Proving Real & Imaginary Parts of Complex Wavefunction

[misogynisticfeminist]

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?

 

[dextercioby]

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.

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...

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 $\vec{r}$ at the moment "t" is:
$$\mathcal{P}(\vec{r},t)=|\Psi(\vec{r},t)|^{2}$$

Daniel.

 

[masudr]

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.

 

[dextercioby]

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...

Daniel.

 

[Physicsguru]

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

 

[masudr]

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.

 

[dextercioby]

That's weid,your words

So we introduce a complex scalar field

don't seem to say it...You should try to be more careful with the terminology,coz people may not understand what u really want to say...

Daniel.

 

[masudr]

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.

 

[dextercioby]

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...

Daniel.

 

[masudr]

Yes you're right , 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.

 

[dextercioby]

the more physicists nit-pick the closer we get to being proper mathematicians

I resent that...No physicist wants to be a mathematician...

Daniel.

 

[masudr]

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.

 

[dextercioby]

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.

 

[selfAdjoint]

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.

 

[caribou]

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

 

[Galileo]

I resent that...No physicist wants to be a mathematician...

Daniel.

**AHHHUUUMMMMM**!

 

[dextercioby]

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

Daniel.

 

[Galileo]

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: )

 

[dextercioby]

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.

 

[Galileo]

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

 

[dextercioby]

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. :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:

 

[ZapperZ]

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.

 

[masudr]

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.

 

[Raparicio]

Hilbert?

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:

 

[ZapperZ]

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)

 

[masudr]

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)

 

[masudr]

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.

 

[Galileo]

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.

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.

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

$$\vec v=\vec v_0 +\vec u \ln\frac{m}{m_0}$$"

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?)

 

[dextercioby]

Does it really matter what type of mathematics u use to get the physically correct result...?I hope not.

Daniel.

 

[misogynisticfeminist]

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...

 

[reilly]

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 achieve 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

 

[caribou]

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.

Infinitesimals are another one, with the "infinitesimals" of scientists and engineers actually being ficticious concepts in the branch of mathematics they use.

 

[Hurkyl]

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)

 

[caribou]

Yeah but as I say, they're not using that branch of mathematics.

 

[selfAdjoint]

Yeah but as I say, they're not using that branch of mathematics.

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.