Constructing Physics

Elementary Physics Course Notes

Adam Johnston
Department of Physics
Weber State University

 

In 1900, Max Planck proposes that energy is quantized.  In other words, he suggests that energy comes in packets of discrete amounts, just like charge.  He doesn't propose this just for the fun of it or to cause trouble; he does it in order to fix some physics that predicted that objects should emit outrageously (and non-evidenced) high intensities of light at high frequencies. (This was known as the "ultraviolet catastrophe.")  This simply doesn't happen.  In other words, the theoretical physics of the time did not predict what was actually observed.
Planck's description of the energy of light and its dependence on frequency is written like so:
E = hf
where "h" is known as Planck's constant.  Basically, all this says is that the higher the frequency waves of light happen to be, the more energy they carry.  At this point, everyone says, "So what?" 
In 1905, Einstein explains the photoelectric effect.  As we see in class, light actually runs into particles (electrons) and bumps them around!  We know this happens because we see ultraviolet light making a negatively charged (excess electrons) electroscope lose its charge.  Also, this happens immediately, suggesting collisions rather than electrons absorbing some energy and then jumping ship later.  Finally, we see that high frequency light (such as ultraviolet) is more apt to kick electrons off than low frequency light (such as visible).
(Note: This doesn't happen if the electroscope is positively charged!  Wow oh wow oh wow!  This tells us that Adam wasn't lying after all: The electrons (negatives) really are the less massive parts of the atoms!)
What does that all mean?  Well, if light is "bumping" into something, then it is behaving as a particle, a discrete package of something solid that can knock into other things.  We call this particle a photon.  So, light has a dual nature.  It acts as a wave (as it does when it interferes) and it acts like a particle (as in the photoelectric effect).  At this point students began to spontaneously combust.  "First you say that space is warped in dimensions that we can't see and now you're telling us that light is made out of particles?"  The thing is, physicists of the early 20th century reacted with the same kind of skepticism.
But, here's the ultra-weirdity:  The electrons that got kicked off of pieces of metal in the photoelectric effect had an amount of energy that was proportional to the frequency of light that was shone.  Einstein showed that this energy was equal to:
E=hf
Look familiar?  Yes, the crazy idea that Planck came up with and the crazy idea that Einstein came up with both gave the same results.  Planck and Einstein were each showing examples where light energy was coming in discrete packages of energy ("quanta"), and each not only explained the observations of the time, but they also confirmed one another's descriptions.
(Note: Planck did his work in 1900, and Einstein in 1905.  But they didn't get their Nobel prizes until 1918 and 1921 respectively.  Why?)
So, at this point, we have three general observations of nature:
1. Light can be observed to behave as waves.
2. Light can be observed to behave as particles.
3. Matter can be observed to behave as particles.
In order to make these 3 statements symmetrical, we could add a fourth statement:
4. Matter can be observed to behave as waves ??????????
Obviously, the last statement isn't directly observed; but we should begin to look for it.  How would we see this?  Well, waves interfere with one another, so we should try to get some matter to interfere with some other matter.  Yeah, right.  In fact, this is what really can happen, and we are led by our logic and against our common sense into things such as the Heisenberg Uncertainty Principle and the Bohr Atom.
Take a deep breath.
But that, statement number 4, is simply stupid, right?  We don't witness "stuff" acting as waves . . . or do we?  Now that we know that light is a particle, we can cite instances where we observe its wave nature.  So maybe all particles (charges, cats, cars, baseballs, etc.) should have a wave nature to them?
That's exactly what deBroglie proposed, and he went so far as to say that the wavelength of all objects is proportional to Planck's constant (h) and inversely proportional to the object's momentum.  This is pretty neat for two reasons:
1. We see Planck's constant pop up again.  Kind of nice that nature keeps using the same numbers over and over again.  It's also nice that this is such a small number.  It suggests that most wavelengths are going to be so small that you'd never notice them, except . . .
2. as the momentum of an object gets really really small, then we should see the wavelength become more noticeable.  So, the only times that you're going to notice the wave behavior of a particle is when its wavelength gets bigger than its particle size.  This doesn't happen for cats and baseballs, because their momentum is so large compared to Planck's constant.  BUT, for things such as electrons, the momentum is very small, increasing the wavelength.  Thus, we should notice the wavelength for very small particles such as electrons; and, in fact, if you fire a bunch of electrons side by side, they are going to spread out like waves and interfere with one another.  We can show this experimentally.  You can't do this with cats, for a number of reasons.

Great, but what does this really DO for us?  Well, think back to our picture of the atom.  Adam (our professor) was panicked about atoms (the little buggers that make all molecules and thus all matter and thus all of us).  His logic was that atoms are made from electrons, which are accelerating charges, and accelerating charges should produce light, and the production of light should carry energy away from atoms, and a loss of energy from atoms should result in electrons spiraling into the nucleus.  This would be an end for all atoms!  This means all matter should "fall" into itself in fractions of a millionth of a second! 
Poof!
But we're still here! 
So, you have some good evidence that atoms don't "fall" into themselves.  Why not?
The Bohr model of the atom shows that electrons can only be at specific energy levels, just like the bore Adam can only stand on the steps in the lecture hall (never in between steps).  We get to witness this by looking at the atomic spectra in class and seeing only specific wavelengths (colors) getting spit out, meaning that electrons are confined to specific energy levels and thus only emitting specific amounts of energy. 
This is explained once you think of the electrons as waves, rather than as particles.  In music, specific frequencies are supported as standing waves on a string or in a tube.  Similarly, electrons-as-waves can only "fit" an atom when there are certain wavelengths.  This explains why only certain energies of electrons are allowed, and thus the electrons never have a chance to spiral into the nucleus because then they would have had to have been in between energy levels, with wavelengths not in resonance with the atom.
Dude . . .
(A disclaimer at this point: You must realize that these are ONLY notes!  You shouldn't be reading any of these notes and expecting to understand it all without some other serious thought and study.  This goes double for anything having to do with quantum mechanics!)
So, fantastic.  If the combination of notes, reading, lecture, and divine intervention has convinced you that particles can be waves, then you can begin to realize that the smallest of particles are kind of "fuzzy".  That is, the electron that as a particle seems to be so pinpointed is now represented as a wave, which is much more spread out.  That means that you aren't so sure exactly where that little electron or other particle is anymore.  This is the thrust of the Heisenberg Uncertainty Principle: You don't know anything.  Specifically, the more detail you try to measure about one thing, the less you know about another detail.  This isn't because you aren't trying hard enough or because you need a better instrument/technology, but because Nature is saying, "Ha ha!  I'm clouding the picture so that you can never really know exactly what the initial conditions of something are!" 
This might be kind of depressing in a way, but it's also freeing:  If we could know everything exactly, then with Newton's laws we could predict every outcome of every situation, including the next letter that I'm going to type (Q) and the grade you're going to receive in this course (D+) and the person you're going to "choose" to marry.  There would be no free will -- it's just the result of a bunch of collisions of particles!  But, if none of the positions and velocities of particles are knowable, then almost anything is possible!  You get your free will back!  (As long as it abides by the rules of physics.)