Home › Core Derivations › Quantum Mechanics in Planck Units

Quantum = Classical Waves

In lattice-Planck units, every quantum mechanical “mystery” becomes a trivial statement about waves on a discrete medium. There is no quantum mechanics — only wave mechanics that humans misidentified as something new.

§1 — The QM ↔ Wave Dictionary

Set lattice-Planck units: a = 1 (lattice spacing), k = 1 (spring constant), η = 1 (inertia). Then c = 1 and ℏ = π/2. Every quantum concept translates to a wave concept:

QM Concept	Wave Reality	Planck Units
Planck’s constant ℏ	Action per half-cycle of a wave	π/2
Wave function ψ	Lattice displacement field φ(x,t)	Dimensionless amplitude
\|ψ\|² (probability)	Wave intensity = energy density	φ²(x)
Superposition	Waves add linearly (Hooke’s law)	φ = φ₁ + φ₂
Quantized energy E = nℏω	Standing wave boundary conditions	E_n = nπω/2
Spin ½	Binary internal state (yin/yang, ±)	s = ±1 (in units of ℏ/2 = π/4)
Photon	Transverse wave packet	E = πω/2
Electron	Zone-edge standing wave (λ = 2a)	m_e = π²/2
Proton	Spherical j₀ standing wave	m_p = 3π⁷
Uncertainty principle	Fourier theorem on a lattice	Δx · Δp ≥ π/4
Tunneling	Evanescent wave through potential barrier	T = e^−2κL
Entanglement	One spatially extended wave, not “two particles”	φ(x₁, x₂) ≠ f(x₁)g(x₂)
Measurement / collapse	Wave–wave interaction redistributes energy locally	No mystery. Detector is also a wave.
Zero-point energy	Lattice ground-state vibration (can’t freeze a discrete medium)	E₀ = πω/4 per mode
Commutator [x, p] = iℏ	Discrete lattice shift ↔ phase rotation	[x, p] = iπ/2

The pattern: Every quantum mechanical concept is a classical wave concept. The factor ℏ = π/2 is not a “quantum of action” — it is the action accumulated by a wave in half a cycle on a lattice with spacing a = 1. Quantum mechanics is what you get when you do classical wave mechanics and forget that the medium is discrete.

§2 — Uncertainty Principle = Fourier Theorem

The Heisenberg uncertainty principle is not a fundamental law. It is Fourier’s theorem applied to waves on a lattice.

Uncertainty in Planck Units Δx · Δp ≥ ℏ/2 = π/4

Any wave localized to Δx lattice spacings must contain a spread of wavenumbers Δk ≥ 1/(2Δx). Since momentum p = ℏk = (π/2)k:

Fourier constraint: Δx · Δk ≥ 1/2 (mathematical identity for any wave)

Multiply by ℏ: Δx · (ℏΔk) ≥ ℏ/2 → Δx · Δp ≥ π/4

Lattice cutoff: The minimum Δx = 1 (one lattice spacing = Planck length). The maximum Δp = π/2 (Brillouin zone edge: p_max = ℏπ/a = π²/2). These are geometric facts, not physics.

What this means: The uncertainty principle is not a “limitation on knowledge” or a sign that “reality is fundamentally probabilistic.” It is the same mathematical identity that prevents a guitar string from being localized to one point while having a single pure frequency. Waves spread. That is all.

Minimum uncertainty in Planck units

The tightest localization possible is one lattice spacing: Δx = 1 (Planck length). At that scale:

Δp ≥ π/4 ≈ 0.785 (Planck momentum units)

This is the lattice’s built-in resolution limit. Below one lattice spacing, the concept of “position” ceases to exist — not because of quantum mysticism, but because there are no nodes closer than a = 1.

§3 — The Schrödinger Equation = Lattice Wave Equation

The fundamental equation on the lattice is the classical wave equation for a discrete elastic medium:

Lattice Wave Equation (Planck units) ∂²φ/∂t² = Σ_nn (φ_neighbor − φ) + V′(φ)

This is Newton’s second law applied to each lattice node: acceleration = restoring force from neighbors + potential force. In the low-energy, long-wavelength limit, this becomes:

Continuum limit (wavelength ≫ a): the discrete Laplacian becomes ∇²φ, giving the wave equation ∂²φ/∂t² = ∇²φ (with c = 1).

Non-relativistic limit: factor out the rest-mass oscillation φ = ψ · e^−imt. The slowly-varying envelope ψ satisfies:

Schrödinger Equation (Planck units) i(π/2) ∂ψ/∂t = −(π/2)²/(2m) ∇²ψ + Vψ

This is exactly the textbook Schrödinger equation with ℏ = π/2. It was not postulated — it was derived as the low-energy limit of the lattice wave equation.

The Schrödinger equation is to the lattice wave equation what the guitar-string equation is to a chain of masses on springs. Same physics, different resolution. At long wavelengths you see smooth Schrödinger waves. At short wavelengths (near a = 1) you see the lattice nodes directly. Quantum mechanics is the infrared limit of classical wave mechanics on a discrete medium.

Dispersion relation

The lattice dispersion relation (exact, all wavelengths):

ω² = (4/a²) sin²(qa/2) → ω² = 4 sin²(q/2) (Planck units)

At long wavelength (q ≪ 1): ω ≈ q (linear, relativistic). At the zone edge (q = π): ω = 2 (maximum frequency). The group velocity v_g = cos(q/2) goes to zero at the zone edge — this is why the electron (zone-edge mode) has mass: it cannot propagate.

The textbook E = p²/(2m) is the long-wavelength approximation of this exact lattice result.

§4 — Quantized Energy = Standing Wave Boundary Conditions

Energy quantization is not a “quantum postulate.” It is what happens when a wave is confined between boundaries.

Standing Wave Condition k_n = nπ/L ⇒ E_n = n²π³/(4L²m)

A wave confined to a region of size L lattice spacings can only sustain wavelengths λ = 2L/n. The allowed wavenumbers are k_n = nπ/L, exactly as for a vibrating string. The energy E = ℏ²k²/(2m) = (π/2)² · (nπ/L)² / (2m) = n²π³/(4L²m).

Hydrogen atom quantum numbers

Every quantum number is a mode count:

Quantum Number	Wave Meaning	Planck Origin
n (principal)	Radial mode count: n nodes in radial direction	Inner BC at r = r_p, outer BC at r → ∞
l (orbital)	Angular mode type: l nodal planes through origin	Spherical harmonic Y_l^m = angular standing wave
m_l (magnetic)	Orientation of angular pattern relative to z-axis	−l ≤ m ≤ +l: (2l+1) orientations
s (spin)	Internal yin/yang state: ± polarization	s = ±1 (binary), always exactly 2 states

Pauli exclusion principle: Two standing waves with identical mode numbers (n, l, m_l, s) would be the same wave. You cannot have two copies of the same wave at the same location — that is just one wave with twice the amplitude. “No two fermions in the same state” = “a standing wave is unique.”

§5 — Spin = Internal Binary State

Each lattice node has a binary internal degree of freedom: it can be displaced in the positive or negative direction along the bond. This is the yin/yang state — the origin of spin.

Spin in Planck Units s = ±ℏ/2 = ±π/4

The “½” in spin-½ is not mysterious. It means the spin angular momentum is half of ℏ = π/2, which is π/4. This is the minimum angular momentum on the lattice — one internal flip per cycle.

From yin/yang to the Dirac equation

Two internal states (±) per node → Pauli matrices σ_x, σ_y, σ_z span the internal space (SU(2))
Three spatial directions (x, y, z) → Clifford algebra Cl(3,0): σ_iσ_j + σ_jσ_i = 2δ_ij
Time as a distinguished direction → extends to Cl(3,1) with γ matrices: {γ^μ, γ^ν} = 2g^μν
The Dirac equation follows: (iγ^μ∂_μ − m)ψ = 0

Every spinor, chirality, CPT property, and antimatter state is a consequence of lattice geometry. The Dirac equation is derived, not postulated.

§6 — The Born Rule = Wave Intensity

The Born rule states that the probability of finding a particle at position x is |ψ(x)|². In wave mechanics, this is not a postulate — it is a definition:

Born Rule (Wave Form) P(x) = |φ(x)|² / ∫|φ|²dx = energy density at x / total energy

Where will a detector click? Where the wave is most intense. A water wave hitting a beach deposits most energy where the amplitude is largest. A lattice wave deposits most energy (triggers detectors) where |φ|² is largest. There is no mystery.

Why squared? Energy in a Hookean medium is proportional to displacement squared: E = ½kφ². This is Hooke’s law, not quantum mechanics. The “probability = amplitude squared” rule is the statement that detectors respond to energy, and energy goes as amplitude squared in any elastic medium.

§7 — Tunneling = Evanescent Waves

A wave encountering a potential barrier V > E does not stop. Its amplitude decays exponentially inside the barrier — an evanescent wave. If the barrier is thin enough, non-zero amplitude emerges on the other side.

Tunneling (Planck units) T = e^−2κL where κ = √(2m(V−E)) / (π/2)

This is identical to frustrated total internal reflection in optics, evanescent coupling in waveguides, and the decay of sound through a wall. All are classical wave phenomena. The lattice adds one feature: the minimum barrier width is 1 (one lattice spacing), setting a maximum tunneling rate.

Cosmological tunneling

The most spectacular tunneling result: each lattice node sits in a double-well cosine potential. The tunneling rate through the barrier determines the age of the universe:

Cosmological Tunneling Rate Γ = e^−1/α = e⁻¹³⁷ ≈ 10⁻⁶⁰ per Planck time

The inverse of this rate is the Hubble time: t_H ≈ d³ · e^1/α Planck times ≈ 10⁶⁰ Planck times ≈ 13.8 Gyr. The universe is old because α is small. Cosmology is quantum tunneling on a grand scale.

§8 — Entanglement = One Wave, Not Two Particles

The “spooky action at a distance” that troubled Einstein has a simple wave explanation: there were never two separate particles. There was always one spatially extended wave.

QM says Two entangled particles in an EPR pair have correlated measurements even when separated by light-years. Measuring one “instantly affects” the other.

Wave reality A single wave occupies a large region of the lattice. Its internal correlations (phase relationships between nodes) are built in from the start. Measuring at one end samples the wave; the correlation was always there. Nothing travels.

Consider a vibrating drumhead. Press one edge down — the opposite edge moves up. Not because a signal traveled across the drum, but because the drum is one connected object with correlated modes. The lattice is the drum.

Bell inequality violations: Bell’s theorem proves that no local hidden variable theory can reproduce QM correlations. But a wave is not a local hidden variable — it is a non-local, spatially extended object. Waves naturally produce the correlations Bell tested, because the wave was never localized to begin with. Bell’s theorem eliminates particle models. It does not constrain wave models.

§9 — Measurement = Wave Interaction

The “measurement problem” is the most famous puzzle in quantum mechanics: how does a superposition “collapse” to a definite outcome? The wave answer: it doesn’t. There was never a collapse.

QM says The wave function collapses upon measurement. Before: superposition. After: definite state. The transition is instantaneous, non-unitary, and unexplained.

Wave reality A detector is also a wave (a large, complex standing wave in the lattice). When the target wave and detector wave interact, energy redistributes locally. The interaction is deterministic — it looks random only because the detector’s internal state (10²³ modes) is unknowable in practice.

No collapse. No non-unitary evolution. No observer-dependent reality. Just two waves interacting on the lattice, governed by the same wave equation before, during, and after “measurement.”

The double slit

A wave passes through two slits. It interferes with itself. An interference pattern forms. This happens for water waves, sound waves, light waves, electron waves, and neutron waves — because they are all waves in the same medium.

“Detecting which slit” destroys the pattern because the detector wave interacts with the target wave at the slit, disrupting the phase relationship needed for interference. This is wave mechanics, not quantum weirdness.

In Planck units: the slit separation d determines the fringe spacing Δy = (π/2)L/(pd), where p = (π/2)k is the wave momentum and L is the distance to the screen. Pure geometry.

§10 — The Hydrogen Atom in d and π

The simplest atom — one proton wave + one electron wave — has every property determined by d = 3 and π:

Quantity	Formula (d, π, α)	d = 3 Value	SI Value
Bohr radius a₀	1/(2d · π^2d−1 · α^d²+d+1)	1/(6π⁵α¹³)	0.529 Å
Ground state E₁	−d · π^2d−1 · α^d(d+1)+2 · E_Pl	−3π⁵α¹⁴ E_Pl	−13.61 eV
Rydberg energy	d · π^2d−1 · α^d(d+1)+2 · E_Pl	3π⁵α¹⁴ E_Pl	13.61 eV
Energy levels E_n	E₁/n²	−3π⁵α¹⁴/n²	−13.61/n² eV
He screening constant	(2d−1)/2^d+1	5/16	5/16
H₂ bond energy	Weinbaum (HL + ionic + Wang)	variational	4.02 eV (−15%)
a₀/r_p ratio	1/(2d · α^d²+d+1 · π^2d−1) × dπ^2d/2	6π⁵/(4α) = 62,923	62,920

The α exponents decoded: The Bohr radius involves α¹³ where 13 = d² + d + 1 = 9 + 3 + 1. The ground-state energy involves α¹⁴ where 14 = d(d+1) + 2 = 12 + 2. Every exponent is a polynomial in d. The hydrogen atom is pure lattice geometry.

§11 — Every QM Postulate, Derived

Textbook quantum mechanics has 5–6 postulates. Every one is a theorem of wave mechanics on a discrete lattice:

Postulate 1 States are vectors in Hilbert space

Derived Wave displacements on the lattice form a vector space. Linearity (superposition) comes from Hooke’s law: F = −kφ is linear. The inner product ⟨φ|χ⟩ = Σ φ_iχ_i is the overlap energy on the lattice. Completeness follows from the finite (but enormous) number of lattice nodes.

Postulate 2 Observables are Hermitian operators

Derived Physical observables (energy, momentum, angular momentum) are generators of lattice symmetries. Translation symmetry → momentum operator. Time-translation symmetry → Hamiltonian. Rotation symmetry → angular momentum. Hermiticity follows from the lattice being a real elastic medium (energy is real-valued).

Postulate 3 Measurement yields eigenvalues

Derived Standing waves have definite frequencies (eigenvalues of the Hamiltonian). When a detector wave resonates with a standing wave, it responds at that frequency. “Measurement yields an eigenvalue” = “a tuning fork responds to its natural frequency.”

Postulate 4 Probability = |⟨a|ψ⟩|² (Born rule)

Derived Energy in a Hookean medium: E = ½kφ². Detectors respond to energy. Fraction of energy in mode |a⟩ = |⟨a|ψ⟩|² / ||ψ||². The Born rule is Hooke’s law. (See §6)

Postulate 5 Time evolution: iℏ d|ψ⟩/dt = H|ψ⟩

Derived This IS the lattice wave equation in the non-relativistic limit, with ℏ = π/2. (See §3)

Postulate 6 Collapse upon measurement

Eliminated No collapse occurs. Wave–wave interaction redistributes energy locally. The deterministic wave equation governs the system at all times, including during “measurement.” Apparent randomness arises from practical ignorance of the detector’s 10²³ internal modes. (See §9)

Score: Postulates 1–5 are derived as theorems. Postulate 6 (collapse) is eliminated entirely. Quantum mechanics has zero independent postulates. It is classical wave mechanics on a discrete 3D lattice with spacing a, stiffness k, and inertia η.

§12 — Every QM Constant in Planck Units

Quantity	SI Expression	Planck Value
Action quantum ℏ	1.055 × 10⁻³⁴ J·s	π/2
Spin quantum ℏ/2	5.27 × 10⁻³⁵ J·s	π/4
Minimum uncertainty ℏ/2	5.27 × 10⁻³⁵	π/4
Fine structure α	1/137.042	d²/(2^d+1 · 120^1/4 · π^11/4)
Electron mass m_e	9.109 × 10⁻³¹ kg	π²/(d−1) = π²/2
Bohr radius a₀	5.29 × 10⁻¹¹ m	1/(6π⁵α¹³)
Rydberg energy	13.606 eV	3π⁵α¹⁴ E_Pl
Compton wavelength λ_C	2.43 × 10⁻¹² m	2π(ℏ/m_ec) = π²/(π²/2) = 2
Zero-point energy (per mode)	ℏω/2	πω/4
Photon energy	ℏω	πω/2
de Broglie wavelength	λ = h/p	λ = π²/p
Magnetic moment (electron)	eℏ/(2m_e)	απ/2 · (2/π²) = α/π

The Punchline

Every quantum mechanical constant, equation, and “postulate” reduces to classical wave mechanics on a 3D discrete lattice. The only inputs are:

d = 3 spatial dimensions
π = ratio of circumference to diameter (wave geometry)
a = 1 lattice spacing (sets the scale)

Quantum mechanics is not a separate theory of physics. It is the name humans gave to classical wave mechanics when they discovered it without knowing the medium existed.