h vs ℏ: A Proof That Planck's Constant Is a Coordinate Choice, Not Physics

J. Rogers, SE Ohio

Abstract

We prove that the choice between h (Planck's constant) and ℏ (reduced Planck's constant) represents a coordinate system choice in measurement space, not a physical distinction. The invariant physics is always the proportionality E ∝ f; the constant (h or ℏ) is purely an artifact of how we graduate the frequency axis. 

1. The Physical Invariant

The fundamental physical relationship is:

E ∝ f

Energy is proportional to frequency. This proportionality is the physics—it is invariant, coordinate-independent, and universal. The constant of proportionality depends entirely on how we choose to measure frequency.

2. The Notation Proof

2.1 The Physical Reality

A photon oscillates. This is a discrete, countable phenomenon. Each complete oscillation represents one quantum of action.

2.2 Measurement in Cycles per Second

Choice: Measure frequency in cycles per second (Hz).

Result:

E = hf

where h = 6.62607015 × 10⁻³⁴ J·s.

This says: "Energy equals h times the number of complete cycles per second."

2.3 Measurement in Radians per Second

Choice: Measure frequency in radians per second (angular velocity, ω).

CRITICAL: ω is not frequency. It is 2π times frequency:

ω ≡ 2πf

This is a pure relabeling: one complete cycle = 2π radians.

The physics is still E ∝ f (energy proportional to cycles), NOT E ∝ ω:

Substitution:

f = ω/(2π)
E = hf                    [The physics: E ∝ f]
E = h(ω/(2π))            [Substitute the relabeling]
E = (h/2π)ω              [Algebraic rearrangement]

Definition:

ℏ ≡ h/(2π)

Result:

E = ℏω

2.4 Analysis

No physics occurred. The fundamental proportionality remains E ∝ f in both cases:

Case 1 (using h): E = hf → Energy is proportional to cycles per second Case 2 (using ℏ): E = ℏω = ℏ(2πf) = hf → Energy is STILL proportional to cycles per second

We performed a "Find and Replace" operation:

• Replace "cycles per second" (f) with "radians per second" (ω = 2πf)
• Replace h with h/(2π) = ℏ to compensate

The invariant physics E ∝ f never changed. We merely decided to use a protractor (radians) instead of a counter (cycles), and adjusted our conversion constant accordingly.

Theorem 1: The physics is always E ∝ f (energy proportional to frequency in cycles). The choice between writing E = hf or E = ℏω is purely notational—both reduce to the same proportionality E ∝ f.

Proof:

1. The physical phenomenon: oscillations occur at frequency f (cycles per second)
2. Case 1: Measure directly in cycles → E = hf → E ∝ f
3. Case 2: Measure in radians, where ω = 2πf → E = ℏω = ℏ(2πf) = hf → E ∝ f
4. Both cases yield identical physics: E ∝ f
5. The choice between h and ℏ is equivalent to choosing how to graduate the measurement axis
6. Since both describe the same invariant proportionality E ∝ f, the choice contains zero physical content. ∎

3. How ℏ Obscures Physical Geometry

3.1 What h Represents

h corresponds to one complete cycle:

• One photon = 1 quantum = 1 complete oscillation
• Action = 1 × h (integer multiple)
• Natural counting in cycles

3.2 What ℏ Represents

ℏ corresponds to one radian:

• One photon = 2π radians of angular motion
• Action = 2π × ℏ (requires factor of 2π)
• Counting requires angular conversion

3.3 The Hidden Geometry Problem

When we use ℏ instead of h, geometric factors in physical laws become obscured because the √(1/2π) scaling factors get absorbed into mass and temperature measurements.

Example: Hawking Temperature

The natural dimensionless ratio (the real physics) is:

T/T_P = m_P/(16π²M)

Using h (non-reduced Planck units):

T_P = (c³h)/(Gk_B)
m_P = √(hc/G)

T = c³h/(16π²GMk_B)

The geometric factor 16π² appears explicitly in the equation.

Using ℏ (reduced Planck units):

T_P_reduced = T_P × √(1/2π)
m_P_reduced = m_P × √(1/2π)

T = c³ℏ/(8πGMk_B)

The geometric factor now appears as 8π.

What happened? The √(1/2π) factors from using ℏ got absorbed:

• T_P absorbed one √(1/2π)
• m_P absorbed another √(1/2π)
• Their product absorbed (1/2π)
• This converts 16π² → 8π in the equation

The real geometry is 16π², not 8π. The reduced form hides this by absorbing scaling factors into the natural unit definitions.

Theorem 2: Using ℏ obscures the true geometric structure of physical laws by absorbing √(1/2π) factors into Planck unit definitions, causing geometric factors like 16π² to appear incorrectly as 8π.

Proof:

1. The dimensionless physics contains geometric factor 16π²
2. Using ℏ introduces √(1/2π) into T_P and m_P
3. These factors combine as (1/2π) in products
4. This converts 16π² = 2 × 8π² into 8π × (1/2π) × 2 = 8π
5. The true geometric factor (16π²) is obscured by the artificial scaling. ∎

4. The Invariant Physics: E ∝ f

4.1 The Fundamental Truth

The physics is always E ∝ f, never E ∝ ω.

Even when we write E = ℏω, the underlying proportionality remains E ∝ f because ω is just 2πf:

E = ℏω = ℏ(2πf) = (2πℏ)f = hf

The physical phenomenon hasn't changed—oscillations still occur at f cycles per second. We've merely chosen to express that same frequency using angular measure (ω = 2πf) and adjusted our constant accordingly (ℏ = h/2π).

Critical insight: ω is not an independent physical quantity. It is a derived measurement convention: ω ≡ 2πf. The fundamental oscillation rate is f (cycles), not ω (radians).

4.2 Geometric Interpretation

Consider a graph with Energy on the vertical axis and Frequency (in cycles) on the horizontal axis. The physical law is a straight line through the origin with fixed slope determined by nature.

Both h and ℏ represent this same line, just with different axis graduations:

If you graduate the frequency axis in Hertz (cycles/second):

Horizontal axis shows: f
Slope = h
E = hf

If you graduate the frequency axis in rad/s:

Horizontal axis shows: ω = 2πf (stretched by factor of 2π)
Slope = ℏ = h/(2π) (compressed by factor of 2π)
E = ℏω = ℏ(2πf) = hf

The slope value changes because you stretched the graph paper horizontally by 2π and the slope compressed by 2π to compensate, keeping the actual line unchanged. The universe didn't change—your axis graduation changed.

4.3 Temperature Analogy

Theorem 3: Asserting that ℏ is "more fundamental" than h is equivalent to asserting that Celsius is "real" and Fahrenheit is "fake."

Proof:

• Temperature T is an invariant physical quantity
• Celsius and Fahrenheit are different coordinate choices for expressing T
• Neither is more "real"—they are affine transformations of the same invariant
• Similarly, the invariant physics is E ∝ f (energy proportional to cycles)
• h and ℏ are different coordinate choices for expressing this same proportionality
• h measures it directly: E = hf
• ℏ measures it with a 2π conversion: E = ℏω = ℏ(2πf) = hf
• Neither constant is more "real"—they are coordinate choices for the same invariant E ∝ f. ∎

Corollary: Just as we don't claim that "temperature is fundamentally proportional to Celsius degrees rather than Fahrenheit degrees," we should not claim that "energy is fundamentally proportional to ω rather than f." The physics is E ∝ f in all coordinate systems.

5. Why h Is the Better Choice

While both h and ℏ are coordinate choices and thus equally "valid" mathematically, h is the better physical choice because:

5.1 Cycle Alignment

h aligns with natural cycle counting:

• 1 photon = 1 cycle = 1 × h of action
• Integer quantum numbers appear naturally
• No 2π factors needed to describe "one quantum"

ℏ requires angular conversion:

• 1 photon = 2π radians = 2π × ℏ of action
• Integer quantum numbers require 2π factors
• Must convert between cycles and angles

5.2 Geometric Clarity: The Hawking Temperature Example

Using h preserves real geometric factors:

The natural dimensionless relationship is:

T/T_P = m_P/(16π²M)

With h, this becomes:

T = c³h/(16π²GMk_B)

The geometric factor 16π² is explicit and visible.

Using ℏ obscures geometric factors:

With ℏ, the same physics becomes:

T = c³ℏ/(8πGMk_B)

The geometric factor appears as 8π because the √(1/2π) factors from T_P and m_P combined to absorb half the geometry.

The real geometry is 16π². Using ℏ hides this by:

1. T_P_reduced = T_P × √(1/2π) absorbs one factor
2. m_P_reduced = m_P × √(1/2π) absorbs another factor
3. Their product absorbs (1/2π), converting 16π² → 8π

This is not just aesthetics—it obscures the actual geometric structure of spacetime that produces Hawking radiation.

5.3 Natural Unit Consistency

Theorem 4: h produces natural units that preserve geometric structure, while ℏ produces natural units that absorb and hide geometric factors.

Proof:

1. Physical laws contain dimensionless geometric factors (like 16π²)
2. Natural units should preserve these factors transparently
3. Using h: geometric factors appear explicitly in equations
4. Using ℏ: geometric factors are partially absorbed into unit definitions via √(1/2π)
5. Therefore, h preserves geometric structure while ℏ obscures it. ∎

6. The Error in Modern Physics

Modern physics textbooks often state or imply that ℏ is "more fundamental" or "more natural" than h. This represents a conceptual error.

6.1 The Claim

"ℏ is the fundamental unit of action in quantum mechanics."

6.2 The Error

This conflates measurement convenience in one domain (angular momentum, where factors of 2π appear naturally) with fundamental physical structure. By insisting on ℏ as fundamental:

1. We obscure geometric factors in physical laws (16π² becomes 8π)
2. We absorb real geometric structure into unit definitions via √(1/2π) factors
3. We hide the natural cycle structure of quantum phenomena behind angular measures
4. We make dimensional analysis less transparent

6.3 The Correction

Proposition: h represents the natural cycle structure of quantum phenomena and preserves geometric clarity; ℏ is a coordinate convenience that obscures geometric structure by absorbing scaling factors.

Justification:

1. Quantum phenomena involve discrete cycles (oscillations, energy levels)
2. h measures complete cycles directly (1 quantum = 1h)
3. ℏ requires angular conversion (1 quantum = 2πℏ)
4. Using ℏ in natural units absorbs √(1/2π) into T_P, m_P, etc.
5. These absorbed factors hide real geometric structure (16π² → 8π)
6. Therefore, h is the clearer choice that preserves geometric transparency

7. Implications

7.1 For Quantum Mechanics

The fundamental quantum postulate is clearer when written:

E = nhf  (n = 0, 1, 2, ...)

rather than:

E = nℏω  (n = 0, 1, 2, ...)

The first makes the cycle structure explicit. The second requires knowing that ω = 2πf to see the underlying E ∝ f relationship.

7.2 For Natural Units

Natural units should be based on h, not ℏ, because:

1. They preserve real geometric factors (16π² not 8π)
2. The √(1/2π) scaling factors don't obscure physical structure
3. Dimensional analysis remains transparent

7.3 For Physics Education

Students should be taught:

• The physics is E ∝ f (energy proportional to cycles)—this never changes
• The constant (h or ℏ) is a measurement convention
• h measures cycles directly
• ℏ measures angular frequency (ω = 2πf), requiring 2π conversion
• Neither is "more fundamental"—they are coordinate choices
• h preserves geometric clarity; ℏ absorbs geometric factors into unit definitions

8. Conclusion

We have proven that:

1. The invariant physics is E ∝ f (energy proportional to frequency in cycles), not E ∝ ω
2. h and ℏ are coordinate choices, representing different ways to measure the same frequency
3. Both reduce to E ∝ f: Even E = ℏω is really E = ℏ(2πf) = hf
4. h preserves geometric structure while ℏ obscures it by absorbing √(1/2π) factors into natural units
5. The Hawking temperature example shows this explicitly: true geometry is 16π², which becomes 8π when using ℏ
6. h is the better choice for transparent physics and geometric clarity

The choice between h and ℏ contains zero physical content. It is purely conventional, like choosing Celsius vs Fahrenheit. The modern insistence that ℏ is "more fundamental" represents a conceptual confusion between:

• Measurement convenience (ℏ simplifies some angular momentum expressions)
• Physical fundamentality (the actual physics is E ∝ f, independent of which constant we use)

The physics is in the proportionality E ∝ f. The constant is just how we measure the Energy scale and frequency scale in our math.

When we use ℏ, we don't change the physics—we just hide geometric factors by absorbing √(1/2π) scaling into our natural unit definitions. This makes equations look simpler superficially (8π instead of 16π²) but obscures the real geometric structure.

The physics is in the proportionality. The constant is just graph paper.

---

Acknowledgments

This work clarifies a conceptual confusion that has persisted since the introduction of ℏ in the 1920s. Planck's original 1900 formulation using h was topologically correct for describing discrete quanta. The later preference for ℏ, while convenient for angular momentum calculations, obscured this discrete structure.

References

Planck, M. (1900). "Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum." Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 237-245.

Dirac, P.A.M. (1926). "The Fundamental Equations of Quantum Mechanics." Proceedings of the Royal Society A, 109(752), 642-653.

---

Note: This paper presents a mathematical and conceptual analysis of existing physics notation. No new physics is proposed. We merely clarify that the choice between h and ℏ is conventional (like choosing units), and that h better represents the discrete topological structure of quantum phenomena.