We begin our series of lessons on quantum theory with a short primer in the historical events that led to the development of the quantum theory, such we can have a full appreciation of the limitations that classical mechanics have in explaning our world.

It was near the end of the 19th century when physicists realised that the classical Newtonian physics that they have used for well over a century was beginning to fail them at explaining some novel concepts (at least at that time). We shall discuss these concepts in order to gain true appreciation of quantum mechanics and how it solves many of chemistry’s problems.

## The Ultraviolet Catastrophe

It is as scandalous as it sounds, the ultraviolet catastrophe was the study that started it all. The term was coined by Ehrenfest in the early 20th century. To understand the significance, we must first learn about black-body radiation. It is the radiation emitte by a body that is in thermal equilibrium. It is governmed by two laws:

- The Stefan-Boltzmann Law:

\[M=\sigma{}T^4\]

Where \(M\) is the excitance, measuring the power of emitted radiation devided by the area of the emitting region, \(\sigma\) is the Stefan-Boltzmann constant, which is independent of the material of the radition-emitting body, and \(T\) is the absolute temperature in Kelvin. The above equation simply tells us the power per unit area of a black body that is emitting radiation at a certain temperature.

The above equation assumes the emitted radiation favours no particular frequency, which is an ideal situation. This brings us to the second equation:

2. Wien’s Displacement Law

\[\lambda_{\text{max}}T=\text{constant}\]

As temperature of a radition-emitting body increases, the shorter wavelengths are favoured, and this equation accounts exactly for that. \(\lambda_{\text{max}}\) is the wavelength at which the intensity of the radiation is greates, \(T\) is temperature, and the constant is of fixed value 2.9 mmK.

Perhaps one of the most challenging physical problems at the end of the nineteenth century was to explain the two laws that govern blackbody radiation. Each one of the two laws focused on finding an mathematical formula for **energy density** \(\mathcal{E}(\lambda)\), the energy in a region divided by the volume of the region. And if we write the contribution \(d\mathcal{E}(\lambda)\) from radiation in the wavelenth range from \(\lambda\) to \(\lambda + d\lambda\) as

\[d\mathcal{E}(\lambda)=\rho_R(\lambda)d\lambda\]

Where \(\rho_R\) is the **spectral density of states **at the wavelength \(\lambda\). Lord Rayleigh, along with James Jeans, proposed the theoretical **Rayleigh-Jeans** **law** for the spectral density:

\[\rho_R(\lambda)=\frac{8\pi kT}{\lambda^4}\]

The more mathematically keen among you will have spotted a problem. Perhaps it is more obvious in the following form:

\[\lim_{\lambda \rightarrow 0} \rho_R(\lambda)= \text{DNE}\]

The above limit suggests, that without regards for temperature, there should always be an infinite energy density at very short wavelengths approaching zero, which is physically absurd. This phenomenon, or perhaps failure, was coined *The Ultraviolet Catastrophe, *by Ehrenfest.

It was around this time that Max Planck made his historic contribution. He suggested that an oscillation of an electromagnetic field (what we perceive as light/radiation) of frequency \(\nu\) can be excited **only** in steps that have the magnitude of \(h\nu\), where \(h\) is the new constant, to be known now as **The Planck’s Constant**. The problem that led to the catastrophe was the assumption that energy is continuous, which led to the problem that even a very high frequency can be excited with a very small energy: that was the source of the problem since classical mechanics seem to suggest that high-energy radiation of short wavelengths can be emitted even at low temperatures).

Planck implemented his suggestion and derived what is now called **the Planck Distribution:**

$$\rho_R(\lambda)=\frac{8\pi hc}{\lambda^5}\frac{e^{\tfrac{-hc}{\lambda{}kT}}}{1- e^{\tfrac{-hc}{\lambda{}kT}} }$$