Schrödinger, Dirac, and the Process of Quantum Becoming

Illustration by R.E. Slater & ChatGPT

Schrödinger, Dirac, and the

Process of Quantum Becoming

A Mathematical Introduction to

Quantum Physics through a Whiteheadian Lens

~ diagrams are placed at the end in the Addendum Section ~

by R.E. Slater & ChatGPT-5

From the murmur of Schrödinger’s wave,
to the mirrored grace of Dirac’s antimatter,
the cosmos writes its story
in the grammar of becoming.

References

Wikipedia - https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

Wikipedia - https://en.wikipedia.org/wiki/Dirac_equation

---

Introduction

Erwin Schrödinger and Paul Dirac were pivotal figures in quantum mechanics, jointly awarded the 1933 Nobel Prize for developing new atomic theories, with Schrödinger creating the wave equation (describing particles as waves) and Dirac formulating a relativistic quantum theory that predicted antimatter (the positron). Their work, though mathematically different, addressed similar problems, establishing the foundations of modern quantum physics and paving the way for quantum field theory and particle physics.

Physicist Erwin Schrodinger

Erwin Schrödinger (Austrian Physicist): Developed the Schrödinger Equation, a cornerstone of quantum mechanics, explaining how the quantum state of a physical system changes over time, treating particles like waves and calculating probabilities.

Physicist Paul Dirac

Paul Dirac (British Physicist): Formulated the Dirac Equation, merging quantum mechanics with special relativity, which predicted the existence of antiparticles (like the positron) and unified quantum theory with electromagnetism.

They together shared the 1933 Nobel Prize in Physics for "the discovery of new productive forms of atomic theory," recognizing their independent yet complementary breakthroughs.

They each held different approaches to their work: Schrödinger's wave mechanics provided a powerful, intuitive description, while Dirac's relativistic approach offered a deeper, relativistic framework, laying groundwork for quantum electrodynamics (QED).

As legacy, both physicists are considered founders of quantum mechanics, with Dirac also pioneering quantum field theory, while Schrödinger's work remains fundamental to understanding atomic structure and quantum behavior.

The Man Who Accidentally Discovered Antimatter

by Veritasium

The Dirac Equation: The Most Important Equation

You’ve Never Heard Of

by Physics Explained

---

Process-Theoretical Background: Quantum Theory as a Language of Becoming

Newtonian Classical physics imagines a world built from enduring particles that possess fixed, intrinsic properties and move through space according to deterministic laws. In this view, reality is ultimately a non-process world: change is merely the rearrangement of already-existing entities, and time functions as a passive stage on which objects act but do not fundamentally become.

In contrast, Quantum physics treats the world as a network of processes whose properties emerge only through interactions. In this respect, it aligns more naturally with Alfred North Whitehead’s process metaphysics, which holds that the basic units of reality are not substances but events. Here, the basic constituents of reality are not static things but dynamic processes, not substances but events.

What something is cannot be separated from how it becomes, because properties arise only within interactions, relations, and transitions.

Thus classical physics operates within a Platonic–substantialist frame, while quantum physics speaks the language of a processual, relational cosmos.

Both quantum and processual frameworks hold that reality unfolds through:

• events (discrete units of occurrence),
• transitions (how states evolve),
• relations (how events influence one another),
• superpositions (multiple potentialities coexisting),
• constraints (structures that shape becoming), and
• actualizations (the final, concrete outcome of a process).

In Whitehead’s terms, each actual occasion moves through three stages:

• Potentiality - a field of unrealized possibilities.
• Process of becoming - the integration of influences and relations (prehension).
• Actualization - a concrete outcome that becomes part of the world.

Quantum mechanics mirrors this structure almost point-for-point:

• The wavefunction $\Psi$ is the field of potentiality.

• The Hamiltonian $\hat{H}$ is the relational pattern shaping its development.

• The time-evolution equation

  $$$$iℏ ∂Ψ∂t=H^Ψ

  is the process of becoming the concrescent flow from possibility toward actuality.

• Measurement corresponds to actualization, the resolution of potential into a concrete event outcome.

Modern physics deepens this picture through special relativity, which adds:

• a universal speed limit c,
• the geometry of spacetime (Lorentz symmetry),
• the existence of antimatter as a structural mirror to matter,
• and the spinor framework describing fermionic particles.

These structures reflect what Whitehead calls the “extensive continuum”: the relational framework that shapes, limits, and enables all becoming.

Thus:

• Schrödinger’s equation describes the internal dynamics of quantum becoming -the evolution of potentials, while

• Dirac’s equation describes the cosmological constraints on becoming - the symmetry conditions, relativistic structure, and bipole of matter/antimatter that any quantum event must respect.

Together, they reveal a universe where to exist is to become, and where becoming unfolds within a structured, relational continuum:

• quantum systems are processes,
• spacetime provides relational structure,
• and actuality emerges from potentiality interacting with constraint.

---

The Schrödinger equation

Quantum theory reveals

that matter is not a solid substance

but an evolving tapestry of possibilities -

shaped by equations, symmetries,

and the creativity of the real.

The Schrödinger equation is the fundamental equation of quantum mechanics, describing how the quantum state (wavefunction, Ψ) of a physical system changes over time, analogous to Newton's laws in classical physics, predicting probabilities of outcomes, not definite paths. It comes in time-dependent (how things evolve) and time-independent (for stationary states/energy levels) forms, using the Hamiltonian operator (Ĥ) and Planck's constant (ℏ) to relate energy (E) to the wavefunction (ĤΨ=EΨ), revealing quantized energy levels in quantum systems.

Key Aspects

• Wavefunction (Ψ): A mathematical function describing the quantum state of a system, containing all information about it.
• Hamiltonian Operator (Ĥ): Represents the total energy (kinetic + potential) of the system.
• Quantization: Solutions reveal that energy levels in quantum systems are discrete (quantized).

The Equations

• Time-Dependent Schrödinger Equation: Describes how the wavefunction evolves over time.

i ℏ (∂Ψ / ∂t) = Ĥ Ψ

• Time-Independent Schrödinger Equation: For systems with constant energy, finding the allowed energy levels.
  
  Ĥ Ψ = E Ψ

What it Does

• Predicts the probability of finding a particle in a certain state or location, not its exact trajectory.

• Explains phenomena like atomic structure, electron orbitals, quantum tunneling, the quantized energy levels in atoms (e.g., the hydrogen atom).

What the Schrödinger Equation Describes

• The probability distribution of a particle’s position or momentum.

• The structure of atoms and molecules.

• The allowed energy levels of quantum systems.

• The interference and superposition properties of quantum waves.

In process terms, it describes the internal evolution of potentiality: how a quantum system develops from moment to moment prior to actualization.

---

The Dirac Equation

In the heart of matter,

- lies a dance of potentials;

in the heart of becoming,

- a symmetry of worlds.

The Dirac equation is a fundamental relativistic wave equation in quantum mechanics that combines quantum theory with Einstein's special relativity, describing fermions like electrons, predicting electron spin, and leading to the discovery of antimatter. Its most compact form isThe Dirac equation is a fundamental relativistic wave equation in quantum mechanics that combines quantum theory with Einstein's special relativity, describing fermions like electrons, predicting electron spin, and leading to the discovery of antimatter. Its most compact form is Its most compact form is:

i γ^μ ∂_μ ψ = m ψ

Here:

• γ^μ = the Dirac gamma matrices

• ∂_μ = the four-gradient (derivatives with respect to time and the three spatial dimensions)

• ψ = a four-component spinor wavefunction

• m = particle mass

• ħ and c are the usual quantum and relativistic constants

Key Aspects

• Relativistic - The Dirac equation correctly incorporates special relativity, unlike the non-relativistic Schrödinger equation.

• Spin - It naturally describes intrinsic angular momentum (spin). The electron’s spin-1/2 value falls out of the mathematics automatically.

• Antimatter - The solutions include negative-energy states. Dirac interpreted these as corresponding to antiparticles, predicting the existence of the positron several years before it was observed.

---

Most Compact / Covariant Form

This is the standard form used in relativistic quantum mechanics.
Physicists often adopt natural units where:

ℏ = c = 1

In these units, the Dirac equation is written:

i γ^μ ∂_μ ψ = m ψ

Where:

• ψ (psi):
  A four-component spinor wave function describing the particle’s state
  (spin-up particle, spin-down particle, spin-up antiparticle, spin-down antiparticle).

• γ^μ (gamma matrices):
  A set of four 4×4 matrices (γ^0, γ^1, γ^2, γ^3) that satisfy the anticommutation relation:
  γ^μ γ^ν + γ^ν γ^μ = 2 η^μν
  These encode the geometry of spacetime and ensure Lorentz invariance.

• ∂_μ (four-gradient):
  The spacetime derivative operator:
  (∂/∂t, ∂/∂x, ∂/∂y, ∂/∂z)

• m:
  The rest mass of the fermion (e.g., the mass of the electron).

• i:
  The imaginary unit (√–1).

---

Common Alternative Forms

1. Expanded Form (Position Space, Natural Units)

This makes each term explicit:

(i γ^0 ∂/∂t

• i γ^1 ∂/∂x

• i γ^2 ∂/∂y

• i γ^3 ∂/∂z
  – m ) ψ = 0

This is the same equation as the compact covariant form, just written component-by-component.

---

2. Hamiltonian Form (Standard Units, Not Natural Units)

This form resembles the familiar time-dependent Schrödinger equation (Ĥ ψ = i ℏ ∂ψ/∂t).

The Dirac Hamiltonian is:

Ĥ = β m c² + c Σ (α_n p_n)

Thus the equation becomes:

(β m c² + c Σ α_n p_n) ψ = i ℏ (∂ψ/∂t)

Where:

• c = speed of light

• ℏ = reduced Planck constant

• p_n = momentum operators (p_x, p_y, p_z)

• α_n and β = alternative 4×4 matrices related to the gamma matrices
  via γ^0 = β and γ^i = β α_i

This form is useful when studying free relativistic particles, energy spectra, or interactions with electromagnetic fields.

---

Processual Interpretation and Integration

What the world is, it becomes;
what it becomes, it shares.
In every quantum event,
creation whispers its unfolding.

From a process perspective:

• Schrödinger’s equation describes the internal development of quantum potentiality.

• Dirac’s equation describes the external relational constraints imposed by spacetime, symmetry, and relativistic structure.

Thus:

• Schrödinger = concrescent evolution of possibilities

• Dirac = cosmic symmetry governing possibilities

Becoming is the interplay of:

• potentiality (Ψ)

• relational structure (Ĥ)

• symmetry (γ^μ)

• bipolarity (matter/antimatter)

• spacetime constraint (Lorentz invariance)

Quantum events emerge as actual occasions in Whitehead’s sense - events whose properties arise from relational process, not static essence.

---

> Dirac’s Mathematical Surprise: Antimatter Emerges

The relativistic energy relation

E = ± sqrt( p² c² + m² c⁴ )

naturally includes negative-energy states.

Schrödinger’s framework could not admit these states. But Dirac's framework embraced them by interpreting them as real particles with opposite charge. This algebraic necessity predicted the positron (discovered 1932), a new form of matter which had emerged from pure mathematics.

Thus:

• Schrödinger describes matter

• Dirac describes matter and antimatter

Dirac’s equation revealed the dipolar structure of the quantum world.

---

> Dirac and the Standard Model

Dirac’s equation introduced the spinor, the mathematical language for fermions:

• electrons
• muons
• quarks
• neutrinos (with modifications)

This framework became the foundation for:

• Quantum Electrodynamics (QED)
• Quantum Chromodynamics (QCD)
• Electroweak Theory

In Quantum Field Theory:

• fields are fundamental
• particles are excitations of fields
• every fermion field includes both particle and antiparticle modes

Antimatter is thus not optional, but built into the mathematics of the universe.

---

> A Metaphor: Schrödinger’s Dream vs. Dirac’s Realization

Schrödinger gives:

• the language of possibility
• the grammar of superposition
• what a particle could be

Dirac adds:

• relativistic becoming
• internal symmetry
• a mirrored realm of anti-beings

In process terms:

• Schrödinger captures the actual occasion forming every becoming
• Dirac reveals the negentropic polarity woven into every becoming

Dirac completes quantum theory by showing:

• every particle has a partner
• every energy has a mirror
• every becoming has a counter-becoming

This is a deeply processual insight.

---

> A One-Sentence Synthesis

Schrödinger describes quantum waves; Dirac completes them by making them relativistic, giving them spin, and revealing that the mathematics of the universe demands antimatter.

---

A Conclusion in Four Voices

Where waves become worlds

and symmetry gives birth to stars,

the universe is not a thing but a becoming -

and every particle is an event in its unfolding.

Scholarly

Schrödinger’s equation governs the non-relativistic evolution of quantum states, revealing quantization and probabilistic structure.

Dirac’s equation extends this framework to the relativistic regime, introducing spin, antimatter, and Lorentz symmetry.

Together they form the foundational architecture of modern quantum theory.

Metaphysical

Schrödinger provides the grammar of becoming; Dirac reveals the geometry of becoming. One describes internal flow, the other external constraint.

Reality is not substance but process - shaped both by creative advance and cosmic symmetry.

3. Theological (Process-Theology)

• Schrödinger shows the world’s creative potentiality;

• Dirac shows the world’s relational order;

• Together they suggest a universe where novelty and structure interweave - a cosmos shaped by both the lure toward creativity and the harmonizing patterns of relational constraint.

4. Poetic

Schrödinger's wave carries whispers of what may be.

Dirac's spinor reflects the symmetry of what must be.

Between them lies the world -

  a tapestry of becoming,

  woven from possibility and relation,

  mirrored by antimatter, and

  lit by the creative advance of the real.

 

---

ADDENDUM

---

1. Diagram: The Schrödinger Equation - Internal Flow of Potentiality

---

2. Diagram: The Dirac Equation - Relational and Symmetry Constraints

---

3. Diagram: Quantum Becoming as Internal Potential + Relational Constraint

This diagram ties the whole document together into one visual:

Or an extremely compact version:

---

Mathematical Progression from Schrödinger → Dirac

---

Introduction: From Waves to Relativity

Schrödinger’s equation (1926) successfully describes quantum systems at low velocities, but it is not relativistic. When electrons were observed traveling at relativistic speeds, it became clear that a new equation was needed - one that respects both quantum principles and Einstein’s special relativity.

Paul Dirac (1928) sought a wave equation that:

1. is first-order in time (to preserve probability),

2. is first-order in space (to preserve Lorentz symmetry),

3. yields the relativistic energy relation,

4. and predicts spin as an intrinsic property.

The result was the Dirac Equation, the first physical theory to predict new matter (the positron) from pure algebra.

Below is the progression of his logic and mathematics.

---

1. Schrödinger’s Equation (Non-Relativistic. 1926)

What it solves: An electron as a non-relativistic wave. The standard quantum wave equation:

$$i\mathrm{\hslash }\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }t}=\widehat{H}\mathrm{\Psi }$$

Visually, the element is a smooth, blue wavefunction evolving over time (as  Ψ(t)).

For a free particle:

$$\widehat{H}=\frac{{\widehat{p}}^2}{2m}\text{with}\widehat{p}=-i\mathrm{\hslash }\mathrm{\nabla }$$

This produces:

$$E=\frac{p^2}{2\mathrm{m}}$$

The Problem: This describes a Galilean world, not a relativistic one. It corrects for slow particles but fails for relativistic electrons.

---

2. Replace the Hamiltonian with Einstien's Special Relativistic Energy Relation

Einstein’s relation:

$$E^{2} = p^{2}c^{2} + m^{2}c^{4}$$

The Visual element is a spacetime light cone behind the equation: "Quantum mechanics must obey this geometry."

Now substitute this formula directly into Schrödinger’s structure:

$$i\mathrm{\hslash }\frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }t}=\sqrt{c^2{\widehat{p}}^2+m^2{c}^4}\mathrm{\Psi }$$

This introduces the operator:

$$\sqrt{-\hbar^{2} c^{2}\nabla^{2} + m^{2}c^{4}}$$

Problem: The square-root operator is nonlocal, nonlinear, and unusable. Hence, Dirac abandoned this approach.

---

3. The Klein-Gordon Equation: A Good Try, But Wrong for Electrons

Square the relation instead:
$$E\to i\mathrm{\hslash }{\mathrm{\partial }}_t,p\to -i\mathrm{\hslash }\mathrm{\nabla }$$
$$<br>(\frac{1}{c^2}{\mathrm{\partial }}_t^2-{\mathrm{\nabla }}^2+\frac{m^2{c}^2}{{\mathrm{\hslash }}^2})\mathrm{\Psi }=0$$

This is the Klein–Gordon equation. It was relativistic but could not describe electrons or probability.

Inherent Problems (Dirac’s objections):

• Negative probability density
• No representation of spin-½
• Second-order in time
• Cannot describe electrons

Dirac required a first-order relativistic wave equation.

---

4a. Dirac imposes these conditions/anticommutators in his algebra

$${\alpha }_i^2=1,{\beta }^2=1,\{{\alpha }_i,\beta \}=0,\{{\alpha }_i,{\alpha }_j\}=2{\delta }_{ij}$$

Visual element: Four 4×4 matrices arranged like cornerstones.

Caption: “Spinor structure must exist. Four components appear.”

• → leads to spin
• → leads to positrons​

These anticommutation relations define the Clifford algebra of spacetime. But these cannot be satisfied by numbers - the operators must be matrices.

---

4b. Dirac’s Insight: Linearize the Energy

Dirac proposed a Hamiltonian linear in momentum:

$$\hat{H} = c\,\boldsymbol{\alpha}\cdot \hat{\mathbf{p}} \,+\, \beta mc^{2}$$

He then inserted this into Schrödinger’s form:

$$i\hbar\frac{\partial\psi}{\partial t} = \left(c\,\boldsymbol{\alpha}\cdot \hat{\mathbf{p}} + \beta mc^{2}\right)\psi$$

To ensure consistency, demand that squaring this reproduces Einstein’s energy.

Take a plane-wave:

$$\psi =u{e}^{i(\mathbf{p}\cdot \mathbf{x}-Et)\mathrm{/}\mathrm{\hslash }}$$

Then:

$$Eu=(c\mathit{\alpha }\cdot \mathbf{p}+\beta m{c}^2)u$$

Square both sides:

$$E^{2}u={(c\mathit{\alpha }\cdot \mathbf{p}+\beta m{c}^2)}^{2}u$$

Expand RHS:

$$=c^2(\mathit{\alpha }\cdot \mathbf{p}{)}^2+m^2{c}^4{\beta }^2+m{c}^3\{\mathit{\alpha }\cdot \mathbf{p},\beta \}$$

To match Einstein:

$$E^2=p^2{c}^2+m^2{c}^4$$

---

5. Emergence of Spinors

The smallest matrices that satisfy the Dirac algebra are 4×4.

Therefore the wavefunction must be 4-component:

$$\psi = \begin{pmatrix} \psi_{1} \\ \psi_{2} \\ \psi_{3} \\ \psi_{4} \end{pmatrix}$$This is the Dirac spinor.

Its components naturally encode:

• spin-up electron
• spin-down electron
• spin-up positron
• spin-down positron

Spin and antimatter emerge from algebra alone.

---

6. The Covariant Dirac Equation

Dirac rewrites α and β in terms of gamma matrices:

$${\gamma }^0=\beta ,{\gamma }^i=\beta {\alpha }^i$$

Using the spacetime derivative:

$${\mathrm{\partial }}_{\mu }=(\frac{\mathrm{\partial }}{\mathrm{\partial }t},\mathrm{\nabla })$$

The equation becomes:

$$\boxed{{\displaystyle i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }\psi =m\mathrm{\psi }}}$$

This form is:

• Lorentz invariant
• First-order in time and space
• Predictive of spin
• Mathematically natural
• Physically profound

It is the cornerstone of quantum field theory.

---

7. Negative Energy and the Discovery of Antimatter

Solving the equation yields the spectrum:

$$E = \pm \sqrt{p^{2}c^{2} + m^{2}c^{4}}$$

Dirac’s interpretation:

• positive solutions = electrons

• negative solutions = new particles with positive mass and opposite charge

These are positrons, discovered experimentally in 1932 by Carl Anderson.

---

This was the first time in history a new particle was predicted by pure mathematics.

---

Summary Diagram

Schrödinger Equation (Non-Relativistic)
→ First-order in time
→ No spin
→ Not Lorentz invariant

Klein–Gordon Equation
→ Relativistic
→ Second-order in time
→ Wrong probability density

Dirac’s Equation
→ First-order in time
→ First-order in space
→ Lorentz invariant
→ Predicts spin
→ Predicts antimatter
→ Foundation of quantum field theory

$$i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }\psi =m\psi$$

---