Complex wavefunction.

  • #26
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I do use Boas in fact, and have come across the very quote before. But as physicists we are not only interested in the result; often the method used to get there is equally as important.

In fact, physicists usually use particularly elegant descriptions, and often they use elegance as a factor for choosing a particular description.

As Dirac said:

This result is too beautiful to be false; it is more important to have beauty in one's equations than to have them fit experiment.
Anyway, my interest in physics has arisen primarily out of the beauty/symmetry of the mathematical descriptions that are employed. Being a first year physics undergraduate student, we are currently going through Electromagnetism, and the course should culminate in Maxwell's equations at the end of the term. Until we get to Maxwell's equations, all the various other equations we have for dipoles, induction, magnetic field etc. seem so haphazard and random.

I have been lucky enough to have read about (and studied) Maxwell's equations (and the underlying mathematics; i.e. vector calculus) indepedently about a year ago, and so I know of the beauty that underlies all of electromagnetism. In fact, as I keep on reading, I learn of the ever more beautiful descriptions of EM (such as differential forms and the Maxwell tensor).

This is a case where my interest has been spurned on primarily by the beauty of the mathematics. Similarly for general relativity - my fascination for it grows with the beauty of the mathematics of it. Unfortunately, my expertise on pseudo-Lorentzian n-manifolds aren't great at the moment, and that's one thing I wish to remedy through extra reading in the next year or two.

But, as I say, the mathematics is not only always a means, but sometimes the end too. To describe the universe with beautiful mathematics (and then use LaTeX to typeset it!)

Masud.

[1] P. A. M. Dirac "The evolution of the Physicist's Picture of Nature" Scientific American 208 (5) (1963)
 
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  • #27
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Raparicio said:
There is another complex space with scalar and vectors?
We have a complex vector space equipped with an inner product, complete with respect to the norm defined by the inner product (i.e. the Hilbert space). Elements of it are vectors. And since it is a complex space, each vector can be multplied by a scalar complex number.

Masud.
 
  • #28
Galileo
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dextercioby said:
Mathematics is an ESSENTIAL tool for a theoretical physicst...

Daniel.

P.S.I would have quoted Feynman,a theorist,not Galileo,an experimentalist...:tongue:
Mathematics is an essential for any physicist, no matter what field you're in.

As the grandfather of modern science, Galileo, like Newton, was both a brilliant theorist and an extremely skilled experimentalist.

ZapperZ said:
But using it ".. to describe nature and to make our results quantitative..." is precisely using it as a "tool". This description of mathematics, as used in physics, does NOT demean nor diminish its importance. Without it, physics has no language and thus, unable to express itself accurately (try describing Gauss's law in words!).

We use human language as a "tool" to communicate with we talk to each other. Most physicists use mathematics as a tool in their work. No one should be offended by this, least of all, mathematicians, considering that without mathematics, physics will be mute.
Fair point. It just didn't feel right to call something essential a 'tool'. Like your analogy; we cannot communicate without some sort of language. Likewise, we cannot do physics without mathematics, therefore it's an integral part of it.
But aw'right, an essential tool is fine with me. :smile:

If we don't do that, we end up NOT doing physics, but end up learning more mathematics than what most math majors would need. Students of physics do not have the time, the patience, nor the inclination to delve into mathematics that deeply - that is why we are not math majors. You are also forgetting that knowing what the "physics" is behind the mathematics allows for the simplification of the problem that isn't obvious from the mathematics.
There's nothing wrong with using physical arguments, quite the contrary. It's the derivation of many equations and such that could be done more carefully.

This could be just me, but whenever there's a step in a derivation I don't precisely understand as to 'why' it is allowed or simply don't quite get it, I get a very uneasy feeling about it. Some sense of incompleteness in my understanding.

Maybe I could conjure an example. Like the equation of motion for a rocket ejecting mass (fuel) out of its rear:

"Well, in an interval between t and t+dt the amount of fuel exhausted is |dm|=-dm (because the mass of the rocket decreases), while the mass of the rocket is m+dm and its velocity [itex]\vec v+d\vec v[/itex].
The momentum of the system at time t is:
[tex]\vec P(t)=m\vec v[/tex]
and the momentum at time t+td is:
[tex]\vec P(t+dt)=\vec P_{\mbox{rocket}}(t+dt)+\vec P_{\mbox{fuel}}(t+dt)=(m+dm)(\vec v+d\vec v)+(-dm)(\vec v+\vec u)[/tex]
[itex]\vec u[/itex] is the velocity of the exhaust gases wrt the rocket.
The change in momentum in the time interval dt is:
[tex]d\vec P=\vec P(t+dt)-\vec P(t)=m d\vec v -\vec u dm[/tex]
where we have dropped the second order term [itex]dmd\vec v[/itex].
Divide by dt to get the change in momentum, which equals the external force.
Rewrite to get:
[tex]m\frac{d\vec v}{dt}=\vec u \frac{dm}{dt}+\vec F[/tex]
In the case of no external force (no gravity in outer space) [itex]\vec F=0[/itex]. We can multiply both sides by dt/m and integrate to find:

[tex]\vec v=\vec v_0 +\vec u \ln\frac{m}{m_0}[/tex]"

I`m not saying the result is wrong or questionable. It's very plausible if you physically interpret this answer.
I find the derivation quite horrid. Things are done I was told that weren't allowed, like treating dm/dt as a fraction. It would be much more elegant to set up a differential equation and solve it. This doesn't even have to be done in a physics class, but in a lecture on DE's.

I`m sure I can think of more examples, but something like this make me go: :uhh: -< (Is this kosher?)
 
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  • #29
dextercioby
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Does it really matter what type of mathematics u use to get the physically correct result...?I hope not.

Daniel.
 
  • #30
dextercioby said:
I may be old fashioned and it's probably not the kinda idea i should be taking in a (hopefully successful) theorist carrier,but i still think that FOR A THEORETICAL PHYSICIST,MATHEMATICS IS A TOOL AND NOT A PURPOSE...

Daniel.
lol, its amazing to see how this thread has evolved from me asking about complex wavefunction to something else.

But personally, I don't treat pure math with much respect and I don't spend much time and effort, say, rigourously going through proofs and derivations in math. But like galileo, to me, i spend almost an equal amount of time going through proofs and derivations of formulas in physics than actually practicing the problems. And yes, i get jittery and uneasy when I don't understand how a particular step fits into the physical situation or if I don't understand why it is like that mathematically.

And yes, i do believe that physicists have to be more careful and concise in proofs and derivations but not really have to study that much in detail of the undelying mathematical structure.

Just my opinion...
 
  • #31
reilly
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Complex numbers and variables have been part of physics, engineering, chemistry, biology for a long time. Why? The best way is to study the math of complex variables and see for yourself. Complex variables provide a depth of analysis hard to acheive with real variables alone. Things like contour integration, conformal mapping and 2-D potential theory are basic in the physics toolbox. You develop intuition about complex variable things by working with them-- over time by continuing to work with them, you will begin to view their use as second nature. So, why use vector spaces, or groups, or other mathematical approaches? They help get the job done. .

What's the job? The job is what physicists say it is -- sometimes physicists are more abstract and mathematical than mathematicians , or were during the heyday of axiomatic field theory. Sometime's its strictly back of the envelope, like Fermi's computation of the TNT equivalent of the first A Bomb at Almogordo -- or his famous orals question: how far can a bird fly?

Physicists do not form a monolithic community, albeit there are certain common threads among the subcultures or schools or groups.... The point is, approaches to math can be all over the map, approaches to research and methodology can be all over the map -- ultimately, at least for the professional, it's a matter of style, convenience and practicality.

And, with regard to rigor, it took quite a few years for the mathematicians to catch up with Dirac and realize the delta function is cool. Intuition is a powerful tool, just as is rigorous logic and math. The style issue: when and how do you blend the use of these tools?

Complex variables expand the language and tool set of physics--they are here to stay.

Regards,
Reilly Atkinson
 
  • #32
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Galileo said:
I`m not saying the result is wrong or questionable. It's very plausible if you physically interpret this answer.
I find the derivation quite horrid. Things are done I was told that weren't allowed, like treating dm/dt as a fraction. It would be much more elegant to set up a differential equation and solve it. This doesn't even have to be done in a physics class, but in a lecture on DE's.

I`m sure I can think of more examples, but something like this make me go: :uhh: -< (Is this kosher?)
Yeah, I also like to try and do things right as well, like not using a "dt" as a "delta t" for example. :smile:

Infinitesimals are another one, with the "infinitesimals" of scientists and engineers actually being ficticious concepts in the branch of mathematics they use.
 
  • #33
Hurkyl
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That's not necessarily true -- there are nonstandard models of analysis that provide honest to goodness infinitessimals.

(But they do take come care to use properly. e.g., a ratio of infinitessimals is only infinitessimally close to the derivative, not exactly equal)
 
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  • #34
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Yeah but as I say, they're not using that branch of mathematics. :wink:
 
  • #35
selfAdjoint
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caribou said:
Yeah but as I say, they're not using that branch of mathematics. :wink:

It's a model of what they're doing. They could represent their equations and integrals in terms of nonstandard analysis if they chose to.
 
  • #36
Sterj
A question, If a particle has a wave function like this f(x,t)=e^i(kx-wt) ,is then the probability to find the particle = |f(x,t)|^2?
 
  • #37
dextercioby
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No,that's the probability density...It's 1,which means that the wave-function,non square integrable Lebesgue,does not describe a physical state of a quantum system...

Daniel.
 
  • #38
Sterj
that means, if f(x,t) want's to be a wave function |f(x,t)| must be equal to 1. And then the probability density is the integral(ff*dV)
 
  • #39
dextercioby
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What?The probability density that the quantum system be found at the moment 't' in the point [itex] \vec{r} [/itex] is [itex] \mathcal{P}=|\Psi(\vec{r},t)|^{2} [/itex] and that's that...

Daniel.
 
  • #40
Sterj
ahh, sure, it has to be so.
A particle can be expressed as a wave, but why the hell it can also be expressed as an oscillator, I mean an oscillator isn't really the same thing as a wave. And why is the ground state energy of a harmonical oscillator the 0-point energy of a particle?
thanks
 
  • #41
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Outer product

masudr said:
We have a complex vector space equipped with an inner product, complete with respect to the norm defined by the inner product (i.e. the Hilbert space). Elements of it are vectors. And since it is a complex space, each vector can be multplied by a scalar complex number.
Masud.
Has HIlbert Space an outer product (grassman)? Could it have this?

:smile:
 
  • #42
dextercioby
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Nope,it doesn't.A Hilbert space is what it is.An inner product Banach space.

Daniel.
 

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