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- Thread starter misogynisticfeminist
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Use the definition,because it is specified there.misogynisticfeminist said:From what I heard, the wavefunction is made up of both real and imaginary parts. How do I prove this?

They're a "necessary evil"...They don't have "physical interpretation".They're just VERY USEFUL mathematical tools...misogynisticfeminist said:Also, what is the physical interpretation of complex numbers?

By Born's statistical interpretation of Schroedinger's wave function.misogynisticfeminist said:How does a complex wavefunction fit into physical reality?

"The probability density to find the particle in the point [itex] \vec{r} [/itex] at the moment "t" is:

[tex] \mathcal{P}(\vec{r},t)=|\Psi(\vec{r},t)|^{2} [/tex]

Daniel.

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So when two wavefunctions add we can have the wave type of destructive/constructive intereference in the complex dimensions. When we need to be brought back to physical reality, we find the length of the complex number (i.e. it's magnitude).

So we use the complex number description because it is a very easy way to describe reality, even though mathematicians invented complex numbers thinking that they would have no real physical counterpart. And it all works out quite well. Isn't physics amazing?!

Masud.

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No connection with any disturbance,whatsoever...

Daniel.

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1. The wavefunction is stipulated to be complex, therefore not only can't you prove it, you aren't supposed to.misogynisticfeminist said:

2. There is no physical interpretation of a complex number.

3. Complex wavefunction doesn't fit into reality well at all, that's why you see everyone trying to convert back to real functions of a real variable.

Regards,

Guru

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I wasn't talking about either classical or quantum field theory (an area where my knowledge only partially extends into); instead I was talking about non-relativistic quantum mechanics, where we give our quantum particles both wave and particulate natures.

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don't seem to say it...You should try to be more careful with the terminology,coz people may not understand what u really wanna say...masudr said:So we introduce a complex scalar field

Daniel.

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I am aware that, even within maths/physics, the word "field" has several meanings (for example we have the definition of a field as two Abelian groups of the same set, with distributive properties over the operations), and I should have made that clear. But what I meant was that every point in space is associated with a complex number.

Masud.

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I know i'm nit-picking you,but the [itex] \Psi(\vec{r},t) [/tex] is a VECTOR...masudr said:When I say scalar field, what I mean is every point in space is associated with a scalar number, as opposed to a vector field where every point in space is associated with a vector (like the electric field).

Daniel.

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And it's good to nit-pick, the more physicists nit-pick the closer we get to being proper mathematicians.

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I resent that...No physicist wants to be a mathematician...masudr said:the more physicists nit-pick the closer we get to being proper mathematicians

Daniel.

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Ok, that is true, but I hate it when physicists are mathematically inaccurate, or when they are sloppy, or take shortcuts etc. e.g. Newton's calculus was not rigorous; it took mathematicians such as Cauchy and Euler to put calculus on rigorous foundations and hence founded Analysis.dextercioby said:I resent that...No physicist wants to be a mathematician...

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I resent that.It surely was,masudr said:Newton's calculus was not rigorous

Daniel.

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selfAdjoint

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Euler and Cauchy weren't rigorous in the modern sense either; it took Weierstrass to put THEIR work on a solid foundation. Newton's ultimate ratio can be mapped into the modern limit concept easily. He wasn't always complete in his demonstrations but they can be made complete without violating his thought.masudr said:Ok, that is true, but I hate it when physicists are mathematically inaccurate, or when they are sloppy, or take shortcuts etc. e.g. Newton's calculus was not rigorous; it took mathematicians such as Cauchy and Euler to put calculus on rigorous foundations and hence founded Analysis.

As I have quoted before, sufficient unto the day is the rigor thereof.

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Hehehe... "infinitesimals".... dx wandering around on its own... ehehehe...

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Galileo

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**AHHHUUUMMMMM**!!!dextercioby said:I resent that...No physicist wants to be a mathematician...

Daniel.

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Galileo,please sustain your point,an honomatopeical interjection will not suffice...

Daniel.

Daniel.

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Galileo

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I love both physics and mathematics and I believe physics should strive to be as rigorous as mathematics. Or in any case, every small step that is made by physical arguments must be justified and thoroughly analyzed.

This is not done in the material presented in the college lectures.

I`m one the those persons who actually checks if interchanging limits, differentiating delta functions, assuming completeness, taking fourier transforms of arbitrary vector fields etc., treating dy and dx as fractions to your fancy, etc, is allowed. (And get a weird from my physics professors at the same time :uhh: )

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Daniel.

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Galileo

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That's what I hear from every physicist. 'Mathematics is just a tool', 'this or that is a purely mathematical result, there is nothing physical about it'.dextercioby said:

Daniel.

I disagree to quite an extend with this view. We honestly cannot do physics without mathematics (practically and probably theoretically).

I`m not saying physics is a branch of mathematics, it is not. The point was that physicist aren't always mathematically rigorous. Fact is: we use mathematics to describe nature and to make our results quantitative, so even if you consider it a tool, you are using it so make sure that what you are doing is mathematically justified. That much seems obvious to me.

Anyway, I stand by it. Mathematics is more than just a tool, it's essential.

- Galileo

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Daniel.

P.S.I would have quoted Feynman,a theorist,not Galileo,an experimentalist...:tongue:

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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!).Galileo said:Fact is: we use mathematics to describe nature and to make our results quantitative, so even if you consider it a tool, you are using it so make sure that what you are doing is mathematically justified.[/I]

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.

Zz.

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I just feel that physicists should not cut corners and make assumptions but rely on mathematical proof much more. I am aware that the counter-argument to this is that one can get lost in all the mathematical details of some equation for example, and lose sight of what the equation is trying to say. For example, it took a great amount of intuition for Dirac to develop his equation, but that is exactly the added skill that makes a physicist different from a mathematician. One still has to be precise.

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Like Hilbert complex space??? There is another complex space with scalar and vectors? :shy:masudr said:So we introduce a complex scalar field and have it oscillate in the complex dimensions.

Masud.

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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 is no point in carrying out an infinite series of terms when one has a clear idea that only 1st or 2nd order terms are necessary. It is where the physics comes in. To quote Mary Boas from her Mathematical Methods text[1]:masudr said:

I just feel that physicists should not cut corners and make assumptions but rely on mathematical proof much more. I am aware that the counter-argument to this is that one can get lost in all the mathematical details of some equation for example, and lose sight of what the equation is trying to say. For example, it took a great amount of intuition for Dirac to develop his equation, but that is exactly the added skill that makes a physicist different from a mathematician. One still has to be precise.

Physicists need to know how to use math correctly. But the means is not the ends. We become physicists because we are interested in the result, now primarily on how or what we use to get there.There is no merit in spending hours producing a many-page solution to a problem that can be done by a better method in a few lines. Please ignore anyone who disparages problem-solving techniques as "tricks" or "shortcuts". You will find that the more able you are to choose effective methods at solving problems in your science courses, the easier it will be for you to master the new material.

Zz.

[1] Mary Boas "Mathematical Methods in the Physical Science, 2nd Ed." (Wiley 1983)

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