Quantum Mechanics – An Introduction – A Guide to Mathematical Studies (Archive)

A lot of quantum physics involved near the end!

Quantum mechanics arose in response to the limitations of classical physics in explaining the behavior of particles at atomic and subatomic levels. Early 20th-century experiments, such as the photoelectric effect and the double-slit experiment, revealed phenomena that challenged classical understanding. The improved understanding of sub-atomic particles prompted new theories, with scientists such as Planck, Einstein, Bohr, Heisenberg, and Schrödinger making significant contributions to quantum mechanics, proposing new theories and mathematical frameworks.

So before we begin, what is quantum mechanics? According to the Department of Energy, “Quantum mechanics is the field of physics that explains how extremely small objects simultaneously have the characteristics of both particles (tiny pieces of matter) and waves (a disturbance or variation that transfers energy).” While classical mechanics handles concepts like SUVAT and Stokes’ law, it fails to explain the behavior of smaller particles such as electrons and photons. In this article, we’ll delve into these features and explore the historic background from which quantum mechanics emerged. Let’s begin!

Part I : The Nature of Light and The Photoelectric Effect

Before delving into quantum mechanics, it’s essential to understand the nature of light, as it played a pivotal role in shaping our understanding of quantum phenomena. From the outset, two distinct theories emerged to explain the nature of light. The first conjecture for such was made by Newton, who proposed the idea of corpuscles, which were particles which interacted with objects, e.g. our retinas to produce color. Such particles would travel in a straight line with a finite velocity and possess momentum. This was backed upon due to the extreme influence Newton held, and was heralded as the theory until the late 19th century where new evidence suggested otherwise.

Huygen on the other hand had a different approach to how light should be treated. In contrast to Newton’s theory, Huygens’ emphasized the wave-like behavior of light, explaining phenomena such as interference and diffraction observed in optics. This however was much overshadowed by Newton’s theory, and was only truly established as a fundamental concept in physics by the mid-19th century.

A true investigation into the nature of light however did not appear until the ultraviolet catastrophe. This was a theoretical problem encountered in the late 19th century when attempting to apply classical physics to the spectral distribution of blackbody radiation. According to classical wave theory, the intensity of radiation emitted by a blackbody should increase indefinitely with increasing frequency, leading to infinite energy at short wavelengths.

This prediction contradicted experimental observations, particularly in the ultraviolet region, where the intensity of radiation did not follow the expected pattern. The failure of classical physics to explain these observations signaled a fundamental flaw in the classical wave theory of light and spurred the development of quantum mechanics.

In response to this discrepancy, Max Planck proposed his groundbreaking idea of quantized energy levels in 1900, which laid the foundation for quantum theory. Planck suggested that energy is quantized and can only be emitted or absorbed in discrete packets called quanta, i.e. $E=nhf$ where $n\in\mathbb{Z}$ , $f$ is the frequency and $h$ is Planck’s constant, derived experimentally as $\approx6.63\times10^{34} \text{Js}$ . This is directly contradictory to what is assumed from wave theory – given the fact that the energy is directly proportional to the amplitude squared of any wave, or in an equation $E=\frac12m(A\omega)^2$ . Despite this, the equation perfectly resolved the ultraviolet catastrophe and marked the birth of quantum physics.

Another key principle of quantum mechanics is the wave-particle duality, which suggests that particles such as electrons and photons can exhibit both wave-like and particle-like properties depending on the experimental context. This concept challenged the classical notion of distinct particles following predictable trajectories and instead introduced probabilistic descriptions of particle behavior – for example quantum tunneling which we will talk about in a bit.

To demonstrate at this let us look at the photoelectric effect – proposed by Einstein, it demonstrated that light existed as particles (photons). When light is shone onto metal, electrons are emitted as they absorb the light and gain kinetic energy. This could be represented by the photoelectric equation $E=\phi+\frac12m_ev^2$ , where $\phi$ is known as the work function and is the minimum amount of energy required to eject an electron.

How light (photons) eject electrons. (Credit : Isaac Physics)

Graph of kinetic energy against frequency.

So apparently, light isn’t just a wave; it can also be viewed as a particle. In other words, light is quantized. However, in 1923, Louis de Broglie suggested that not only did waves exhibit particle-like aspects but the reverse was also true – all material particles displayed wave-like properties as well. This is known as the de Broglie wavelength, and is given by the formula $\lambda=\frac{h}{mv}$ – deriving this from the above principles and the mass-energy equivalence principle $E=mc^2$ is left as an exercise to the readers.

Part II : Schrödinger’s Equation

Now we are equipped to go onto deriving the Schrödinger’s Equation from scratch. There are a few things that we will have to define here, but the general derivation is pretty straightforward.

We first define wavefunctions – these are mathematical descriptions of how a wave behaves, and takes two parameters in this case, displacement $x$ and time $t$ . The wave function can thus be expressed by $\Psi(x, t)$ . This also relates to simple harmonic motion, in the sense that the general SHM equation is given by $x(t)=Acos(\omega{t}+\phi)$ , while the wave function we consider is of the form $\Psi = A \cos(kx - \omega t + \phi)$ , or in complex form $\Psi = Ce^{i(kx - \omega t)}$ .

Another thing we need to define is the angular frequency and the wavenumber of a particle. The angular frequency is denoted $\omega=2\pi{f}$ and the wavenumber $k=\frac{2\pi}{\lambda}$ . As a factor of $2\pi$ is introduced we can also denote a new constant called the reduced Planck’s constant which is given by $\hbar=\frac{h}{2\pi}$ . As such, we can rewrite our equations as $E=\hbar\omega$ and $p=\hbar{k}$ .

With the above we are able to derive Schrödinger’s equation.

Consider the energy conservation principle :

$E_{\text{total}}=E_{\text{kinetic}} +E_{\text{potential}}$

We can rewrite the above with the substitution $E_\text{total}=\hbar\omega$ , and substitute the kinetic energy by $E_\text{kinetic}=\frac{p^2}{2m}$ (deriving this is simple and thus also left as an exercise for the readers). This gives us the below formula :

$\hbar\omega=\frac{p^2}{2m} +V(x, t)$

Where $V(x, t)$ in this case is the potential energy function. By rewriting $p=\hbar{k}$ and multiplying through with a wave function, we obtain the following :

$\frac{{\hbar^2}}{{2m}} k^2 \Psi(x, t) + V(x, t) \Psi(x, t) = \hbar \omega \Psi(x, t)$

We would also like to replace $k$ and $\omega$ , which refer to the wave characteristics of the particle, by differential operators acting on the wavefunction $\Psi(x, t)$ . By considering different partial derivatives, we obtain the following :

$\frac{\partial}{\partial x}\Psi=-kA\sin(kx - \omega t + \phi)$

$\frac{\partial^2}{\partial^2 x}\Psi=-k^2A\cos(kx - \omega t + \phi)=-k^2\Psi$

Thus we can rewrite $k^2$ as $-\frac{\partial^2}{\partial^2 x}$ . In the same fashion, by taking the partial derivative with respect to $t$ , we find that $\omega\rightarrow{i}\frac{\partial}{\partial t}$ (Hint : differentiate $\Psi = Ce^{i(kx - \omega t)}$ instead).

By rewriting the above, we finally arrive at the following :

$-\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V(x, t) \Psi = i\hbar \frac{\partial \Psi}{\partial t}$

This is known as the one-dimensional Schrödinger equation. This is easily generalized to more dimensions with del notation :

$-\frac{\hbar^2}{2m} \nabla^2 \Psi(\mathbf{x}, t) + V(\mathbf{x}, t) \Psi(\mathbf{x}, t) = i\hbar \frac{\partial \Psi(\mathbf{x}, t)}{\partial t}$

$\nabla^2$ is known as the Laplacian and is written in cartesian coordinates as $\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ ( $\nabla$ is also known as the gradient function and is of utmost importance in multivariable calculus).

Part III : Hydrogen States and Potential Energy Wells

In a hydrogen atom, electrons can occupy different states known as orbitals. Orbitals represent regions in space where electrons are likely to be found, based on their probability distributions.

With all of this let us look at an example of how we can apply all these – the hydrogen atom (in this case we shall only consider $H^1$ ). It is only composed of 1 proton (2 up quarks and 1 down quark) and 1 electron, and is the simplest element to exist. As in the photoelectric effect, electrons can be provided with energy, causing them to become ‘excited’ and move to higher energy levels, represented by the principal quantum number $n$ where $n\in\mathbb{Z}$ (the set of integers). This process creates distinct energy gaps in the electron’s energy levels, particularly noticeable in the visible spectrum. When these excited electrons transition back to lower energy levels, they release energy in the form of photons, resulting in the emission of light. These transitions between energy levels produce additional gaps in the spectrum corresponding to the emitted photons, which is shown below.

Hydrogen absorption / emission spectrum. This corresponds to the Lyman series.

In fact, using the Schrödinger’s equation, we can explicitly solve for the energy of electrons in the Hydrogen atom in the Bohr model. This is given by the equation

$E_n = -\frac{{m_e e^4}}{{8 \varepsilon_0^2 \hbar^2}} \frac{1}{{n^2}}=-\frac{13.6}{n^2}\ eV$

From this we can also derive the Rydberg formula, which is left as an exercise for the readers. This equation relates the wavelength of photons absorbed/emitted when electrons move between energy levels with the Bohr equation derived above.

$\frac1{\lambda_{a\rightarrow b}} = -\frac{13.6}{hc}(\frac1{b^2}-\frac1{a^2})$

From this we can derive the first ionization energy for H, where we can consider the principal quantum number to be infinity. Starting at a state of 1, we attain the value $13.6\ eV$ for our energy required. Multiplying by Avogadro’s constant, we achieve our answer of $1350 \ kJmol^{-1}$ .

For the different energy levels, we can also view them as stationary waves due to wave-particle duality, and thus the circumference must be of the form $n\lambda=2\pi r$ . By using de Broglie’s wavelength ( $\lambda=\frac{h}{mv}$ ), we know that $\frac{nh}{mv}=2\pi r \Longrightarrow n\hbar=mvr=I$ where $I$ is known as the angular momentum. This quantization leads to the equal spacing of energy levels in atoms and molecules, where consecutive levels require or release the same amount of energy. This fundamental aspect of quantum mechanics underlies the stability of atomic orbits and the discrete nature of energy spectra.

To end today’s discussion, I would like to mention the effect of quantum tunneling. Mathematically, we can derive a energy square well with the following equations :

Analysing via traditional means, particles with energy less than 20 eV are never going to be able to pass through the wall due to conservation of energy – however, since the particles could behave like waves, there is a probability that a particle with sufficient energy could pass through the well. This, espressed (again) through the general equation for Schrödinger’s equation, we obtain the following :

For more information on Quantum Tunneling, you may find the video useful.

In our exploration of quantum mechanics, we have delved into phenomena that challenge classical physics. It all began with Planck’s discovery of quantized energy levels and the subsequent revelation of wave-particle duality. Schrödinger’s Equation furthered our understanding, allowing us to calculate fundamental properties such as the energy states of the hydrogen atom and the Rydberg formula. These principles not only shed light on the behavior of particles at the quantum level but also showcases concepts like quantum tunneling, which plays a pivotal role in various fields of science and technology.

To end this all, here is a simple exercise – derive the Heisenberg uncertainty principle $\Delta{x}\Delta{p}\geq\frac{\hbar}2$ ¹ – and as always, thank you for reading and I will see you next month!

https://brilliant.org/wiki/heisenberg-uncertainty-principle/ ↩︎

Quantum Mechanics – An Introduction

Comments

Leave a Reply Cancel reply