1. The Abstract Nature of the Schrödinger Equation

The Schrödinger equation

iℏ∂ψ/∂t = Ĥψ

is the cornerstone of quantum mechanics, but its essence is a highly abstract mathematical tool rather than a direct description of physical reality. This abstraction manifests in two key aspects:

1. Non-observability of the Complex Wave Function

The wave function ψ is a complex-valued function, while all physical measurement results are real numbers, so it cannot express a physical quantity. This indicates that ψ itself is not directly observable, serving only as an intermediate computational tool for deriving probability distributions of observables.

2. Non-physicality of High-dimensional Configuration Space

For a system of N particles, ψ is defined in 3N-dimensional configuration space, not in our familiar three-dimensional physical space. This "dimensional explosion" clearly shows: the wave function doesn't describe waves in physical space, but rather an abstract mathematical structure encoding statistical information of many-body systems.

Historical Origins from Optical Analogy

The abstract form of the Schrödinger equation originated from Schrödinger's borrowing from classical optics in 1926:

Helmholtz equation in optics: ∇²E + k²E = 0

Schrödinger stationary equation: -ℏ²/2m ∇²ψ + Vψ = Eψ

This analogy enabled Schrödinger to successfully explain stable states of electrons in atoms, but also planted seeds of conceptual confusion: we borrowed the mathematical form describing light waves (collective oscillations of electromagnetic fields) to characterize individual electron behavior. This mathematical success masked fundamental incompatibilities in physical interpretation.

In quantum mechanical physical systems, the real physical waves are electromagnetic waves, indicating that the Schrödinger equation is a mathematical abstraction of the dominant electromagnetic wave behavior generated by massive numbers of particles participating simultaneously.

2. Atoms as Electromagnetic Systems: The Essence of Collective Behavior

Atoms are essentially pure electromagnetic systems. Interactions between protons and electrons are entirely governed by Coulomb force:

F = ke²/r²

Their stability, energy level structure, and chemical properties all originate from electromagnetic interactions.

More critically, atoms exhibit typical collective behavior characteristics:

• Electron clouds are not individual electron trajectories, but statistical distributions of numerous measurements;

• Atomic orbitals reflect stable patterns of electromagnetic fields, not classical electron motion paths;

• Chemical bonds are collective effects of electromagnetic field interactions between multiple atoms.

Global Fields vs. Local Particles

Electromagnetic fields are fundamentally global—permeating all space, with local disturbances triggering overall responses; while the particle concept is fundamentally local—particles can only be at one position at any moment.

This "global-local" distinction is the root of many conceptual confusions in quantum mechanics. When we use equations describing global electromagnetic fields (Schrödinger equation) to characterize local particles, profound conceptual conflicts inevitably arise.

3. The Electromagnetic Origin of Light and Spectra

Mechanism of Spectral Line Generation

Atomic spectra directly reveal the electromagnetic nature of quantum systems. When atoms transition from high energy state E₂ to low energy state E₁:

E₂ - E₁ = hν = ℏω

The physical essence of this process is reconfiguration of electromagnetic field patterns:

• Initial state: Electron in high-energy orbital, corresponding to specific electromagnetic field distribution;

• Transition: Dynamic adjustment of electromagnetic field, releasing electromagnetic radiation;

• Final state: System enters new stable field configuration.

Origin of Light's Wave Nature

Light's wave nature originates from electromagnetic field oscillations described by Maxwell's equations:

∇²E⃗ - (1/c²)∂²E⃗/∂t² = 0

This is a genuine wave equation, describing electromagnetic field propagation in space. Crucially: it describes collective field behavior, not individual photon trajectories.

4. The Incompatibility of Describing Particles with Wave Equations

Misalignment of Scale and Level

Quantum mechanics' core dilemma lies in: we use mathematical forms describing collective behavior to characterize individual particles. This misalignment is particularly evident in the double-slit experiment:

• Single electron: Each produces only one point on the screen (particle nature);

• Many electrons: Accumulate to form interference patterns (wave nature);

• Wave function: Seemingly describes "single electron wave through double slits," but actually only gives probability distribution.

In reality, interference patterns only appear after numerous repeated experiments. Individual electron behavior is random and discrete, with no actual "wave."

Confusion of Descriptive Levels

The root problem lies in confusing two descriptive levels:

• Microscopic individual level: Single measurements are random, discrete events;

• Statistical collective level: Distribution of multiple measurements follows wave patterns.

The Schrödinger equation essentially operates at the statistical level, providing probability amplitudes, not definite trajectories of individual particles.

5. The Physical Origin of Quantum Probability

Physical Significance of Probability Amplitude

The wave function as probability amplitude has core significance:

|ψ(x)|² represents probability density of finding a particle at position x;

Probability amplitudes can coherently superpose: |ψ₁ + ψ₂|² ≠ |ψ₁|² + |ψ₂|²

This interference originates from coherent superposition of electromagnetic field modes. When two field configurations superpose, their probability distribution exhibits interference patterns.

Since the physical waves corresponding to the Schrödinger equation are electromagnetic waves, abstract probability amplitudes correspond to electromagnetic wave amplitudes. Driving a physical event, like exposure or photoelectric effect, the likelihood of occurrence depends on energy obtained from the environment. Electromagnetic wave amplitude gives local energy disturbance magnitude, thus determining the probability of an observable event occurring.

6. The Essence of Wave-Particle Duality: Complementary Manifestation of Descriptive Levels

Not a Mysterious Property of Particles

"Wave-particle duality" is often misunderstood as an intrinsic mysterious property of particles. Actually, it reflects hierarchical differences in descriptive frameworks:

Particle manifestations:

• Local detection (definite position/momentum)

• Discrete events (like photoelectric effect)

• Indivisibility (never observed "half an electron")

Wave manifestations:

• Interference and diffraction patterns

• Frequency-energy relation E = hν

Unified Understanding Framework

The correct understanding should be:

• Single events: Always particle-like (local, discrete);

• Statistical distributions: Exhibit wave-like properties (continuous, global);

• Electromagnetic fields: Provide probability background, determining likelihood of local measurements.

Therefore, wave-particle duality isn't an internal contradiction of matter, but the natural manifestation of individual behavior and collective statistics at different descriptive levels.

Physical Basis of Complementarity Principle

Bohr's complementarity principle isn't philosophical mysticism but has clear physical foundations:

• Measuring position requires local interaction, destroying momentum certainty;

• Measuring momentum requires prolonged observation of wave behavior, sacrificing position precision.

This complementarity is rooted in the nature of electromagnetic interaction: any measurement relies on electromagnetic coupling, and this process necessarily disturbs the system state.

Returning to the Real Physical Origins of Quantum Mechanics

Through examining these six core issues, we can derive the following key insights:

1. The Schrödinger equation is an abstract mathematical tool, borrowing the form describing collective electromagnetic field oscillations, used to calculate probability distributions, not to depict physical reality.

2. Quantum systems are essentially electromagnetic systems, their collective nature requires field language description, fundamentally contradicting particle locality.

3. Spectra directly reflect electromagnetic essence, energy level transitions are electromagnetic field pattern reconfigurations, light generation is the natural result of electromagnetic oscillation.

4. Descriptive incompatibility is the source of confusion: Using collective wave equations to describe single particles inevitably causes conceptual confusion.

5. Probability originates from electromagnetic field energy density distribution: Global electromagnetic field energy distribution triggers local measurements, producing quantum probability.

6. Wave-particle duality manifests hierarchical differences: The complementary unity of individual behavior (particle nature) and statistical laws (wave nature).

Key point: Quantum mechanics' "weirdness" isn't nature's mystery, but conceptual incompatibility arising when we attempt to understand individual particle behavior using mathematical frameworks describing collective electromagnetic field behavior.

Once recognizing this point, quantum mechanics' "mysterious veil" naturally dissipates—due to incomplete information, it's essentially a statistical method describing microscopic electromagnetic collective behavior involving massive numbers of charged particles.