Applying Schradinger's Equation to Real-World Problems

ELECTRON TRANSPORT IN DNA HELIX CAN BE MODELLED WITH SCHRODINGER'S EQUATION Schrödinger's equation can be used to model electron transport within DNA, and its solution can reveal the DNA's response to external fields. This is a fascinating area of research that combines quantum mechanics, biology, and chemistry. Here's a detailed explanation: 1️⃣Electron Transport in DNA DNA (deoxyribonucleic acid) is a complex biomolecule that contains the genetic instructions for life. It's composed of two strands of nucleotides that twist together in a double helix structure. Electron transport refers to the movement of electrons along the DNA molecule, which plays a crucial role in various biological processes, such as DNA replication, repair, and gene expression. 2️⃣Schrödinger's Equation and DNA Schrödinger's equation can be applied to model electron transport in DNA by treating the electrons as quantum particles moving through the DNA molecule. The equation describes the time-evolution of the electron wave function, which encodes information about the electron's position, momentum, and energy. In the context of DNA, the Schrödinger equation can be written as: iℏ(∂ψ/∂t) = Hψ where ψ is the electron wave function, H is the Hamiltonian operator, i is the imaginary unit, ℏ is the reduced Planck constant, and t is time. 3️⃣Hamiltonian Operator The Hamiltonian operator (H) describes the energy of the electron in the DNA molecule. It includes terms that account for the electron's kinetic energy, potential energy, and interactions with the DNA's nuclei and other electrons. 4️⃣Solution of Schrödinger's Equation Solving Schrödinger's equation for the electron transport in DNA yields the electron wave function (ψ), which contains information about the electron's probability density, phase, and energy. The solution can be obtained using various numerical methods, such as the finite difference method or the density functional theory (DFT). 5️⃣DNA Response to Fields The solution of Schrödinger's equation can be used to study the DNA's response to external fields, such as electric or magnetic fields. By applying a field, the electron transport in DNA can be modulated, leading to changes in the DNA's structure and function. For example, an electric field can induce a current in the DNA molecule, which can affect gene expression, DNA replication, and repair. The Schrödinger equation solution can help predict the DNA's response to such fields, providing valuable insights into the underlying mechanisms. 6️⃣Implications and Applications The application of Schrödinger's equation to electron transport in DNA has far-reaching implications for our understanding of biological processes and the development of new technologies. Some potential applications include: 1. DNA-based electronics 2. DNA repair and gene expression 3. Quantum biology

Option Pricing with Quantum Mechanical Methods I first encountered a formal treatment of pricing financial derivatives using the framework of quantum mechanics in Baaquie’s book Quantum Finance when it was published. Over the years, the term “quantum finance” has appeared more frequently in literature. I paid limited attention to this line of work until the paper discussed below, which caught my interest by addressing a well-known problem using the language of quantum mechanics. The paper proposes an option pricing model that converts the Fokker–Planck equation into the Schrödinger equation, yielding both the return distribution and a closed-form solution for European options. The model shows that S&P 500 returns follow a Laplace distribution with power-law tails and that quantum methods outperform GBM-based models in explaining return dynamics and put option prices. Findings: -The paper proposes an option pricing model inspired by quantum mechanics to address the long-standing puzzle of overpriced put options. -The authors reformulate the stock return dynamics by transforming the Fokker–Planck equation into a Schrödinger equation. -This framework yields an explicit probability density function for stock returns and a closed-form solution for European option prices. -Empirical results suggest that S&P 500 index returns follow a Laplace distribution with power-law tail behavior rather than a Gaussian distribution. -The quantum-mechanics-based model outperforms traditional GBM-based models in fitting both index returns and observed put option prices. -The findings indicate that high put option prices observed in the market are close to fair value when modeled within this quantum framework. Reference: Minhyuk Jeong, Biao Yang, Xingjia Zhang, Taeyoung Park & Kwangwon Ahn, A quantum model for the overpriced put puzzle, Financial Innovation (2025) 11:130 Join a community of 7,000+ quants—subscribe to the newsletter! https://lnkd.in/gVFDBTCK #options #volatility #quantitativefinance ABSTRACT Put options are known to be priced unusually high in the market, which we refer to as the overpriced put puzzle . This study proposes a quantum model (QM) that can explain such high put option prices as fair prices. Starting from a stochastic differential equation of stock returns, we convert the Fokker–Planck equation into the Schrödinger equation. To model the market force that always draws excess returns back to equilibrium, we specify a diffusion process corresponding to a QM with a delta potential. The results demonstrate that stock returns follow a Laplace distribution and exhibit power law in the tail. We then construct a closed-form solution for European put option pricing, determining that our model better explains the returns of the S&P 500 index and its corresponding put option prices than do geometric Brownian motion-based models. This study has significant implications for investors and risk managers,...

The Schrödinger Equation Gets Practical: Quantum Algorithm Speeds Up Real-World Simulations Quantum computing has taken a major leap forward with a new algorithm designed to simulate coupled harmonic oscillators, systems that model everything from molecular vibrations to bridges and neural networks. By reformulating the dynamics of these oscillators into the Schrödinger equation and applying Hamiltonian simulation methods, researchers have shown that complex physical systems can be simulated exponentially faster on a quantum computer than with traditional algorithms. This breakthrough demonstrates not only a practical use of the Schrödinger equation but also the deep connection between quantum dynamics and classical mechanics. The study introduces two powerful quantum algorithms that reduce the required resources to only about log(N) qubits for N oscillators, compared to the massive computational demands of classical methods. This exponential speedup could transform fields such as engineering, chemistry, neuroscience, and material science, where coupled oscillators serve as the backbone of real-world modeling. By bridging theory and application, this research underscores how quantum computing is redefining problem-solving in physics and beyond. With proven exponential advantages and the ability to simulate systems once thought computationally impossible, this quantum algorithm marks a milestone in quantum simulation, Hamiltonian dynamics, and real-world physics applications. The findings point toward a future where quantum computers can accelerate scientific discovery, optimize engineering designs, and even open new frontiers in AI and computational neuroscience. #QuantumComputing #SchrodingerEquation #HamiltonianSimulation #QuantumAlgorithm #CoupledOscillators #QuantumPhysics #ComputationalScience #Neuroscience #Chemistry #Engineering

Researchers have developed a quantum algorithm that significantly accelerates the simulation of coupled oscillator dynamics, systems fundamental to various physical phenomena, including molecular vibrations and electrical circuits. Key Insights:  Quantum Algorithm Efficiency: The new algorithm simulates the dynamics of \( N \) coupled oscillators using only \( \log(N) \) quantum bits (qubits), a substantial reduction compared to classical methods that require resources proportional to \( N \).  Mapping to Schrödinger Equation: Researchers translated the behavior of coupled oscillators into a form solvable by the Schrödinger equation, enabling the application of advanced Hamiltonian simulation techniques.  Exponential Speedup: This approach offers an exponential speedup over classical algorithms, enhancing the efficiency of simulations for complex systems. Implications:  Broad Applicability: The algorithm can be applied to a wide range of systems modeled by coupled oscillators, such as mechanical structures, electrical networks, and even certain biological processes.  Advancing Quantum Computing: This development underscores the potential of quantum computing to tackle complex simulations more efficiently than traditional computing methods. Funding and Collaboration: The research was supported by the Department of Energy Office of Science, National Quantum Information Science Research Center, Codesign Center for Quantum Advantage (C2QA), with additional grants from Google Quantum AI and the Australian Research Council. For a detailed overview, refer to the original publication: https://lnkd.in/gbUABmMz