909
Download Presentation The PPT/PDF document "10/10/2014" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
1
PHY
7
11 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 20:
Summary of mathematical methods
Sturm-
Liouville
equations
Green’s function methods
Laplace transform
Contour integrationSlide2
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
2Slide3
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
3
Sturm-
Liouville
equation (assume all functions and constants are real):
We can prove as a general property of the Sturm-
Liouville
system
,
the
eigenfunctions
f
n
(x)
are
orthogonalSlide4
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
4
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:Slide5
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
5
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
where
is
a variable function which satisfies thecorrect boundary values. The ``proof'' of this inequality isbased on the notion that can
in principle be expandedin terms of the (unknown) exact eigenfunctions
f
n
(x
):
where
the coefficients
C
n can be
assumed
to be
real.Slide6
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
6
Estimation of the lowest eigenvalue – continued:
From the
eigenfunction
equation, we know that
It follows that:Slide7
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
7
Rayleigh-Ritz method of estimating the lowest eigenvalueSlide8
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
8
Recap: Sturm-
Liouville
equation:Slide9
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
9Slide10
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
10Slide11
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
11Slide12
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
12Slide13
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
13Slide14
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
14Slide15
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
15Slide16
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
16Slide17
10/10/2014
PHY 711 Fall 2014 -- Lecture 20
17