Derivation of the Schrodinger equation from fundamental principles

This paper presents a formal derivation of the Schrödinger equation for a particle's wave function by treating it as a probability amplitude and applying the fundamental relations $E=\hbar\omega$ and $p=\hbar k$ to the associated probability wave, thereby offering a rigorous alternative to Schrödinger's original heuristic approach.

The Gist

Imagine you are trying to understand how a tiny, invisible particle (like an electron) moves through space. For a long time, scientists were stuck in a puzzle: sometimes these particles acted like solid marbles, and other times they acted like ripples in a pond.

This paper by Wenzhuo Zhang and Anatoly Svidzinsky is like a detective story. It asks: "Can we figure out the exact rulebook (the Schrödinger equation) that governs these particles, starting from just a few basic clues, rather than just guessing?"

Here is the story of the paper, broken down into simple concepts and everyday analogies.

1. The Old Way: Guessing and Intuition

First, the authors explain how Erwin Schrödinger originally found his famous equation in 1926.

• The Analogy: Imagine Schrödinger was a chef trying to invent a new soup. He didn't have a recipe book. Instead, he tasted the ingredients (physics experiments), looked at the flavors (mathematical patterns), and said, "Hmm, if I mix these two things, it feels like it should taste right."
• The Result: He created the "Schrödinger Equation," which works perfectly. But because he "guessed" it based on intuition, he couldn't explain why it worked from the ground up. It was like having a working engine but not knowing how the pistons move.

2. The New Way: Building from the Ground Up

The authors of this paper want to rebuild that engine from scratch using three fundamental "bricks":

1. The Wave-Particle Connection: Particles are also waves (De Broglie's idea).
2. Energy and Frequency: High energy means a fast vibration (Planck's idea).
3. Probability: We can't know exactly where a particle is, only the chance of finding it there (Born's rule).

The Analogy: Think of the particle not as a solid ball, but as a cloud of fog.

• The density of the fog at any spot tells you how likely you are to find the particle there.
• The ripples moving through the fog tell you how the particle is moving.

3. The Derivation: How They Solved the Puzzle

The authors show that if you treat this "fog" mathematically, the Schrödinger equation pops out naturally. Here is the step-by-step logic they used:

Step A: The "Local" View

In classical physics, a car has a specific speed and location at every moment. In quantum physics, the "fog" has a local speed and local energy everywhere.

• They calculated that the "speed" of the fog is related to how the phase (the rhythm of the wave) changes.
• They found that the energy of the particle isn't just its motion; there is an extra "hidden" energy caused by the fog's own shape. They call this the Quantum Potential.
• Metaphor: Imagine a crowd of people running. If they run in a tight, organized group, they move fast. If they are scattered and confused, there is "friction" or "tension" in the crowd. That tension is the Quantum Potential. It's an energy that only exists because the particle is a wave, not a solid ball.

Step B: The Conservation of Fog

You can't create or destroy fog; you can only move it around. In physics, this is called the Continuity Equation.

• If the fog gets thicker in one spot, it must have flowed there from somewhere else.
• The authors combined the "Energy Rule" (Step A) with the "Fog Flow Rule" (Continuity).

Step C: The Magic Equation Appears

When they mashed those two rules together, the math forced them into a single, elegant formula.

• The Result: The Schrödinger Equation!
• It turns out this equation is just the mathematical way of saying: "The way this probability fog changes over time is determined by its energy and how it flows."

4. Why This Matters (The "So What?")

The paper argues that we shouldn't just teach students the Schrödinger equation as a magic spell to memorize. Instead, we should show them it's a logical consequence of how the universe works.

• The "Symmetry" Argument: The authors compare this to how we understand gravity. Einstein didn't just guess gravity; he realized that if space and time are linked in a specific way (symmetry), gravity must exist.
• Similarly, if you accept that particles are probability waves, the Schrödinger equation must exist. It's not a random rule; it's the only way the math can stay consistent.

5. The Big Picture: A New Vision of Reality

The paper ends with a fascinating thought experiment.

• The Current View: We think the universe is made of particles and fields.
• The Authors' Hint: They suggest that if we change our fundamental assumptions (like assuming the universe has a fixed background geometry), we might get a different theory of gravity that explains things like "Dark Energy" better than Einstein did.
• The Takeaway: Science isn't just about collecting facts; it's about finding the deepest, simplest rules that make those facts inevitable.

Summary in One Sentence

This paper takes the mysterious Schrödinger equation—which describes how quantum particles move—and proves that it is simply the natural result of treating particles as flowing waves of probability, rather than just a lucky guess made by a physicist in 1926.

Technical Summary

1. Problem Statement

The Schrödinger equation is the cornerstone of non-relativistic quantum mechanics, yet it is traditionally introduced in textbooks as a postulate or derived via heuristic arguments based on physical intuition rather than rigorous deduction from fundamental principles. Historically, Erwin Schrödinger's original derivation (1926) relied on analogies between optics and mechanics and variational principles without a clear, axiomatic foundation.

The authors address the need to derive the Schrödinger equation from first principles, specifically starting from the probabilistic nature of quantum mechanics, the de Broglie relations, and the conservation of probability, rather than treating the equation as an unproven axiom.

2. Methodology

The paper employs a deductive approach to reconstruct the Schrödinger equation by combining the following fundamental concepts:

• Probabilistic Interpretation: The wave function $\Psi(t, \vec{r})$ is defined as a probability amplitude, where $|\Psi|^2$ represents the probability density (Born rule).
• de Broglie-Planck Relations: The association of particle energy and momentum with wave frequency and wave vector: $E = \hbar\omega$ and $\vec{p} = \hbar\vec{k}$.
• Fourier Analysis: The wave function is expanded in terms of momentum eigenstates (Fourier series) and energy eigenstates to relate local physical quantities to the wave function's phase and amplitude.
• Hydrodynamic Formulation: The authors utilize the Madelung transformation, expressing the wave function in polar form $\Psi = |\Psi|e^{iS}$, where $S$ is the phase. This allows the separation of the real and imaginary parts of the resulting equations.
• Conservation Laws: The derivation enforces the continuity equation for probability conservation.
• Superposition Principle: The requirement that the governing equation must be linear.

3. Key Contributions and Derivation Steps

The authors present two distinct pathways to derive the Schrödinger equation:

Pathway A: From Classical Energy Conservation and Quantum Potentials

1. Local Energy Definition: Starting from the classical energy relation $E = E_{kin} + V$, the authors define local values for momentum and kinetic energy based on the wave function's phase $S$.
   • Local momentum: $\vec{p} = \hbar\nabla S$.
   • Local kinetic energy: $E_{kin} = \frac{p^2}{2M} - \frac{\hbar^2}{2M}\frac{\nabla^2|\Psi|}{|\Psi|}$.
2. Quantum Potential: The term $-\frac{\hbar^2}{2M}\frac{\nabla^2|\Psi|}{|\Psi|}$ is identified as the "quantum potential" (or Bohm potential), representing the energy of particle confinement. This term distinguishes quantum mechanics from classical mechanics.
3. Hamilton-Jacobi Analogy: By substituting these local definitions into the energy conservation equation, the authors derive a quantum analog of the classical Hamilton-Jacobi equation:
   $$-\hbar\frac{\partial S}{\partial t} = \frac{\hbar^2}{2M}\left((\nabla S)^2 - \frac{\nabla^2|\Psi|}{|\Psi|}\right) + V$$
4. Continuity Equation: The conservation of probability is expressed as $\frac{\partial|\Psi|^2}{\partial t} + \nabla \cdot \vec{j} = 0$, where the probability current $\vec{j}$ is derived from the local velocity $\vec{u} = \frac{\hbar\nabla S}{M}$.
5. Synthesis: Combining the quantum Hamilton-Jacobi equation and the continuity equation yields the complex Schrödinger equation:
   $$i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2M}\nabla^2\Psi + V\Psi$$

Pathway B: From Superposition and Continuity

1. The authors assume the superposition principle (linearity) and the continuity equation.
2. By substituting the expression for probability current into the continuity equation and manipulating the terms, they arrive at a relation where a function $V$ must be real and independent of $\Psi$.
3. By analyzing the limit of a free particle (plane wave solution), they identify $V$ as the potential energy, thereby recovering the Schrödinger equation.

4. Results

• Derivation of the Equation: The paper successfully derives the time-dependent Schrödinger equation for a non-relativistic particle in an external potential $V(t, \vec{r})$ without assuming the equation a priori.
• Identification of Quantum Potential: The derivation explicitly isolates the quantum potential term, demonstrating how it arises from the spatial variation of the probability amplitude ($\nabla^2|\Psi|/|\Psi|$).
• Hydrodynamic Interpretation: The results confirm that the Schrödinger equation is mathematically equivalent to a set of coupled fluid dynamic equations (Madelung equations): a continuity equation for mass density and a modified Hamilton-Jacobi equation for the phase.
• Classical Limit: The paper shows that for macroscopic objects or high-momentum particles, the quantum potential becomes negligible, and the equations reduce to classical mechanics, recovering the classical Hamilton-Jacobi equation and deterministic trajectories.

5. Significance

• Foundational Clarity: The paper challenges the notion that the Schrödinger equation must be accepted as a postulate. It demonstrates that the equation is a necessary consequence of combining the probabilistic interpretation of matter waves with the conservation of probability and energy.
• Bridging Classical and Quantum: By highlighting the role of the quantum potential, the derivation provides a clear physical mechanism for the transition between quantum and classical regimes.
• Pedagogical Value: The authors argue that this derivation should replace heuristic introductions in physics education, offering students a logical path from fundamental principles (de Broglie relations, Born rule) to the central equation of quantum mechanics.
• Broader Context: The conclusion places this derivation within the context of the "symmetry protocol" in modern physics. The authors suggest that just as Maxwell's equations and General Relativity can be derived from symmetry principles (gauge invariance, Lorentz invariance), quantum mechanics can be derived from fundamental probabilistic and continuity principles. They also briefly touch upon alternative gravity theories (vector gravity) to illustrate how changing fundamental postulates alters physical predictions, reinforcing the importance of deriving theories from first principles.

In summary, Zhang and Svidzinsky provide a rigorous, tutorial-style derivation that demystifies the Schrödinger equation, grounding it firmly in the probabilistic nature of the wave function and the conservation laws of physics.