Chapter 4. Heisenberg's Uncertainty Principle

Werner Heisenberg stated that the minimal errors of measurement of the simultaneous position and momentum of an electron--for example, in one direction of movement--is the product of those errors, which cannot exceed the value of Planck's constant.

A. Location and Momentum

Heisenberg emphasized that no limitation exists on the precision of either the location, x, or the momentum, p_x, of the electron, yet one does exist for their product. The Uncertainty Principle for direction x is stated as:

Δx Δp_x > or = h (15

The dimensions of the left side are L for the position and MLT^-1 for the momentum giving a product of ML²T^-1. Substituting each dimension with its comparable unitary electronic attribute gives m_e·λ_e²·t_e^-1, which is Planck's constant. It also can be stated to be c·m_e·λ_e, where c is the speed of light and cannot change. The displacement L cannot change by an interval less than the electronic attribute, λ_e, and the mass M, less than the electronic attribute m_e. Therefore, we can state Heisenberg's Uncertainty Principle in a more-exact way: Not only cannot the products of the precisions of the displacement and the momentum be less than Planck's constant, the precision of both the displacement and the momentum of the electron are constrained, also.

B. Energy and Time

In a different formulation of Heisenberg's Uncertainty Principle, if E is the energy of a photon, and t is the time of emission, the principle states:

ΔE Δt > or = h (16

The dimensions of (E·t) are the same as for (x·p_x). Therefore, the result is the same--Planck's constant.

C. Frequency and Time

In yet another formulation, the frequency, ν, and arrival time, t, of a beam of photons gives:

Δν Δt > or = 1 (17

The term, (ν·t), is a dimensionless TT^-1, and h is replaced by the number 1, which is actually t_e·t_e^-1.

The interpretation of the meaning of Heisenberg's Uncertainty Principle is easier when viewed in this manner. Intuitively, the product of the minimal errors cannot be smaller than the combination of the pertinent, quantum attributes of the phenomenon observed. Otherwise, the indivisible quantized electronic attributes would need to be separated into fractional parts, which, of course, is impossible in the discrete, discontinuous Quantum World.

D. Quantum Momentum

On the quantum level, an elementary particle, such as an electron, cannot possess momentum. It possesses mass and a position at a particular moment in time. Momentum is a macroscopic concept, where quantum length-unit positions of a mass relate, respectively, to contiguous quantum time units.

For instance, an electron can travel one λ_e in one t_e or remain stationary. It is, therefore, either travelling at light speed or not at all. During a light-speed quantum jump, the electron's momentum is undefined. This is why the limiting factor, with regard to location and momentum or to energy and time, in Heisenberg's Uncertainty Principle, is Planck's constant. In essence, this FUPCON establishes the boundary between the Real and Quantum Worlds. Of course, when using the SE of unit measure, its value is one, as follows:

Δx Δp_x > or = 1 ΔE Δt > or = 1 Δν Δt > or = 1

Each delta, Δ, factor in the above formulas and, even more restrictively, each contributing dimension in those delta factors cannot take on fractional values if they are measured using the attributes of elementary particles as units of measure. This is what Heisenberg, in a disguised fashion, is saying in the Uncertainty Principle. No doubt, at the time, Heisenberg's lack of knowledge of what Planck's constant consists is responsible for this ambiguity. Now, however, we realize that the Uncertainty Principle uses the various quantum attributes of individual elementary particles as limiting factors rather than an assemblage of them in the guise of Planck's constant.