﻿ Planck's Constant: Its Roles and its Limit in Subatomic Physics
San José State University

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Thayer Watkins
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Planck's Constant:
Its Roles and its Limit
in Subatomic Physics

Here are the various ways that Planck's constant h or its reduced form h arose in subatomic physics:

• As a parameter in the frequency distribution for black-body radiation discovered by Max Planck

#### f(ν) = (2hν³/c²)/[exp(hν/kT)−1]

• In the formula for the energy of a photon asserted by Albert Einstein,

#### E = hν

• As the quantification of angular momentum in Niels Bohr's solar system model of the hydrogen atom

#### L = mωr² = nh

where ω is the rate of rotation in radians per second, r is orbit radius and n is an integer.

• As a key parameter in Louis de Broglie's 1924 thesis on the wave nature of material particles. The wave length λ of such a matter wave is given by

#### λ = h/p

where p is the momentum of the particle.

• As a parameter in Erwin Schrödinger's equations for a quantum system

#### ih(∂ψ/∂t) = H^ψ (time dependent)

where i is the imaginary unit, ψ is the wave function and H^ is the Hamiltonian operator for the system.

• As the key parameter in Werner Heisenberg's Uncertainty Principle

#### σpσx ≥ ½h

where σp and σx are the standard deviations for momentum and location, repectively.

• As the key parameter in the Energy-Time Uncertainty Principle

#### σEσt ≥ ½h

where σE and σt are the standard deviations for energy and time, repectively.

• As the quantification of angular momentum in the model of nuclear rotation formulated by Åge Bohr and Dan Mottelson

#### Jω = h(I(I+1))½

where J is the moment of inertia of the nucleus, ω is the angular rate of rotation in radians per second and I is a positive integer.

Clearly black-body radiation and photons involve the electromagnetic field. In an atom the electrons are held in orbit by the electric field. The de Broglie relations are not derive but asserted by analogy. The attempts to derive Schrödinger equations start with Maxwell equations and hence also pertain to electromagnetic fields. Tthe Uncertainty relations are derived from a Schrödinger equation and hence pertain to electromagnetic fields.

Planck's constant does not seem to be the appropriate constant empirically for the Bohr-Mottelelson formula for nuclear rotation. See Nuclear Rotation. What appears to be the case is that Planck's constant applies for phenomena concerning the electromagnetic field but another constant applies in phenomena concerning the force involved in nuclei. This nuclear constant is about 1/137 of the magnitude of Planck's constant. This would apply to the constant in de Broglie's relation.

This ratio is justified in that a quantum constant like Planck's constant is inversely proportional to the coupling constant of its field. The coupling constant for the electromagnetic field, which is usually called the fine structure constant, is approximately 1/137. The coupling constant for the nuclear force is approximately 1.0. Thus the quantum constant for the electromagnetic field should be 137 times the quantum constant of the nuclear force field.

Thus the Bohr-Mottelson formula should be

#### Jω = q(I(I+1))½

where q is the quantum constant for the nuclear force field and qh/137.