Wave-particle duality (aka ‘Why the flip do we need quantum mechanics, and what is it?’)
It’s obviously understandable that the majority of the world (including quantum physicists themselves), don’t actually understand what quantum physics actually is, and why it’s necessary. Surely, Newton’s very simple laws of motion tell us all that we need to know about our world? We can calculate the position, displacement, velocity, and acceleration using the kinematic equations; we can calculate the force required to accelerate a moving body; we can explain why springs work the way that they do; we can explain heat transfer using the laws of thermodynamics; we can explain electricity. And so, to both the trained and untrained eye, it seems that quantum mechanics is of no great importance really. After all, in our macroscopic world, the most we would like to do is to understand the macroscopic world around us. And even if we wished to examine the microscopic world, we would need to look no further than the atom, which is, to the layman, a fundamental particle. That is, without it, nothing could exist.
Now, coincidently enough, the entire reason we need quantum mechanics, is so atoms can exist. The atom, as it is traditionally presented in high school chemistry and physics classes is depicted by the standard image of a compact nucleus consisting of protons and neutrons, surrounded by rings used to represent the orbit of electrons about the center of the atom. For the purposes of explaining simple chemical reactions of course, this model is appropriate enough to use. One of the first things learned in high school chemistry is that reactions occur such that atoms have a certain number of electrons in their outermost orbit (also referred to as a valence shell). However, the caveat that high school chemistry fails to mention is that as electrons orbit around a nucleus, that they radiate energy. And it is here, friends, where we begin our discussion of quantum mechanics. I'll use a table of contents, so you can easily jump between topics by using the CTRL+F browser function, and typing in the section code (much in the manner game walkthroughs are written).
Centripetal acceleration [CA]
Photoelectric effect and Planck's constant [PE]
De Broglie's dissertation and the wave nature of matter [DB]
Quantized energy levels [QL]
Centripetal acceleration [CA]
As aforementioned, we think about an atom, the Rutherford model comes to mind. You know... that picture with the circle in the middle (the nucleus), surrounded by rings (electron orbits). Well... I'm here to tell you that is wrong. Electrons can't orbit around a nucleus, because if they did, we'd all be dead. Or rather, we'd never have existed in the first place. Nothing would have existed actually. Quite a large fallacy for something which is so integral to our existence. So another model had to be proposed. Now we know this alternate model to be quantum physics. But first thing is first -- why can't an electron orbit around a nucleus? Because we'd all die is the short answer; because an accelerating point charge emits energy (in the form of electromagnetic radiation, which I will refer to as EMR), is the long answer.
So... What the flip does that mean? Well usually, when people think about acceleration, they imagine a car speeding up/slowing down. What most fail to realize/acknowledge is that velocity is a vector, which has both magnitude and direction. Acceleration, which is then defined as a change in velocity per unit time, not only depends upon the magnitude of the velocity (i.e. speeding up/slowing down), but also direction. If an object is orbiting another body, it is fair then to say that the direction of the object changes at every instant. And so, an electron is indeed accelerating constantly around the nucleus, though with a constant magnitude of velocity. This phenomenon is referred to as centripetal acceleration, and wouldn't actually matter so much if the electron did not have a charge associated with it. If it were just a random flipping thing floating around another random flipping object, we'd have no problem accepting Rutherford's model.
The problem however is then, electrons do possess a charge. And all point charges generate an electric field around them. (There's a considerable bit of vector/multivariable calculus involved with this derivation -- so I won't bother going into it.) Just imagine, that an electron is a lightbulb. A lightbulb does not emit light from just one point. It emits light spherically/radially around it. The same is true of an electron. Furthermore, an electron moving with constant velocity (so in a straight line), also emits a radial magnetic field around it. This is cool and all, but what we are interested in is what happens when a point charge changes velocities/accelerates. Well, first, we need to understand that an electric/magnetic field is infinite. It goes on forever and ever and ever, through space. When a particle is moving at constant velocity (i.e. not accelerating), every individual point along this field (you can think of a field as an infinite number of points around a point charge, each point of this field is moving with some constant velocity. And the field looks the same way. However, when a point charge begins to accelerate, what happens then? If a point charge begins to accelerate (in this case, change direction), each individual point in its field must experience an acceleration of the same magnitude. However, acceleration begins first at the point charge, meaning that for some infintesimally short period of time, that only the center of the electron is accelerating. After this time has passed, then the field points in the immediate vicinity begin to accelerate. Following this line of thinking, we see that that the farther away from the point charge we get, the later the field begins to accelerate. This leads to a "break", in the electric field. But like I said, electric/magnetic fields are infinite. You cannot break them. That's just not how it works. So instead we get a picture which looks like this:
The contents within the innermost circle are the points in the field experiencing acceleration. The particles outside the bigger circle have not yet begun to experience acceleration. And so, if we didn't have those lines connecting the inner and outer fields, we would have broken an electric/magnetic field, which is something we never want to do. So what are those line? Those lines are EMR, aka energy.
Now, we can look to Einstein's very famous equation: \(E = mc^2\). What this is actually saying is that because of the absence of field "particles" within this ring, that something must be filling in the gap. All Einstein's equation does, is give us a means by which to interconvert matter and energy. That is, the internal energy possessed by an object is proportional to its mass times the speed of light squared. And similarly, that a wave, of energy \(E\), has a mass of \(\frac{E}{c^2}\). Basically, matter is energy, and energy is matter. And therefore, in the absence of matter, energy must exist. So if there are no field particles, there must be field energy to compensate. Which is nice and all... But what does that say about the electron? If we take a look at Einstein's formula again (\(E = mc^2\)), and the electron is particle with a mass of around \(10^{-18}\) g (don't quote me on that), that it has around \(10^{-8}\) J of energy. Each time it accelerates (which during orbit is constant), it loses some of this energy to compensate for the absence of field particles. Eventually, once it has lost all of its energy, it no longer has mass either. If \(E = 0\), then \(m\) must also be \(0\). And thus, matter ceases to exist (it's slightly more complex than that... but we won't deal with that). Now that we have proven that if the Rutherford model were correct that nothing would exist, we have to propose an alternate model. This comes in the form of quantum mechanics, which is surprisingly based on very simple postulates.
Photoelectric effect and Planck's constant [PE]
We then move to a discussion of the photoelectric effect. This is simply a fancy term given to the phenomenon which occurs when light is shined upon a metal. Essentially, at frequencies above, some frequency w, electrons are discharged from the middle. Shining light of a lower frequency releases no electrons; shining light of a higher frequency releases the same number of electrons, the only difference being that they possess a different kinetic energy (KE). This implies then, that the energy levels within an atom must be discrete. That is, electrons can only exist with certain energies within an atom. All other energies are inpermissible. It also suggests that light, is not a wave but rather a discrete packet of energy (called a photon). The energy possessed by a photon is given by, \(E = hf\), where \(h\) is Planck's constant (simply the slope of KE vs. frequency) = \(6.626\times10^{-34}\), and \(f\) is the frequency of light. I've also included a plot of KE against KE below (just in case you're curious).
That's cool. But what about our whole electron problem? Well. Let's go back to Einstein's equation \(E = mc^2\). We can rewrite this as \(m = \frac{E}{c^2}\). Simple enough, right? Now, we can introduce another quantity, momentum, denoted \(p = mv\). The momentum is the collision force of a particle (likelihood to remain in motion), and is the product of a particle's mass and its velocity. But because \(m = \frac{E}{c^2}\), we can also write momentum as, \(p = \frac{E}{c^2}\times{c}\). And we end up with \(p = \frac{E}{c}\). Which can be further simplified as \(p = \frac{hf}{c}\). Given that \(c = f\lambda\), where \(f\) is frequency and \(\lambda\) is wavelength, we can rewrite this as \(p = \frac{h}{\lambda}\). If we consider all matter instead of just EMR, with velocity, \(v\), we get \(\lambda = \frac{h}{p} = \frac{h}{mv}\). What we have just proven here, is that matter... with mass. Has a wavelength. Which means that it must exhibit wavelike properties.
De Broglie's dissertation and the wave nature of matter [DB]
The equation we just derived in the previous section is referred to as De Broglie's wavelength equation, and was also the topic of his rather short PhD thesis. Just to recap, De Broglie's equation is given by, \(\frac{h}{mv}\), and suggests that all matter has an associated wavelength. So, why can't we see this in the macroscopic world? Well. Think about a car with velocity, \(10000\) m/s and mass, \(100\) kg. If we calculate the wavelength of a car's motion using the equation we just derived... we get. \(6.626\times10^{-40}\) m. Now, that's infinitesimally small. Much smaller than an electron itself. In the microscopic world of atoms however, this is slightly more observable and can be proven through experimentation.
Quantized energy levels [QL]
Returning to the electron, which we previously believed was matter, we now must admit, can also be considered a wave. And so electrons do not "orbit" around the nucleus, but radiate around the nucleus of a wave.
What about the whole notion of discrete energy levels? Can't electrons just "radiate" around the nucleus at any point in space... No. We've already been over the whole, we can't break waves/fields deal. So actually, we need the electron, with a wavelength, \(\lambda = \frac{h}{mv}\), to radiate around the nucleus, such that the wave begins and ends at the same point along the circumference of radiation. The circumference of a circle, is as we know, \(2{\pi}r\). Therefore, we can say that \(n\) wavelengths must add up to \(2{\pi}r\), and thus, \(n\lambda = 2{\pi}r\). Granted that we cannot change the wavelength of an electron which is fixed, all we may do is change the radius of orbit such that \(n\) is an integer value... And so electrons can only exist at certain radii \(r\), away from the nucleus. And hence energy is most certainly quantized, and matter most certainly exists... excluding the whole business with quantum field theory.
Obviously, what has been presented in this discussion is extremely simplified so as to not confuse the average reader with the mathematical rigor of some of these derivations. If you'd like me to elaborate on this post/make another post examining quantum mechanics from an intensely mathematical perspective, let me know. As always, I hope you enjoyed reading this as much as I enjoyed writing it; questions are always welcome!
