Development of Quantum Physics V – We Cannot be Sure About the World, Physics Says

The previous article left many questions unanswered.

We’ve seen that we are able to describe classical particles (like electrons, neutrons, atoms,…) by the same formalism which we used to describe photons quite some time before. (Photons are the tiniest energy packets of the electromagnetic field.) As a logical consequence, former particles are associated with a characteristic wavelength – the so-called de Broglie wavelength. Kind of crazy, right?!
I’ve additionally mentioned that electrons and other particles (I shouldn’t call them particles actually, but you know what I mean, don’t you?) give same experimental results under same conditions. That is, electrons cause an interference pattern behind a double slit similar to photons.
Hence, the question arises: Can this wave-like description of matter be the true and natural one? Do we have the right impression of the fundamental building blocks of our world? And I’ll spoil at this point: No, we haven’t found it yet. But after making some modifications to traditional ideas we will come closer to a working concept. Be warned: Our understanding of the world will be turned upside down again.

You can find all the other articles of this series here.

 * * *

In order to describe a classical particle as a wave we simply have to characterize it in the same mathematical manner as a photon, which is a “particle” of light. (Don’t worry, I’ll spare you the mathematical details and explain all the math tricks in plain words instead. Disadvantage: You have to believe me. 😉 )
In doing so, we realize that there are discrepancies between our assumption and reality. Such matter waves, for instance, get wider with time and spread throughout space. We don’t observe such behaviour on things we call particles! We know from experience that an atom is more or less localized somewhere in space.
We can fix this issue by introducing so-called wave packets. Another quite appropriate name would be “wave group” (or something). In doing so, we superpose infinitely many waves of similar, however a bit varied frequencies. (This may sound far from reality, but within the framework of mathematics it makes sense indeed!) As a result, this superposition of waves suddenly has maximum amplitudes at certain spatial points. Or as we could say in other words: The wave is extra “big” at some places and quite “small” at others. The matter wave is localized.

Propagation of a one-dimensional wave packet. (The wave which we have discussed above corresponds to the envelope of this packet and spreads with the wave’s “group speed”. The propagation speed of the narrower waves inside the envelope is called “phase speed”.

Fantastic! – That is what we wanted, right?
Yep, it is. We’ve found a function which is capable of effectively describing a “particle” as a matter wave.
The idea of this superposition of infinitely many similar waves (= a wave packet) is actually not such a bad thing:

  • A wave packet, which can describe a massive particle, is, contrary to the previously assumed concept of a wave, localized. (It has maximal amplitudes at certain spatial points. By the way, this maximum is at least as wide as the particle’s de Broglie wavelength, as it can be shown mathematically.
  • The speed of the wave packet is equal to the speed of the particle. (This is, too, the result of a mathematical derivation.)
  • We can also derive the particle’s momentum from the function of the wave packet directly.

But as aesthetic this wave-like description of a particle might seem, it creates problems as follows:

  • Possibly, the function of a wave packet hasn’t only positive and real values which would be relevant for the physical reality, but also negative and comlex ones. Complex values cannot be linked to actual, physical measurands.
  • The width of the wave packet increases as time goes on. Hence, a narrow, well-located packet spreads non-negligibly as it moves around the room. This is contradictory to classical particles, which generally keep their shape.
  • The model of a wave packet describes particles like, e.g., electrons. Electrons are indivisible, but a wave packet isn’t. We can divide a wave packet easily in a beam splitter or in other things – just like we can divide a water wave. After splitting, each of the two (or maybe more) parts of the wave (or wave packet) moves in different directions.

Well, we still haven’t found a particle model which fits reality.

Due to this major discrepancies, the German physicist Max Born felt compelled to suggest a statistical interpretation of a matter wave.

Max Born

Imagine a situation as follows: A wave (e.g., a light wave) comes upon a boundary surface. The wave is partly reflected and partly transmitted. This also applies to a matter wave. (That’s logical, because a wave packet is just some kind of a wave too.) But it’s impossible for a particle to be both reflected and transmitted, as it is indivisible!
Born therefore suggested that we have think of the reflected or transmitted parts of the matter wave as the probabilities for reflection or transmission.

The following animation shows an example of a probability wave which comes upon a boundary surface where it gets partly reflected and transmitted. (Here, transmission can only happen because of the tunnel effect. You can also see interference effects near the surface.)

Splitting of a wave function at a boundary surface
(Credit: Jean-Christophe Benoist)

A little recap:

Every massive particle can be described in terms of a wave packet, a function which depends on location and time. Via this probability density (technically, via the square of the absolute value) we can find out how likely we will detect a particle at a certain place at a given time. The quantum mechanical randomness, which describes lots of phenomena in nature, appears for the first time. Great characters like, e.g., Albert Einstein simply disliked this idea of randomness on the fundamental level. By now, we have largely accepted this property of nature and we can actually deal with it extremely well.

Finally, I’d like to motivate you to reflect about the following fact:
The probability of a particle to be found somewhere in infinite space is equal to 1 (= 100 per cent). This is obvious, because the particle just has to be somewhere. But as soon as we ask about the probability of the particle being in a finite volume of space, we cannot be 100 per cent sure if we will actually find the particle in this volume, because the borders of the matter wave aren’t sharp-edged. Admittedly, the probability is biggest near the center of the wave packet, but it is zero nowhere in space!
Hence, it’s just impossible to localize a particle at a precise place. The localization is always associated with a uncertainty!

With this at the back of your mind, I’d like to refer to the upcoming article, in which we will discuss this uncertainty and its consequences more carefully. Next time’s topic is the Heisenberg Uncertainty Principle, as you’ve maybe guessed already.

__________

Previous part: How We Found the Wavy Particles Next part: Heisenberg Challenged Our Worldview …Again!
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About tempse

I think about physics, other stuff, and physics. Besides, I share some thoughts on the internet.

Posted on September 21, 2013, in history, physics, quantum mechanics, science and tagged , , , , , , . Bookmark the permalink. 8 Comments.

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