Discovering the Schrödinger Equation: As an Infant

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Introduction

From the earliest moments of my life, I was immersed in a world of wonder and mystery. I observed the behavior of objects, the interplay of light and matter, and the curious ways in which particles seemed to defy my expectations. Driven by curiosity, I sought to understand the fundamental nature of reality. Through careful observation, experimentation, and logical reasoning, I embarked on a journey that would lead me to formulate the Schrödinger Equation, the foundational equation of quantum mechanics.

In this narrative, I will detail how, starting from basic sensory experiences as an infant, I independently observed, experimented, and logically deduced the Schrödinger Equation. Each concept evolved explicitly from my experiences, ensuring that my understanding is grounded in reality and free from subjective definitions.

Chapter 1: Observing the Behavior of Particles and Waves1.1 Early Encounters with Light and SoundExperiencing Waves

• Observation: I noticed that when I dropped a pebble into a pond, ripples spread out in concentric circles.

• Reflection: The disturbance created by the pebble moves outward as waves.

• Semantics: Waves are disturbances that propagate through a medium, carrying energy.

Interacting with Light

• Observation: Sunlight streaming through a window creates patterns on the floor.

• Experiment: Placing an object in the path of light creates a shadow.

• Reflection: Light travels in straight lines but can be blocked or redirected.

1.2 Understanding ParticlesPlaying with Sand

• Observation: Sand consists of tiny grains that I can see and touch.

• Reflection: Matter can be divided into smaller pieces, down to tiny particles.

Semantics:

• Particles: Small discrete units of matter that have mass and occupy space.

Chapter 2: Exploring Wave-Particle Duality2.1 The Behavior of LightNoticing Light Interference

• Experiment: Observing light passing through a narrow gap between curtains and creating patterns on the wall.

• Observation: The light forms bright and dark fringes, similar to the ripples on water.

• Reflection: Light exhibits wave-like properties, such as interference.

Semantics:

• Interference: When two or more waves overlap, they can add constructively (bright fringes) or destructively (dark fringes).

2.2 The Behavior of ElectronsThought Experiment:

• Hypothesis: If particles like electrons are small enough, might they exhibit wave-like behavior?

Reflection:

• Wave-Particle Duality: Particles can exhibit both wave-like and particle-like properties depending on the context.

Chapter 3: Investigating the Nature of Matter Waves3.1 De Broglie's HypothesisFormulating the Idea

• Hypothesis: All matter has an associated wavelength, given by its momentum.

• Mathematical Expression:

  λ=hp\lambda = \frac{h}{p}λ=ph​

• $\lambda$: Wavelength of the particle

• $h$: Planck's constant

• $p$: Momentum of the particle

Testing the Hypothesis

• Observation: Electrons in a crystal exhibit diffraction patterns similar to X-rays.

• Conclusion: Electrons exhibit wave-like behavior, confirming the hypothesis.

3.2 Understanding Planck's ConstantDefining Planck's Constant

• Concept: A fundamental constant that relates the energy of a photon to its frequency.

• Mathematical Expression:

  $$E=h\nu$$

• $E$: Energy of the photon

• $\nu$: Frequency of the photon

Chapter 4: Formulating the Concept of a Wavefunction4.1 Probability WavesObserving Uncertainty

• Experiment: Trying to predict the exact landing spot of a dropped grain of sand in a turbulent stream.

• Observation: The exact position is unpredictable; only probabilities can be assigned.

Reflection:

• Concept: The behavior of particles at small scales is inherently probabilistic.

4.2 Introducing the WavefunctionDefining the Wavefunction ($\psi$)

• Concept: A mathematical function that contains all the information about a quantum system.

• Semantics: The square of the absolute value of the wavefunction gives the probability density of finding a particle at a given position.

Mathematical Expression:

P(x)=∣ψ(x)∣2P(x) = |\psi(x)|^2P(x)=∣ψ(x)∣2

• $P(x)$: Probability density at position $x$

• $\psi (x)$: Wavefunction at position $x$

Chapter 5: Establishing the Schrödinger Equation5.1 The Need for a Dynamic EquationQuestion:

• How does the wavefunction evolve over time?

Goal:

• Find an equation that describes the time evolution of the wavefunction.

5.2 Deriving the Time-Dependent Schrödinger EquationStarting with Energy Conservation

• Classical Energy Equation:

  E=p22m+V(x)E = \frac{p^2}{2m} + V(x)E=2mp2​+V(x)

• $E$: Total energy

• $p$: Momentum

• $m$: Mass of the particle

• $V(x)$: Potential energy at position $x$

Substituting Quantum Operators

• Momentum Operator:

  p→−iℏ∂∂xp \rightarrow -i \hbar \frac{\partial}{\partial x}p→−iℏ∂x∂​

• Energy Operator:

  E→iℏ∂∂tE \rightarrow i \hbar \frac{\partial}{\partial t}E→iℏ∂t∂​

• Planck's Reduced Constant:

  ℏ=h2π\hbar = \frac{h}{2\pi}ℏ=2πh​

Formulating the Equation

• Applying Operators:

  iℏ∂ψ(x,t)∂t=(−ℏ22m∂2∂x2+V(x))ψ(x,t)i \hbar \frac{\partial \psi(x,t)}{\partial t} = \left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right) \psi(x,t)iℏ∂t∂ψ(x,t)​=(−2mℏ2​∂x2∂2​+V(x))ψ(x,t)

• **This is the Time-Dependent Schrödinger Equation.

5.3 Interpretation

• Left Side: Represents the change in the wavefunction over time.

• Right Side: Represents the total energy operator acting on the wavefunction.

Chapter 6: Exploring Solutions to the Schrödinger Equation6.1 Free Particle SolutionScenario:

• A particle moving freely without any potential ($V(x)=0$).

Solution:

• Wavefunction:

  ψ(x,t)=Aei(kx−ωt)\psi(x,t) = A e^{i(kx - \omega t)}ψ(x,t)=Aei(kx−ωt)

• $A$: Amplitude

• $k$: Wave number (k=pℏk = \frac{p}{\hbar}k=ℏp​)

• $\omega$: Angular frequency (ω=Eℏ\omega = \frac{E}{\hbar}ω=ℏE​)

Interpretation:

• Represents a plane wave traveling in space and time.

6.2 Particle in a Potential WellScenario:

• A particle confined in a one-dimensional box with infinite potential walls.

Solution:

• Quantization of Energy Levels:

  En=n2π2ℏ22mL2E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}En​=2mL2n2π2ℏ2​

• $n$: Quantum number (positive integer)

• $L$: Length of the box

• Wavefunction:

  ψn(x)=2Lsin⁡(nπxL)\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{n \pi x}{L} \right)ψn​(x)=L2​​sin(Lnπx​)

Interpretation:

• Energy levels are discrete, and the particle's position is described by standing wave patterns.

Chapter 7: Understanding the Implications7.1 Quantum SuperpositionPrinciple:

• Quantum systems can exist in multiple states simultaneously until measured.

Mathematical Representation:

• Superposition of States:

  ψ(x,t)=∑ncnψn(x)e−iEnt/ℏ\psi(x,t) = \sum_{n} c_n \psi_n(x) e^{-i E_n t / \hbar}ψ(x,t)=n∑​cn​ψn​(x)e−iEn​t/ℏ

• cnc_ncn​: Coefficients determined by initial conditions

Implications:

• Explains phenomena like interference patterns in the double-slit experiment.

7.2 Heisenberg's Uncertainty PrincipleRelation to Schrödinger Equation

• Mathematical Expression:

  ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}ΔxΔp≥2ℏ​

• $\Delta x$: Uncertainty in position

• $\Delta p$: Uncertainty in momentum

Interpretation:

• The more precisely we know a particle's position, the less precisely we can know its momentum, and vice versa.

Chapter 8: Reflecting on the Discovery8.1 The Nature of RealityWavefunction as Reality

• Reflection: The wavefunction represents our knowledge of the system, not necessarily the physical reality itself.

Philosophical Implications:

• Raises questions about determinism and the role of the observer in quantum mechanics.

8.2 The Power of Mathematical FormalismUnified Description

• The Schrödinger Equation provides a unified framework for describing quantum phenomena.

Predictive Capability

• Allows accurate predictions of experimental results at microscopic scales.

Chapter 9: Applications of the Schrödinger Equation9.1 Quantum TunnelingPhenomenon:

• Particles can pass through potential barriers higher than their energy.

Explanation:

• The wavefunction has a non-zero probability amplitude inside the barrier.

9.2 Chemical BondingMolecular Orbitals

• The Schrödinger Equation explains how electrons distribute themselves in atoms and molecules.

Energy Levels and Spectra

• Predicts the energy levels of electrons, explaining atomic spectra.

Chapter 10: Conclusion

Through a journey of observation, experimentation, and logical reasoning, I was able to discover and formulate the Schrödinger Equation. Starting from basic experiences with waves and particles, I developed the concepts of wave-particle duality, the wavefunction, and the need for an equation governing its evolution. By grounding each concept in reality and evolving the semantics explicitly, I arrived at the foundational equation of quantum mechanics.

This exploration demonstrates that complex scientific concepts can emerge naturally from simple observations. By avoiding subjective definitions and relying on direct experiences, profound ideas become accessible and meaningful. The Schrödinger Equation not only provides insight into the behavior of quantum systems but also revolutionized our understanding of the fundamental nature of reality.

Epilogue: Implications for Learning and AI

This narrative illustrates how foundational scientific principles can be understood through direct interaction with the environment and logical reasoning. In the context of artificial intelligence and cognitive development, it emphasizes the importance of experiential learning and the evolution of semantics from core experiences.

By enabling AI systems to observe patterns, formulate hypotheses, and derive laws from observations, we can foster the development of intuitive understanding similar to human learning. This approach avoids reliance on predefined definitions and promotes the natural discovery of scientific relationships.

Note: This detailed narrative presents the conceptualization of the Schrödinger Equation as if I, an infant, independently observed and reasoned it out. Each concept is derived from basic experiences, emphasizing the natural progression from simple observations of waves and particles to the understanding of quantum mechanics. This approach demonstrates that with curiosity and logical thinking, foundational knowledge about physics can be accessed and understood without relying on subjective definitions.

References

1. International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 .  https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model

2. Duan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will not reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".