Wave-particle duality and special relativity

[lntz]

hi,

i am in year 13 of school, so i would appreciate it if any responses to this question weren't too far over my head. saying that i have a fairly decent back ground knowledge of some of these ideas.

so, we have learned about the whole lead up to deciding photons etc. are not waves or particles but rather a mix that contains properties of both. eg young's double slit experiments and the photo electric effect. we also have learned about relativity, but in less detail than i would have liked.

so my question is this; what is the significance of wave-particle duality for special relativity? the two ideas seem closely linked, but all the discussions i have read quickly indulge in quantum physics that i have very limited knowledge of. so any explanations as to why these two ideas are linked and what there implications upon each other, would be really helpful :)

thanks, sorry if that was a bit long and waffley.

 

[Mr_Physicist]

They are indeed linked. A famous physicist named Paul Dirac discovered what is now called the Dirac equation. It is essentially an updated version of Schrodinger's equation, which is one of the starting points of quantum mechanics and is built upon that idea of wave-particle duality. The Dirac equation is basically the Schrodinger equation written to obey special relativity (but not general relativity). You can read about why the Dirac equation is important on the wikipedia page dedicated to it. Unfortunately, you will probably have quite a bit of trouble with getting a working knowledge of even basic quantum mechanics without a solid knowledge of calculus. But books like the Feynman Lectures on Physics Volume 3 will still make for an interesting read even if you ignore the math. Good luck on your understanding.

 

[lntz]

well my calculus understanding doesn't go beyond the second differential, differentiating trig functions and similar for integration. how much more advanced is the calculus involved in quantum ideas? (i know that it is probably way more advanced, but is it worth me trying to understand it, or is it simply beyond me at the moment?)

thanks for the reply by the way

 

[James Leighe]

Look into getting entry level book on quantum mechanics. (some non-relativistic introduction to quantum mechanics)

If you have a reasonable understanding of calculus you should be able to tackle most if not all of it.

 

[Mr_Physicist]

The math in quantum mechanics really comes down to an understanding of the Fourier transform (particularly the complex Fourier transform) and some differential equations. The calculus you have is a good foundation and a more complete understanding is probably within your grasp, but it will require quite a bit of hard study. It will be particularly difficult without someone to talk to about it; quantum mechanics is a formidable subject and it only gets harder as you move into some of the more fascinating subjects, such as quantum electrodynamics, quantum chromodynamics or quantum computation. I would therefore suggest that, at your level (that is, as anyone who is not a graduate student studying one of these subjects), you rely upon books written with the general population in mind. I have read and highly recommend "Quantum: A Guide for the Perplexed" as a place to start. It is listed on amazon for about $12. You also can't go wrong with anything by Richard Feynman, one of the most famous scientists ever. Try "Elementary Particles and the Laws of Physics: The 1986 Dirac Memorial Lectures" or "Q.E.D." to help get your feet wet. As I said, I believe that you could begin to develop the ability to do the math involved in quantum mechanics, but I strongly suspect that you would most enjoy learning quantum mechanics without worrying about having to do the math. It is the math that is, in truth, the most frustrating part. If you get to the point where you want to get into the more complicated aspects of it, however, you would probably be best served by an textbook on introductory Quantum Mechanics, one intended for an undergraduate course.

 

[lntz]

thanks Mr physicist i shall look into some books by feynmann, he has written extensively on the subject by the looks of things. I'm sure between my physics teacher and the rest of internet i can grasp at least a basic understanding of quantum ideas.

i'm sure i'll have more to ask once i have read a book on the subject.

thanks again, lntz

 

[Born2bwire]

I would say in some ways they aren't connected, in others they are.

Basically you can still see wave-particle duality in non-relativistic quantum mechanics. So you do not need special relativity to see it arise. In chemistry you probably would have discussed the electron orbitals and how they are probability densities or "clouds." These orbitals are found without taking into account special relativity yet we see the wave-particle duality nature.

However, non-relativistic quantum mechanics is limited in the kinds of problems that you can solve. Most strikingly, you cannot do adequate high-energy physics or a lot of quantum electrodynamics. The former because special relativity is needed to account for the creation and destruction of particles. The latter because electrodynamics automatically follows special relativity. There have been many attempts to update non-relativistic quantum mechanics with special relativity. One of the earliest was the Klein-Gordon equation and shortly afterwards the Dirac equation. But both of these were not fully successful. It wasn't until the 1950's that the solutions to this problem really picked up speed. So the major way that special relativity has been integrated into quantum mechanics is in quantum field theory. Right now, the major challenge is trying to integrate general relativity into quantum mechanics.

 

[lntz]

please forgive any ignorance i show in this reply because i more than likely don't know what i am talking about.

in chemistry we have talked about electrons as an electron density or cloud, but isn't that more to do with the Heisenberg uncertainty principle, that you can't know it's momentum and position at the same time? or at least with any degree of accuracy. I'm failing to see how this is an argument for electrons behaving as waves and particles. please explain

 

[Born2bwire]

The Heisenberg Uncertainty Principle is a consequence of the physics that give rise to properties like wave-particle duality, it is not to be taken as a cause. There is an equivalent "uncertainty" if we talk about any wave system.

That is, there is an uncertainty in knowing the position of a wave in time and its frequency. This can be manifested in the idea of displaying the frequency content of a wave within a time window. For example, when you damn kids play your damn music on a computer, you usually have a frequency spectrum displayed. Now this spectrum is essentially the frequency content over a window in time of your music. But the smaller the window you use, the less frequency information you get and vice-versa. So this is another manifestation of the uncertainty principle. The more you know about the frequency content of a signal, the less you know about what it looks like in the time domain.

So how do these electron clouds demonstrate wave-particle duality? It's done by the fact that you have to solve a wave equation to find these clouds. What happens is that these clouds turn out to be something like standing waves. If you were to solve the problem of a free electron traveling in empty space, the solution would be a traveling wave (which would be something not too dissimilar to the deBroglie momentum-wave that was a precursor to all of this). One of the early pseudo-quantum ideas of solving the problem of the electron orbitals was that they were standing deBroglie momentum-waves. The Bohr model of the hydrogen atom has energy levels where the matter wave can be a successful standing wave (though it should be admitted that Bohr's model predate's deBroglie's thesis).

EDIT: Of course, if you would like a more direct example, we can do Young's Double Slit experiment with particles other than photons, like electrons or even large molecules like Buckyballs. In this instance, we can solve for the fringes that result using non-relativisitic quantum mechanics.

 

[lntz]

thanks, that was a really clear explanation with useful examples.

i hadn't thought of youngs' double slit experiment and i should have done really!

i didn't know that the experiment produced fringes when using buckyballs though... sounds interesting