Some Physics – Part 2

Imagine a drop of water splashing into a small bowl of water. When the drop hits, it will create water waves that will propagate through the bowl. If we place a screen with two small openings in the path of the waves (far away from where the drop hit the water, we want the waves to “look” like planes), we will observe a phenomena known as diffraction. If you haven’t seen diffraction before, I’ve included this
Image courtesy of Wikimedia commons. Author: Loomang graphic from Wikipedia, or there are plenty of pictures on Google and I’m sure some demonstration videos on youtube. The key point is this: if we we were to look far away from the slits, we would observe a pattern on the wall; a “distribution” if you will, of the water. We could (mathematically) write down the expression that gives the “density” of water at each point on our observation plane. This would be some form of a probability density function, that we could integrate to find the total amount of water in some subset of our observation panel. Diffraction is one example of what is referred to in physics as a wave phenomena. Like last post, I won’t rigorously define what I mean by a wave, but I think the reader should have some intuition built up as to what I mean.

Now, diffraction patterns in water waves are one thing, but perhaps one of the most important experiments of early 19th century physics was Thomas Young’s famous double slit experiment, where Young demonstrated that light, when properly manipulated, exhibits the wave property of diffraction. In other words, Young provided experimental evidence that light is a wave. Now here’s where the story gets interesting. If we fast-forward to the early 20th century, we find Einstein’s Nobel prize winning work regarding the photoelectric effect (shockingly, Einstein’s Nobel prize was not awarded for his theory of relativity). In this (rather simple) experiment, light strikes a metal surface. The light hits some of the loose electrons that are near the surface of the material and causes them to be ejected from the metal. The electron output is then measured by some form of detector. If light were (purely) a wave, then it can be shown that the electron output should be proportional to the intensity of the incumbent light. Unfortunately for the “wave theory” advocates, the experimental results did not match this prediction. It was found that the electron output was proportional to the frequency of the light wave. This is where Einstein made his contribution. He theorized that light actually consisted of discrete quanta called photons, each of which had energy proportional to the frequency of the wave. In other words, light is both a wave and a particle! This amazing idea revolutionized how science perceived light and reconciled the seemingly contradictory results of diffraction and the photoelectric effect. (I should be careful, the idea of discrete light had actually shown up before this in black-body radiation, but this is probably the more famous application and the one that really led to the discovery of quantum mechanics).

It is this “duality” principle that is at the heart of quantum mechanics. Once physicists began to accept this double nature of light, it was only natural that they should ask the same question about matter. Of course, until that time, it was assumed that matter theory was one of particles; discrete “point” masses if you will. However, de Broglie (1925 I believe) hypothesized that matter could also exhibit wavelike behavior, and before long physicists were designing experiments to test this idea. Sure enough, in 1927, a successful experiment was performed in which diffraction patterns were observed after a beam of electrons was fired at a specially designed crystal. This showed the wave-particle duality of matter, not just light. So why don’t diffract when we pass through a door? Well, it turns out there is an important physical constant, called Planck’s constant (denoted by h, or sometimes \hbar:=h/2\pi) that shows up in many quantum equations. Planck’s constant is “small” compared to scales we experience in our everyday life which tends to block out any quantum effects in our everyday life. But, sure enough, if one looks at the molecular and atomic level, evidence of the dual nature of matter abounds. I’ll now state de Broglie’s famous equations; I’m not sure if we’ll need them in the future, but they’re fun to write down. If you want to, you can use them to calculate your wavelength! For a particle with total energy E and momentum \mathbf{p}, the particle’s wavelength \lambda and frequency \nu are given by

E=h\nu=\hbar\omega\hspace{5mm}\lambda=h/p=\hbar k

where \omega is the angular frequency and k is the wavenumber, for those of you who know about waves, otherwise these are just constant multiples of the parameters listed above.

Perhaps one way to best illustrate this “duality” (and introduce an important mathematical function) is in the following example: consider the diffraction experiment above, performed with electrons. Suppose we slowed the electron emission rate so that we let out one electron every, say, 5 seconds or so (i.e. we have a very sparse “beam”, we can observe where individual electrons “go”). One might expect the electron to simply hit the barrier we placed in front of it (the wall with slits) and reverse its direction (this is the conclusion if we assume a classical, particle trajectory). Turns out the particle will pass through the slits and will strike some place on the wall. The next electron will do the same, but will (likely) end up in a different place. If we repeat this experiment many times and record the density of electrons as a function of position along the observation plane, we will eventually gain enough data to piece together of probability density function, call it |\psi(x)|^2, so that |\psi(x)|^2\hspace{1mm}dx represents the probability of finding the next electron emitted in the location dx (or if that is too physicist-ish for you, \int_a^b |\psi(x)|^2\hspace{1mm}dx is the probability of finding the next electron in the interval (a,b)). It turns out we have no way of measure what “path” the electron took to get to the wall: we can only measure where it ended up. This is was I meant in the last post by losing the idea of a trajectory. The particle (electron) existed in a so called “quantum state” until we performed a measurement (seeing where it ended up) on it, causing it to enter an allowed (eigen) state and it stayed in that state from thereon.

These ideas are the core principles of quantum mechanics (at least the Copenhagen interpretation of quantum mechanics, which is currently the “accepted” version of QM, perhaps Zach would like to write about the philosophical implications of QM sometime?). Given some quantum mechanical object, there is some function \psi(\mathbf{r},t), called the wavefunction, that contains all the information there is to know about that particular object. As above, |\psi(\mathbf{r},t_0)|^2 represents the probability density function for finding the particle at location \mathbf{r} at some fixed time t=t_0. The wavefunction evolves deterministically in time according to Schrodinger’s equation (which I will write down next time). The particle can exist in any “quantum state” until a measurement is performed on it, in which case the wavefunction “collapses” to an allowed state. Mathematically, we will work in an abstract Hilbert space, where the wavefunction will correspond to some vector in this space. A measurement corresponds to an operator being applied to the wavefunction, and the “allowed” measurements correspond to eigenvectors for this operator. The “collapse” of a the wavefunction correponds to a (singular) event, where the wavefunction instantaneously “changes” from its state before measurement to one of these eigenstates after measurement (what really happens here is unknown. We will take this for granted as a postulate and only worry about the mathematical evolution of the system before and after observation).

Individual trajectories no longer exist, as measuring position can only tell us where the object “ended up”. Of course, since we have no trajectories, we do not have force either. Instead we must replace forces with potentials V(x) like in the harmonic oscillator in the previous post. It turns out if we know the potential function, we can figure out the wavefunction from Schrodinger’s equation. I’ll talk more about this in the next post. I think that’s enough for now though. This post is obviously rather wordy and quite long, but I think it’s good to read through a couple times and think about if you are really trying to get something out of my series of posts. The examples, though simple, do a good job to illustrate quantum principles in a way that hopefully is accessible to someone without a lot of physics background. I hope this gives you some idea of what we’re working towards and how it came about in the scientific community. Anyways, next time I’ll talk a bit about Schrodinger’s equation and then introduce some physics notation for inner products. Unfortunately I’m about to get busy with end of the year exams and such, so it might be a while before I write again. Until next time, good luck with any work/exams you may have!

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About Ryan

I'm a software developer at Hudl where I work on awesome software. Before that, I was a grad student in mathematics, interested in probability theory as well as analysis, more on the side of functional analysis and less on the side of PDEs. Apart from that I'm pretty lame. Though I do enjoy watching football, playing golf, and playing the trumpet.
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2 Responses to Some Physics – Part 2

  1. Jay Cummings says:

    Neat stuff!

    But if I may make one suggestio: go ahead and embed those cool experiment graphics into your post rather than just linking to them! They’re good to see, it visually breaks up the post, and feels like we’re all doing an experiment together. :-)

  2. Ryan says:

    There… you made me relearn how to do images in HTML =P

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