Disclaimer: I am aware that the de facto model for financial markets is the Black-Scholes equation, however this is an intellectual study exploring a stock price by the Schrödinger wave equation.

Thesis

While the stock market is considered to be purely random price movements, any day trader will tell you there are indicators and price action, driven by both volume and momentum, that can influence the price of a stock. Just as the precise position of a fundamental particle cannot be pinned down with certainty, we accept the probabilistic nature by determining the highest probability of its being somewhere: the collapse of the wave function. I hypothesize that the Laplacian operator is the closest analog we have to the gradient measurement in price action at any given moment.

A quick primer on Quantum Mechanics

Venturing into the realm of quantum mechanics demands a departure from conventional logic. As the renowned physicist Richard Feynman aptly expressed, "If you think you understand quantum mechanics, you don't understand quantum mechanics." This domain challenges our preconceived notions of the universe's fundamental laws and it's been thought that by stumbling upon quantum mechanics, we've opened the door to a realm we are still hundreds of years behind mathematically and logically in understanding. Here, the precise position of a particle remains elusive, and we can only describe the probability of its existence within a certain region. What was once thought of as the indivisible atom has revealed its complex internal structure, unveiling a universe within. The scale at which these mechanics unfold is pivotal, each domain governed by its unique interpretation of the fundamental laws.

Fundamental particles, the universe's most elementary building blocks, are characterized by the electric charge they carry. This charge, in turn, exerts a transformative influence on the space they occupy, changing the very geometry it inhabits. Particles bearing electric charge alter the charge space, those carrying isospin modify the isospin space, and the enigmatic "color" charge (not to be confused with the colors we perceive) reshapes the color space. Additionally, particles carrying energy-momentum, often referred to as mass, impart their influence upon spacetime. This intricate interplay between particles and the geometry of space forms a symbiotic relationship akin to a flywheel.

Within the world of fundamental particles, we encounter a diverse array of entities, including quarks, leptons, and bosons. Quarks, fundamental constituents of protons, neutrons, and other hadrons, manifest in six distinct "flavors": up, down, strange, charm, bottom, and top. Leptons make their presence felt through the electron, neutrino, muon, tau, and their respective neutrinos. Bosons, the mediators of fundamental forces, encompass photons, gluons, W and Z bosons, and the Higgs boson, the last of which was revealed to the world in 2011. This journey through the world of fundamental particles allows us to continue unraveling the profound mysteries of the universe.

The dual nature of fundamental particles, exhibiting both wave-like and particle-like characteristics, is a foundational principle in quantum mechanics. According to the theory, particles can exist in a state of superposition, where they simultaneously occupy all possible locations until an observation is made to determine their position. However, this concept goes even further, as the act of observation doesn't yield a definitive result; instead, it leads to what is known as the "collapse of the wave function." In this state, the particle is associated with a probability distribution of being in a specific position, rather than a single, precise location. This phenomenon underscores the inherent probabilistic nature of quantum systems, challenging classical intuitions and forming the basis for quantum mechanics' unique and often perplexing characteristics.

Erwin Schrödinger proposed the famous thought experiment known as "Schrödinger's Cat" in 1935 as a way to illustrate some of the counterintuitive aspects of quantum superposition and entanglement, particularly highlighting the idea that a quantum system can exist in multiple states simultaneously until observed. Werner Heisenberg then built upon these ideas, making significant contributions to quantum mechanics like the uncertainty principle, highlighting the uncertainty of a particle's exact location, warranting the need for its probability.

We will begin with the Schrödinger equation, a cornerstone of quantum mechanics named after Austrian physicist Erwin Schrödinger, who formulated it in 1926. But before delving into its intricacies, let's establish a foundation by revisiting some key concepts.

A partial differential equation is a mathematical construct involving two or more independent variables, an unknown function dependent on these variables, and partial derivatives of the unknown function concerning the independent variables. The Laplacian operator, signified by the nabla, plays a crucial role in our journey and provides a means to calculate the divergence of a function's gradient.

The Schrödinger equation comes in two flavors, one for non-relativistic quantum mechanics (which is Schrödinger's time-dependent equation) and another for relativistic quantum mechanics (the Dirac equation). The non-relativistic version is the more commonly encountered use case, so we'll explore that one as shown here:

If a particle is constantly moving at high speeds in a random motion, how do we ponder the probability of that particle's next location, a concept profoundly intertwined with the wave function? This probability hinges on the diffusion constant (multiplied by the imaginary unit "i"), the divergence of gradient vectors, and the probability of motion. Diffusion is what Joseph Fourier was studying when he devised the Fourier series as a way to break down any wave into its constituent harmonics. It's no accident that the property of waves inherent in the Fourier series is also present within the wave-like nature of fundamental particles. If you're not convinced the quantum world bypasses any sound logic, the fact that it is expressed as a complex number which is inherently imaginary should drive the point home.

This equation looks intimidating but in reality, we fear what we don't know and we can learn about this equation by slowly dipping our toes. In the most simplest of terms with this equation:

• Left side → tells us how a quantum state changes with time.
• Right side → tells us it's due to the particle spreading out in space while also feeling the influence of its surroundings.

Now going a level deeper, the equation is telling us: the left side represents how the quantum state changes over time as a complex number, while the right side represents the particle's spread in space, considering its quantum wavelength and the influence of potential energy.

The h-bar is the reduced Planck constant, which relates quantum properties to classical analogs, emphasizing quantum limitations. The evolution of the quantum state over time is represented by:

V is the potential energy which accounts for external forces' that impact the quantum particle throughout its motion. M is the mass of the particle and connects the particle mass to its kinetic energy and momentum. We covered the Laplacian operator which quantifies the spatial variation and wave function curvature.

Solving this equation allows us to determine the allowed energy levels and corresponding wave functions for quantum systems, which in turn, helps predict and explain various physical phenomena, such as the behavior of electrons in atoms and molecules.

The wave function denoted by psi, describes the quantum state of a particle. It was Max Born's interpretation of the wave equation that stated the square of its absolute value represents the probability density, indicating the likelihood of finding the particle at a specific location.

In other words, the more concentrated the wave function is in a particular region of space, the higher the probability of finding the particle there upon measurement.

Because quantum systems can exist in superpositions of multiple states, the Schrödinger equation allows these superpositions to evolve over time (which is what the partial derivatives and its time-dependence nature grants us). This results in a complex probability distribution, where a particle may simultaneously occupy multiple possible positions until a measurement is made, causing the wave function to "collapse" to a specific state.

To fully immerse oneself into the world of quantum mechanics, we use a new type of geometry known as "internal spaces", which typically refers to the abstract spaces that represent the quantum states of particles or systems. These internal spaces don't correspond to the physical dimensions of space and time that we encounter in classical physics but rather to the mathematical spaces used to describe quantum properties. While these spaces are abstract, one could ponder on its inherent "reality", since it is upheld mathematically although not thought to be "physically" real. The same was said for imaginary numbers and yet we find their influence entrenched in the quantum world.

This special geometry in these internal spaces is different from the familiar Euclidean geometry of everyday experience and is often described using complex vector spaces. The concept of inner products plays a crucial role in defining quantum states, operators, and "observables". One of those internal spaces used is called Hilbert spaces, named after the German mathematician David Hilbert. These spaces provide a rigorous and elegant way to model the properties and behavior of quantum particles.

At its core, a Hilbert space is a complex vector space with certain key properties. It is a vector space because it allows for the addition and scalar multiplication of vectors, which represent quantum states. These vectors represent the possible states a quantum system can be in, encompassing the probabilistic nature of the states of these particles.

One defining feature of Hilbert spaces is the concept of inner products, which is a mathematical operation that assigns a complex number to pairs of vectors, capturing the notion of "overlap" or "similarity" between quantum states. The inner product is used to define the notion of orthogonality and, by extension, the probability amplitudes for finding a quantum system in a specific state. Hilbert spaces also have the property of completeness, meaning that they encompass all possible quantum states for a given system. This ensures that all conceivable quantum states are mathematically represented within the space. The infinite dimensionality of Hilbert spaces accommodates the vast range of quantum possibilities, making them a versatile and powerful tool to traverse the strange world of quantum mechanics.

Particle mass as an individual stock

There’s a reason quants are often physics and mathematics PhDs. They have a firm grasp on non-linear, multi-dimensional dynamics and can model them accordingly. The same science that goes into the study of fundamental particles can be applied to a stock price, so it’s not a radical departure to associate a stock price with a fundamental particle.

In fact, both are said to be random movements which should automatically put us in the probabilistic mindset. Just as each quantum state would correspond to a different stock price, the wave function which typically used to describe a particle's quantum state, could represent the probability of the stock having a particular price. The diffusion equations incorporate the external environment that acts on the movement of the particle, which in our case could be the constant battle between bulls and bears we call price action.

The time-dependence of the Schrödinger wave equation can be used to simulate how the probability distribution of the stock price changes over time and one could add variables to incorporate factors like market volatility, trading volume, and other financial data.

The Planck constant was used to limit the range at the quantum level, you could allude to indicators like Bollinger bands to make the same comparison. These all act as various inputs that help improve the probability that a wave would collapse in a certain position, or in our case that the stock would coalesce towards a single price in that given moment.