Historical Origins of Quantum Mechanics

Where do correct ideas come from? Do they drop from the sky? No! Are they innate in the mind? No! They come from social practice, and from it alone.

— Mao Zedong, Draft Decision of the Central Committee of the Chinese Communist Party on Certain Problems in Our Present Rural Work

☝️ actual quote the Greensite Lecture began Chapter 2 with btw

Atoms… As anyone who’s watched a pop-science video ever can tell you, we owe the name “atom” to the Greek thinker Democritus who lived five centuries before Christ. But it was not until the 1800s that we began to see real evidence for this—particularly from thermodynamics. Here it was, that if we were to assume gases were made of atoms, one could derive the state for ideal gasses via $PV=nRT$ (Pressure $\times$ Volume $=$ the Number of Moles of Gas $\times$ the Ideal Gas Constant $\times$ Temperature). And, assuming that solids were also made of atoms, one could figure out their specific heats which agreed pretty well with the experimental values at high temperatures.

By the early 1900s improvements in technology and the discovery of radioactivity enabled physicists to study the internal structure of atoms, the mass ($m_e$) and charge ($q_e$) the electron, and the interaction of atoms with light. But certain aspects of physics which emerged from these early investigations were puzzling and even paradoxical, in the sense that the observed behavior of electrons, atoms, and light seemed to contradict the known laws of mechanics and electromagnetism.

These aspects fell roughly into three categories:

1. The Particle-like Behavior of Light Waves Black-body radiation, the photoelectric effect, the Compton effect.
2. The Puzzling Stability of the Atom Why doesn’t the electron fall into the nucleus? What is the origin of atomic spectra?
3. The Wave-like Behavior of Particles Electron diffraction.

And so quantum mechanics emerged to try and explain what was going on. Let’s begin.

Black-Body Radiation

Isaac Newton thought that light was made of particles. He reasoned that all wave motion exhibits interference and diffraction effects, which are the signature of any phenomenon involving waves. He looked for these effects by passing light through small holes, but didn’t see them. Thus he concluded that light is a stream of particles.

Now, one of Newton’s contemporaries, Christian Huygens, believed in the wave theory of light. Huygens pointed out that the refraction of light could be explained if light moved at different velocities in different media, and that Newton’s inability to find diffractive effects could simply have been because of the insensitivity of his experiments. Interference effects are most apparent when wavelengths are comparable to, or larger than, the size of the holes. If the wavelength of light were very small compared to the size of the holes used by Newton, interference effects would be very hard to observe. Huygens turned out to be right. More sensitive optical experiments by Young (1801) and Fresnel demonstrated the interference and diffraction of light, and measurements by Foucault (1850) showed that the speed of light in water was different from the speed of light in air, as required to explain refraction.

Then in 1860, Maxwell by unifying and extending the laws of electricity and magnetism, demonstrated that electric and magnetic fields would be able to propagate through space as waves, traveling with a velocity …

$$v=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}$$

speed of the wave (light in a vacuum) is equal to one over the square root of the permeability of free space (how magnetic fields behave in a vacuum) times the permittivity of free space (how electric fields behave in a vacuum)

…which turned out to be equal within experimental error, the known velocity of light. Experimental confirmation of the existence of electromagnetic waves followed shortly after, and by the 1880s the view that light is a wave motion of the electromagnetic field was universally accepted.

But, ironically, evidence began to accumulate that light was, after all, a stream of particles. Now, here, the Greensite Lecture notes talk about “black-body” radiation. I think it is first useful to define it. And also, why we care:

I should add pictures to this. Also might be worth adding an interactive "quiz" on what each variable in every formula mentioned means...

When light comes into contact with an object, it can either absorb it, reflect it, or transmit it. Reflection is what gives off the appearance of color. For example, plants absorb all light that isn’t green, which is why they appear to be that color. Transmission is when light passes through something (think glass). Black-bodies are these (idealized?) objects that only absorb light, giving off a black appearance. But this is interesting. Most objects we like to model as “black-bodies” are not what you’d. Like, an idealized black body would have an emissivity ($ϵ$) of $1$. Compare that to this table

Material	Emissivity
Human Skin	0.97–0.98
Water	0.95–0.96
Wood	0.90–0.95
Stainless Steel	0.10–0.30
Phenolic Impregnated Carbon Ablator	0.85–0.90
The Sun	0.99

Stainless steel, with such low emissivity is used as a heat shield on Spacecraft. (Sidebar: You may be surprised at the emissivity of PICA-X, given that it’s also used as a heat shield. It actually works by pyrolysis: the material chars and breaks down under intense heat, and the gases produced carry the energy away with it.) Anyways, stars, which are bright, are pretty close to ideal “black-bodies.” I found this confusing but I think it should help to clarify the rest of the section about it.

The way this works is that atoms in any body are always in thermal motion — electrons and nuclei constantly jostling and colliding. These are charged particles, and oscillating charges radiate, per Maxwell. So a body at temperature $T$ is emitting light just by virtue of being warm, no incoming light required. The hotter the body, the more energetically the charges oscillate $T_{body}\uparrow$, the more power gets radiated $P_{light}\uparrow$. That’s black body radiation: thermal emission, driven entirely by temperature. (The “black” part just refers to the idealization that the body absorbs everything that hits it — it says nothing about the color it actually glows.)

We can calculate this light via

$$P_{\text{light}}=\sigma A{T}^4$$

Where $\sigma$ is the “Stefan–Boltzmann constant,” $T$ is the temperature and $A$, the surface area

The spectrum of light emitted by an object that that absorbs all incoming radiation depends solely on the temperature, not the material. Hotter objects emit more total energy peaking at a shorter, bluer, wavelengths. Any object, at any finite temperature, emits electromagnetic radiation at all possible wavelengths. The mechanism is simple: atoms are made of negatively charged electrons and positively charged nuclei. Upon collision with other atoms, these charges oscillate in some way— like the the electron cloud oscillates relative to the nucleus. According to Maxwell’s theory, oscillating charges emit (and can also absorb) electromagnetic radiation. So it is not really a mystery that if we have a metallic box whose sides are kept at some constant temperature $T$, the interior of the box will be filled with electromagnetic radiation, which is constantly being emitted and reabsorbed by the atoms which make up the sides of the box. There was, however, some mystery in the energy distribution of this radiation as a function of frequency.

The energy density of the radiation in the box, as a function of frequency is easily worked out using the equipartition principle of statistical mechanics. (By the way, the equipartition principle states that for a system in thermal equilibrium at an absolute temperature $T$, every independent “way” a system can store energy—known as a degree of freedom—carries the same average amount of energy: $\frac{1}{2}{k}_{B}T$, where $k_B$ is the Boltzmann constant.) So the total energy, meaning the total energy of the $\text{rad}$iation inside the box, is

$$\begin{array}{rl}{E}_{rad}& =\text{no. of degrees of freedom}\times \frac{1}{2}kT\\ & =2\times \text{(no. of standing waves)}\times \frac{1}{2}kT\end{array}\text{\hspace{1em}(2.1)}$$

where $k$ is Boltzman’s constant and $T$ is the temperature of the box. By the way in classical statistical mechanics, each quadratic degree of freedom in the Hamiltonian contributes $\frac{1}{2}kT$ to the average energy, by the equipartition theorem. For an electromagnetic standing wave (mode) in a cavity.

An electromagnetic field in a box can be thought of as a superposition of an infinite number of standing waves; the “degrees of freedom” are the amplitudes of each distinct standing wave. The factor of $2$ comes from the fact that each standing wave can be in one of two possible polarizations. As we will see later, the number of standing waves that can exist in a cubical box of volume $V$, for frequencies in the interval $[f,f+\Delta f]$, is

$$N(f)\Delta f=V\frac{4\pi }{c^3}{f}^2\Delta f\text{\hspace{1em}(2.2)}$$

Why the $\Delta f$

The $\Delta f$ means we’re not counting waves at a single exact frequency, rather we’re counting waves whose frequencies fall within a tiny window of frequencies. Like If you have a box of waves, their frequencies aren’t all exactly the same. They’re spread out. So instead of asking: “How many waves have frequency exactly 500 Hz?” (which is impossible to answer precisely), you ask: “How many waves have frequency between 500 Hz and 501 Hz?” That interval —from 500 to 501— is $\Delta f$. Its width is 1 Hz. As the box gets bigger, the allowed frequencies get closer and closer together. At some point, it’s easier to talk about how many waves exist in a small range rather than listing each one individually. So $\Delta f$ is just the width of that small range. If you want the total number of waves between frequency $f_1$ and $f_2$, you’d add up (integrate) $N(f)\Delta f$ over that range.

Then, the energy of radiation in this range of frequencies will be

$$\begin{array}{rl}\Delta E_{rad}& =2N(f)\Delta f\times kT\\ & =\frac{8\pi kT{f}^2}{c^3}V\Delta f\end{array}\text{\hspace{1em}(2.3)}$$

Why $kT$ and not $\frac{1}{2}kT$

Each electromagnetic mode behaves like a harmonic oscillator with two quadratic degrees of freedom: one for the electric field amplitude, one for the magnetic. Equipartition gives $\frac{1}{2}kT$ to each, so the average energy per mode per polarization is $\frac{1}{2}kT+\frac{1}{2}kT=kT$. Equation (2.1) lists $\frac{1}{2}kT$ as a schematic placeholder for “energy per degree of freedom” — the factor of 2 out front accounts for polarizations, but you still need to apply equipartition correctly to the mode itself.

The energy density per unit frequency $\mathcal{E}(f,T)$ is therefore

$$\mathcal{E}(f,T)\equiv \frac{\Delta E_{rad}}{V\Delta f}=\frac{8\pi kT{f}^2}{c^3}\text{\hspace{1em}(2.4)}$$

which is known as the Rayleigh-Jeans law.

What

Here $\mathcal{E}(f,T)$ is a function that tells you how much energy is stored per cubic meter of the box per hertz of frequency, depending on the frequency $f$ and the temperature $T$. The quantity $\Delta E_{rad}$ is the energy contained in a small slice of frequencies from $f$ to $f+\Delta f$, not the total energy from all frequencies. Dividing by $V$ removes the dependence on the size of the box, and dividing by $\Delta f$ removes the dependence on how wide the frequency window is, leaving a quantity that depends only on the color of light ($f$) and how hot the box is ($T$).

The Rayleigh-Jeans law can be tested by making a hole in the box, and measuring the intensity of radiation emitted from the box as a function of frequency; this intensity is directly proportional to $\mathcal{E}(f,T)$. Radiation from a small hole in a cavity is known as “black-body radiation”, because any radiation falling into the hole is not reflected out the hole, but is ultimately absorbed by the walls. The experimental result, compared to the prediction, is shown in $\text{Fig. [2.1]}$. Theory disagrees with experiment at high frequencies. In fact, it is clear that there had to be something wrong with theory, because the total energy is predicted to be

$$\begin{array}{rl}{E}_{rad}& =2\times \text{(no. of standing waves)}\times \frac{1}{2}kT\\ & =\infty \end{array}\text{\hspace{1em}(2.5)}$$

simply because the range of frequencies is infinite, so there is an infinite number of different standing waves that can be set up in the box. But obviously The energy of a box is finite, because otherwise its mass, according to special relativity, would be infinite. Thus, this result cannot possibly be correct.

Planck’s contribution to this problem was a masterpiece of what is known in physics as phenomenology.

def. looking at experimental data and finding a mathematical formula that fits it without necessarily knowing why it works, then trying to find a physical justification afterwards

The first step of phenomenology is to stare at the data, in this case the experimental curve shown in $\text{Fig. [2.1]}$, and try to find some simple analytical expression that fits it. Planck found that

$$\mathcal{E}(f,T)=\frac{8\pi h{f}^3}{c^3}\frac{1}{e^{hf/kT}-1}\text{\hspace{1em}(2.6)}$$

would do nicely, if one chose the constant $h$ to be

$$h=6.626\times 10^{-34}\text{ J}\cdot \text{s}\text{\hspace{1em}(2.7)}$$

The second step is to try to derive the analytical expression for the data, starting from some simple physical assumptions about the system. Planck took aim at the equipartition principle. This principle is only valid if the energy associated with each degree of freedom can take on any value between $0$ and $\infty$, depending on the physical state. In electromagnetism, the energy of a standing wave of a given wavelength is proportional to the square of its amplitude, which can certainly be any number in the range $[0,\infty ]$. Planck’s suggestion was that, for some unknown reason, the oscillating charges in the walls could only emit or absorb energy in multiples of $hf$, where $f$ is the frequency of the oscillator. This means that the energy of radiation of frequency $f$ in the box could only have the possible values

$$E_n=nhf\text{\hspace{1em}(2.8)}$$

where $n$ is an integer. This assumption, combined with the rules of statistical mechanics, is enough to deduce the Planck distribution (2.6).

Note the appearance in Planck’s formula of the constant $h$, known as Planck’s constant. It is one of the three most fundamental constants in physics, sharing the honor with $c$, the speed of light, and $G$, Newton’s constant. All theoretical predictions of quantum physics, to the extent that they disagree with classical physics, have Planck’s constant $h$ appearing somewhere in the expression.

The Photoelectric Effect

The success of Planck’s idea immediately raises the question: why is it that oscillators in the walls can only emit and absorb energies in multiples of $hf$? The reason for this was supplied by Albert Einstein in 1905, in connection with his explanation of the photoelectric effect.

It was found by Lenard, in 1900, that when light shines on certain metals, the metals emit electrons. This phenomenon is known as the photoelectric effect, and what is surprising about it is that the energy of the emitted electrons is independent of the intensity of the incident light.