| Topic: |
Science > Physics |
| User: |
"Lasse" |
| Date: |
09 Jan 2004 06:40:08 AM |
| Object: |
Symmetry, group theory and Differential equations |
Hi, I would like to get some help.
Could somebody point me to a book/books, where I can find something
that wovens the above topics together ?
I am interested of the theoretical foundations of the solving
procedure of Schrödinger equation in condensed matter systems.
I am a layman in this field. I am just in it for fun.
I am interested in the following questions:
1. Why can a differential eqn. like Schrödinger's be treated as an
eigenvalue eqn. ? Can all diff. equations be treated as
eigenvalue-eigenvector problems ?
2. Why can symmetry be used in the solving procedure ? One see people
-expand the the assumed solution in a set of basis functions that have
the same symmetry as the system considered, and solve for the
coefficients as system of linear equations.(Gauss-elimination)
The basis functions of the one-dimesional representaion of the group
that the system belongs to, are the basis functions that are used. Why
? Because the symmtry operators and the hamiltonian operator "commute"
(?????????)Help !
3. How does "variational theory" (and what is it ?) come into the
solving theory of Schrödinger eqn. ?
4. Perturbation theory in quantum mechanics. Is it one of these
none-proven things ? Is it used in mechanics ? I never saw it my
newtonian mechanics class.
When you suggest books, consider that I have the education of an
electronics engineer. No fancy mathematics courses like functional
analysis, etc.
So maybe you have to suggest a book for the basics also....
I don't ask for help finding the books because of lazyness. I don't
have a decent library near me.
Kindest regards,
Lasse
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| User: "Bjoern Feuerbacher" |
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| Title: Re: Symmetry, group theory and Differential equations |
09 Jan 2004 07:36:35 AM |
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Lasse wrote:
[snip]
I am interested in the following questions:
1. Why can a differential eqn. like Schrödinger's be treated as an
eigenvalue eqn. ?
An eigenvalue equation is, essentially, an equation of the type
"linear operator acting on a vector gives the same vector, multiplied
with a constant". The linear operator in Schroedinger's equation is
-hbar^2/2m Laplace + V, the vector is the wave function it is acting on.
(in coordinate space; if you are familiar with the Dirac notation, it
becomes
even more obvious that Schroedinger's (time-independent!) equation is an
eigenvalue equation)
Can all diff. equations be treated as
eigenvalue-eigenvector problems ?
No, only the ones of the type outlined above (for example, a requirement
is that the differential operator has to be linear).
2. Why can symmetry be used in the solving procedure ?
If an equation has a certain symmetry, one can show that the *set* of
its solutions has to have the same symmetry. This in general greatly
simplifies finding the solutions.
One see people
-expand the the assumed solution in a set of basis functions that have
the same symmetry as the system considered,
Well, usually the basis functions have *not* the same symmetry as the
system... (for example, not all wave functions for the H atom are
spherically symmetric!)
and solve for the
coefficients as system of linear equations.(Gauss-elimination)
The basis functions of the one-dimesional representaion of the group
that the system belongs to, are the basis functions that are used. Why?
Sorry, I'm not very familiar with this...
Because the symmtry operators and the hamiltonian operator "commute"
(?????????)Help !
When one can find operators for certain symmetries which commute with
the Hamiltonian, one usually chooses the wave functions to be
eigenfunctions to these operators, too. Again, this is done because it
usually simplifies finding the solutions (example: for solving the H
atom, it is *very* helpful to try to find a wave function which is an
eigenstate for angular momentum, too - because then one already knows
that the wave function has to be a multiple of a spherical harmonic!)
3. How does "variational theory" (and what is it ?) come into the
solving theory of Schrödinger eqn. ?
AFAIK, the Schroedinger equation usually can't be "solved" by the
variational theory - one can only find *approximate* solutions. What it
is? Well, the idea (in the case of the Schroedinger equation) is to use
some "test" functions and calculate the expectation value for the energy
for them. It is easy to prove that these expectation values have to be
greater or equal to the ground state energy, hence by looking for the
lowest possible energy among all of the test functions, one gets an
upper bound for the ground state energy. (this is only the tip of the
iceberg, but I think you get the general idea...)
4. Perturbation theory in quantum mechanics. Is it one of these
none-proven things ?
I think a physicist would say that it is proven, but a mathematician
would disagree. ;-)
Short answer: it works, often with astounding accuracy.
Is it used in mechanics?
Yes, although in completely different form. Example: for studying the
influence of the other planets when trying to calculate the orbit of one
planet around its sun.
I never saw it my newtonian mechanics class.
Well, I think in many mechanics classes, this is considered to be an
advanced topic.
When you suggest books, consider that I have the education of an
electronics engineer. No fancy mathematics courses like functional
analysis, etc.
But apparently you have some experience with differential equations and
matrices, don't you? What about the general definition of "vector
space"? What about the notion of "differential operators"?
[snip]
Bye,
Bjoern
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| User: "Herman Jurjus" |
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| Title: Re: Symmetry, group theory and Differential equations |
09 Jan 2004 08:35:09 AM |
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"Bjoern Feuerbacher" <bfeuerba@ix.urz.uni-heidelberg.de> wrote in message news:3FFEAE63.69E0AAC7@ix.urz.uni-heidelberg.de...
Lasse wrote:
[snip]
2. Why can symmetry be used in the solving procedure ?
If an equation has a certain symmetry, one can show that the *set* of
its solutions has to have the same symmetry. This in general greatly
simplifies finding the solutions.
You might be interested in:
Peter Olver, Applications of Lie Groups to Differential Equations,
Second Edition, Springer-Verlag, 1993.
[snip]
When you suggest books, consider that I have the education of an
electronics engineer. No fancy mathematics courses like functional
analysis, etc.
O, then Olver probably is not for you (or perhaps it is...).
But in case anyone else is interested, i give the reference anyway.
Regards,
Herman Jurjus
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| User: "Lasse" |
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| Title: Re: Symmetry, group theory and Differential equations |
11 Jan 2004 11:14:50 AM |
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Hi all, and thanks Bjoern and Herman
2. Why can symmetry be used in the solving procedure ?
If an equation has a certain symmetry, one can show that the *set* of
its solutions has to have the same symmetry....
Do you know the name of the person who proved that ?
One see people
-expand the the assumed solution in a set of basis functions that have
the same symmetry as the system considered,
Well, usually the basis functions have *not* the same symmetry as the
system... (for example, not all wave functions for the H atom are
spherically symmetric!)
and solve for the
coefficients as system of linear equations.(Gauss-elimination)
The basis functions of the one-dimesional representaion of the group
that the system belongs to, are the basis functions that are used. Why?
Sorry, I'm not very familiar with this...
Because the symmtry operators and the hamiltonian operator "commute"
(?????????)Help !
When one can find operators for certain symmetries which commute with
the Hamiltonian, one usually chooses the wave functions to be
eigenfunctions to these operators, too. Again, this is done because it
usually simplifies finding the solutions (example: for solving the H
atom, it is *very* helpful to try to find a wave function which is an
eigenstate for angular momentum, too - because then one already knows
that the wave function has to be a multiple of a spherical harmonic!)
Any references about the theory of the commutation ?
3. How does "variational theory" (and what is it ?) come into the
solving theory of Schrödinger eqn. ?
AFAIK,
AFAIK ?? You've lost me.
the Schroedinger equation usually can't be "solved" by the
variational theory - one can only find *approximate* solutions. What it
is? Well, the idea (in the case of the Schroedinger equation) is to use
some "test" functions and calculate the expectation value for the energy
for them. It is easy to prove that these expectation values have to be
greater or equal to the ground state energy, hence by looking for the
lowest possible energy among all of the test functions, one gets an
upper bound for the ground state energy. (this is only the tip of the
iceberg, but I think you get the general idea...)
Easy introductory text?
4. Perturbation theory in quantum mechanics. Is it one of these
none-proven things ?
Well, I think in many mechanics classes, this is considered to be an
advanced topic.
When you suggest books, consider that I have the education of an
electronics engineer. No fancy mathematics courses like functional
analysis, etc.
But apparently you have some experience with differential equations and
matrices, don't you? What about the general definition of "vector
space"? What about the notion of "differential operators"?
Sure Bjoern, I've read some intriductory quantum mechanics books, and
they often have a section of functions whicj are orthogonal to each
other, the concept of Hilbert space. A differntial operator must be
that I can write
a diff. eqn. like (d^2/dt^2)y + y=0, like (D^2 +1)*y=0.
I've calculated these type of equations in the calculus courses. But I
have no clue of the behindlying theory. It is the theory I want to dig
into....
regards,
Lasse
.
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| User: "Herman Jurjus" |
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| Title: Re: Symmetry, group theory and Differential equations |
12 Jan 2004 04:41:34 AM |
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"Lasse" <Lasse.Karagiannis@swipnet.se> wrote in message news:6aae2a48.0401110914.7dfcea3d@posting.google.com...
[snip]
When one can find operators for certain symmetries which commute with
the Hamiltonian, one usually chooses the wave functions to be
eigenfunctions to these operators, too. Again, this is done because it
usually simplifies finding the solutions (example: for solving the H
atom, it is *very* helpful to try to find a wave function which is an
eigenstate for angular momentum, too - because then one already knows
that the wave function has to be a multiple of a spherical harmonic!)
Any references about the theory of the commutation ?
Not sure if this is what you want to know, here, but
perhaps you're interested in the 'Poisson bracket' (see Google).
[snip]
I've calculated these type of equations in the calculus courses. But I
have no clue of the behindlying theory. It is the theory I want to dig
into....
Then i'd say: Lie groups, Lie algebras and the Peter Olver book.
But don't expect easy goings, here. Most of the texts are either
focussed on theory, or on applications, without making the
connection between the two explicit.
BTW, if you want to dig into theory, functional analysis is
quite worthwhile (and doesn't need to be *that* hard).
Cheers,
Herman Jurjus
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| User: "Bjoern Feuerbacher" |
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| Title: Re: Symmetry, group theory and Differential equations |
12 Jan 2004 04:36:45 AM |
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Lasse wrote:
Hi all, and thanks Bjoern and Herman
2. Why can symmetry be used in the solving procedure ?
If an equation has a certain symmetry, one can show that the *set* of
its solutions has to have the same symmetry....
Do you know the name of the person who proved that ?
No, sorry. That type of proof can be found in many standard book of
Quantum Mechanics, but AFAIK, there is no particular name associated
with it.
I'll give you a simple example here. Look at the one-dimensional,
time-independent Schroedinger equation
-hbar^2/2m psi''(x) + V(x) psi(x) = E psi(x)
and imagine that you've got a potential which is an even function of x:
V(-x) = V(x).
Now simply change x to -x everywhere in the equation:
-hbar^2/2m psi''(x) + V(-x) psi(-x) = E psi(-x)
Using the fact that the potential is even, this becomes
-hbar^2/2m psi''(x) + V(x) psi(-x) = E psi(-x)
Hence obviously psi(-x) is also a solution to this equation. This leaves
you with two possibilities:
1) There is no degenaracy, psi(x) is the only solution - then,
obviously, psi(-x) has to be identical to psi(x), or, in other words,
the only solution function has the same symmetry as the differential
equation itself.
2) There is a degeneracy, psi(-x) is different from psi(x). But then at
least the *set* of all solutions has the same symmetry as the
differential equation itself: choosing an arbitrary solution and turning
x into -x gives you another solution.
[snip]
Because the symmtry operators and the hamiltonian operator "commute"
(?????????)Help !
When one can find operators for certain symmetries which commute with
the Hamiltonian, one usually chooses the wave functions to be
eigenfunctions to these operators, too. Again, this is done because it
usually simplifies finding the solutions (example: for solving the H
atom, it is *very* helpful to try to find a wave function which is an
eigenstate for angular momentum, too - because then one already knows
that the wave function has to be a multiple of a spherical harmonic!)
Any references about the theory of the commutation ?
I don't think there is an entire "theory" about commutation; but lots of
information about commutation can be found in most books on Quantum
Mechanics or Linear Algebra.
If you have got any specific questions, feel free to ask!
3. How does "variational theory" (and what is it ?) come into the
solving theory of Schrödinger eqn. ?
AFAIK,
AFAIK ?? You've lost me.
Sorry. As Far As I Know. (a standard abbreviation used on usenet -
well, at least AFAIK ;-) )
the Schroedinger equation usually can't be "solved" by the
variational theory - one can only find *approximate* solutions. What it
is? Well, the idea (in the case of the Schroedinger equation) is to use
some "test" functions and calculate the expectation value for the energy
for them. It is easy to prove that these expectation values have to be
greater or equal to the ground state energy, hence by looking for the
lowest possible energy among all of the test functions, one gets an
upper bound for the ground state energy. (this is only the tip of the
iceberg, but I think you get the general idea...)
Easy introductory text?
Not in English, sorry. I've studied physics in Germany and have used
mostly German books...
The only English book on QM I can think of in the moment are by Sakurai
and by Dirac - but both are not by any means "introductory texts"... :-(
If you want to see the above mentioned proof that the expectation value
for the energy is always at least as great as the ground state energy, I
can give it to you - it's very short, requires only a few lines.
However, it requires some familarity with the concept of "complete
systems of functions". But judging from what you say below, you appear
to know this concept...
[snip]
When you suggest books, consider that I have the education of an
electronics engineer. No fancy mathematics courses like functional
analysis, etc.
But apparently you have some experience with differential equations and
matrices, don't you? What about the general definition of "vector
space"? What about the notion of "differential operators"?
Sure Bjoern, I've read some intriductory quantum mechanics books, and
they often have a section of functions whicj are orthogonal to each
other, the concept of Hilbert space.
Oh, nice! So you have some of the basic knowledge for Quantum Mechanics
available.
A differntial operator must be that I can write
a diff. eqn. like (d^2/dt^2)y + y=0, like (D^2 +1)*y=0.
Right. (although I wouldn't write "*" in the last equation)
I've calculated these type of equations in the calculus courses. But I
have no clue of the behindlying theory. It is the theory I want to dig
into....
So you need information about mathematics, not about physics?
Have you seen the post by Herman Jurjus, where he recommended a book?
Bye,
Bjoern
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