| plot dirac delta function? [message #49366] |
Wed, 19 July 2006 03:30  |
Nic
Messages: 5
Registered: June 2006
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Junior Member |
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Hi all!
I am new to IDL and learning how to do non analytic plots such as a
dirac delta function or a finite square well. Does anybody have ideas
of what tools I should use or keywords to get started in plotting
these?
thank you
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| Re: plot dirac delta function? [message #49517 is a reply to message #49366] |
Sun, 30 July 2006 07:32  |
James Kuyper
Messages: 425
Registered: March 2000
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Senior Member |
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swingnut@gmail.com wrote:
> kuyper@wizard.net wrote:
...
>> No, it is not. I'm very well versed in the use of the dirac delta
>> function in physics, and the value of delta(0) is never used in any
>> meaningful sense. Any equation which attempts to make use of the value
>> at zero is meaningless. The dirac delta function only becomes
>> meaningful after you've integrated over it.
>
> FYI: the physical meaning of delta is used ALL the damn time when
I was not referring to the physical meaning of delta, which is that it
represents a point source. I was referring specifically to the actual
value of the Dirac delta function at 0; any equation that involves
evaluating that function at that point, rather than integrating over
it, to calculate a physically meaningful quantity is an error.
> you're talking about the spatial distribution of point particles (think
> electrons and other extensionless subatomic particles here). If the
> particle has no extension, as is generally believed to be true for,
Yes, it's considered quite likely that quarks and leptons are true
point particles, which means that their physical extent is accurately
described by a dirac delta function. However, any physically meaningful
quantity you can calculate is not directly related to the value of that
function at its center, but is instead calcuated directly or
indirectlly from an integral over that function.
> e.g. electrons due to quantum considerations (classical radius of the
> electron and arguments like that), then the only way to describe a mass
> distribution is by summing up a bunch of things that are zero except
> for at a single point in "configuration space" with no physical
> extension. Following the logic, in any phase space, when the object's
> parameters are point values in that space, you get the same behavior
> for those parameters: a Dirac for the object's state. To say that
> delta(x-a) just means that the particle is at x=a in that phase space;
> x=0 refers to the origin of the phase space. These things are used all
> the time or underly other calculations that are used all the time. If
> you are using statistical mechanics at all, you should be seeing this
> regularly.
I haven't done any statistical mechanics in more than a decde, but when
I was studying it, we used dirac delta functions all over the place.
It's a very useful concept, and I've never denied that fact. But we
never used the value of that fucntion at zero for any meaningful
purpose. It only became meaningful after integration.
> Another physical application is as the Green's function corresponding
> to Guass's law for a point charge, the mathematics of which gets
> quickly generalized for minimum variance packet in quantum mech. The
> fun just never stops.
Again, that's an example where a Dirac delta function is a very useful
tool - but only inside an integral, which is how Green's functions are
always used. It's meaningless without the integral.
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| Re: plot dirac delta function? [message #49518 is a reply to message #49366] |
Sun, 30 July 2006 07:18 |
James Kuyper
Messages: 425
Registered: March 2000
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Senior Member |
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swingnut@gmail.com wrote:
> Ah, I see we're about to head off into the realm of defining functions,
> ever a popular discussion (not the least of which is because I relish
> the opportunity to learn more about it, not being a mathematician). Why
> do you say this? In plasma kinetic theory a while back, the prof give
> us the pop quiz about the value of particle distributions made from
> sums of delta functions. As with the "How do you put an elephant into
> the refrigerator?" test, everyone jumped to the wrong answer since the
> integral of the Dirac delta is 1. The function wasn't a sum of
> integrals, its a sum of deltas, so the value of the distribution at a
> particle's parameters in the phase space we were working in is
> infinity.
That's just a test question about a intrinsically meaningless
intermediate step in a calculation. The only thing you can ever
meaningfully do with such a particle distribution function is integrate
over it, one way or another. The value of such a function at a given
point is meaningless, except in those places where it's zero, in which
case it's meaningful only as a simplified approximation to reality.
It's value at any point where it's non-zero is not a measurable
quantity. Can you show me any actual calculations where the value of a
dirac delta function at 0 is actually used to calculate something
meaningful, without being integrated over?
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| Re: plot dirac delta function? [message #49520 is a reply to message #49406] |
Sat, 29 July 2006 22:27 |
swingnut
Messages: 30
Registered: September 2005
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Member |
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kuyper@wizard.net wrote:
> swingnut@gmail.com wrote:
>> FYI, while the definition can be approached a number of equivalent
>> ways, the value of the Dirac IS "well-defined" at delta(x).
>> Technically, it's "well-defined" at the value such that its argument is
>> zero (which here is x=0).
>>
>> The value is indeed infinity. At least, that's how it's used in
>> physics.
>
> No, it is not. I'm very well versed in the use of the dirac delta
> function in physics, and the value of delta(0) is never used in any
> meaningful sense. Any equation which attempts to make use of the value
> at zero is meaningless. The dirac delta function only becomes
> meaningful after you've integrated over it.
FYI: the physical meaning of delta is used ALL the damn time when
you're talking about the spatial distribution of point particles (think
electrons and other extensionless subatomic particles here). If the
particle has no extension, as is generally believed to be true for,
e.g. electrons due to quantum considerations (classical radius of the
electron and arguments like that), then the only way to describe a mass
distribution is by summing up a bunch of things that are zero except
for at a single point in "configuration space" with no physical
extension. Following the logic, in any phase space, when the object's
parameters are point values in that space, you get the same behavior
for those parameters: a Dirac for the object's state. To say that
delta(x-a) just means that the particle is at x=a in that phase space;
x=0 refers to the origin of the phase space. These things are used all
the time or underly other calculations that are used all the time. If
you are using statistical mechanics at all, you should be seeing this
regularly.
Another physical application is as the Green's function corresponding
to Guass's law for a point charge, the mathematics of which gets
quickly generalized for minimum variance packet in quantum mech. The
fun just never stops.
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