Heisenberg Uncertainty Principle

In Quantum Mechanics, we use of probability density functions to describe waves and particles and the fact that we only know things probablisticly leads to the uncertainties. To arrive at h/2'pi, the uncertainty represents the smallest width (of space or time) to cover 50% of the possibilities.

To derive this from CTE considerations, consider that a CTE photon exists simultaneously across a region of space proportional to its wavelength and a region of time proportional to its period. This results as the SOE stretches a point in space-time into a helix, whose pitch and diameter is a wavelength. In this case uncertainties arise due to existence being spread out across time and space.

If a photon exists simultaneously across time and space, a point in time does not correspond to a point in space, but to a region. Similarly, a point in space corresponds to an interval in time. This introduces a potential measuring error proportional to the extent that time and space are spanned relative to the other dimension being fixed.

If the location of a photon is measured at a single point in time, the potential error will be proportional to its wavelength. If the time when a photon arrives at a particular point in space is measured, it will have a potential error proportional to its period. Because of the fixed speed of light, these proportionality constants are the same.

If we take the ratio of the spatial uncertainty to the temporal uncertainty, ignoring the proportionality constant, which cancels anyway, we get the speed of light.

    Us = λ

    Ut = λ / c	    (the period)

    Us/Ut = c

We want to quantify the product of the uncertainty in the measurement of the momentum of a photon and the uncertainty in a simultaneous measurement of its position in space. For these purposes, we will consider that the uncertainty is the average uncertainty or the smallest width of a range which contains half of the existence in that dimension (time or space).

The average uncertainty in location is relatively simple and is equal to λ/2. The worst case is λ and the best case is 0 and there is a symmetric distribution between these extents, giving us an average uncertainty of λ/2.

Momentum is the mass times the velocity. A photon is massless, but we can substitute the mass energy equivalent to arrive at an expression for the momentum of a photon.

    p = m v, v = c

        E = m c**2 = h ν

            h ν
        m = ----
            c**2

        h ν   E
    p = --- = -
         c    c

This momentum expression represents the average across a period. The uncertainty is quantified within a period, so we must arrive at expressions for the time varying components of momentum in order to compute its uncertainty.

We can define this time varying momentum function, P(t), such that when integrated over a period, it will result in the required average momentum.

    2π
    ∫ P(t) dt = p
    0

We know that p is a constant over time since Planck's constant, the speed of light and the frequency are all time invariant. This allows us to express P(t) as,

    P(t) = Kp p f(t)

    2π
    ∫ f(t) dt = 1 / Kp
    0

For f(t) was of the form sin**2(t), the definite integral over the relevant period would be π. Similarly, for the form sin**2(x)/x**2, the relevant function integrals would also be π. For the integral of f(t) to be equal to π, Kp must be equal to 1/π. These forms represent the forms of the solutions for the behavior across both time and space, both of which require the same value of Kp. Note that justifications for the use of these specific functions are missing.

Kp also sets the uncertainty in the momentum. It does this by establishing the worst case and average error for any measurement. The uncertainty is proportional to Kp and the worst case equal to 2*Kp, which with a normal distribution gives results in an average of Kp.

If we take the expression for the average uncertainty in the momentum of a photon and multiply it by the average uncertainty in its position, we get,

    U = (average_momentum_uncertainty) (average_location_uncertainty)

    U = (Kp p) (λ/2)

            h ν
        p = ---
             c 

        λ = c / ν

        pλ = h

    U = Kp h/2

        Kp = 1/π

        h
    U = --
        2π

(C) 1997-2004 George White, All Rights Reserved