Complex wavefunction.

  • #1
From what I heard, the wavefunction is made up of both real and imaginary parts. How do I prove this? Also, what is the physical interpretation of complex numbers? How does a complex wavefunction fit into physical reality?
 

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  • #2
dextercioby
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misogynisticfeminist said:
From what I heard, the wavefunction is made up of both real and imaginary parts. How do I prove this?
Use the definition,because it is specified there.

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


misogynisticfeminist said:
How does a complex wavefunction fit into physical reality?
By Born's statistical interpretation of Schroedinger's wave function.
"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.
 
  • #3
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A complex wavefunction arises because we need a way to describe the wave nature of particles, without having them actually be a disturbance in some medium. So we introduce a complex scalar field and have it oscillate in the complex dimensions.

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.
 
  • #4
dextercioby
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The reason for the introduction of COMPLEX scalar fields into classical and quantum field theory is because the "COMPLEX" attribute allows for a correct description of ELECTRIC charge,as the the field of complex numbers is the field of scalars over which the fields determine an associative algebra with involution,the involution in this case being the complex conjugation...
No connection with any disturbance,whatsoever...:wink:

Daniel.
 
  • #5
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misogynisticfeminist said:
From what I heard, the wavefunction is made up of both real and imaginary parts. How do I prove this? Also, what is the physical interpretation of complex numbers? How does a complex wavefunction fit into physical reality?
1. The wavefunction is stipulated to be complex, therefore not only can't you prove it, you aren't supposed to.

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
 
  • #6
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Daniel,

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.
 
  • #7
dextercioby
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That's weid,your words
masudr said:
So we introduce a complex scalar field
don't seem to say it...:wink:You should try to be more careful with the terminology,coz people may not understand what u really wanna say...

Daniel.
 
  • #8
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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).

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.
 
  • #9
dextercioby
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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).
I know i'm nit-picking you,but the [itex] \Psi(\vec{r},t) [/tex] is a VECTOR...:wink:

Daniel.
 
  • #10
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Yes you're right :smile: , although I'm pretty sure we're both correct, because we have a complex valued function (which assigns a complex number to each point in spacetime) which also inhabits a vector space. So the function itself is a vector, but not part of a vector field.

And it's good to nit-pick, the more physicists nit-pick the closer we get to being proper mathematicians.
 
  • #11
dextercioby
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masudr said:
the more physicists nit-pick the closer we get to being proper mathematicians
I resent that...No physicist wants to be a mathematician...

Daniel.
 
  • #12
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dextercioby said:
I resent that...No physicist wants to be a mathematician...
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.
 
  • #13
dextercioby
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masudr said:
Newton's calculus was not rigorous
I resent that.It surely was,for that time .It's unfair to judge the past with the mind of a contemporary indivdual.Mathematics was not rigurous then...After Leibniz it began to be...

Daniel.
 
  • #14
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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.
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.

As I have quoted before, sufficient unto the day is the rigor thereof.
 
  • #15
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Hehehe... "infinitesimals".... dx wandering around on its own... ehehehe... :biggrin:
 
  • #16
Galileo
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dextercioby said:
I resent that...No physicist wants to be a mathematician...

Daniel.
**AHHHUUUMMMMM**!!!
 
  • #17
dextercioby
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Galileo,please sustain your point,an honomatopeical interjection will not suffice...:wink:

Daniel.
 
  • #18
Galileo
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What's there to say? I`m a counterexample to your statement.
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: )
 
  • #19
dextercioby
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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.
 
  • #20
Galileo
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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.
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'.
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.

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth.

- Galileo
 
  • #21
dextercioby
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The way you've written this message makes me think that u've misunderstood my assertion and thought that i was diminishing the importance of mathematics.:surprised:Wrong...Mathematics is an ESSENTIAL tool for a theoretical physicst...

Daniel.

P.S.I would have quoted Feynman,a theorist,not Galileo,an experimentalist...:tongue:
 
  • #22
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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]
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.

Zz.
 
  • #23
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Ok that is all fair enough, but at my university (Oxford) where, during the interview/selection process, I was told that there was heavy emphasis on the mathematical side of physics, but I have still found, to my annoyance, that my physics lecturers take shortcuts, and are generally sloppy with notation, whereas all my maths lecturers/tutors are generally much more precise and pay more attention detail.

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.
 
  • #24
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Hilbert?

masudr said:
So we introduce a complex scalar field and have it oscillate in the complex dimensions.
Masud.
Like Hilbert complex space??? There is another complex space with scalar and vectors? :shy:
 
  • #25
ZapperZ
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masudr said:
Ok that is all fair enough, but at my university (Oxford) where, during the interview/selection process, I was told that there was heavy emphasis on the mathematical side of physics, but I have still found, to my annoyance, that my physics lecturers take shortcuts, and are generally sloppy with notation, whereas all my maths lecturers/tutors are generally much more precise and pay more attention detail.

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.
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]:

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.
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.

Zz.

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

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