The Uncertainity Principle

I am fascinated by the linking between the atomic world and the universe, as discussed in the previous post. For long I’ve been trying to quantify the linkage with some strong quantifiable connections. I set out to prove Heisenberg’s Uncertainity Principle (which is very evident on the small scale – but largely insignificant on the astronomic level) using the Infinitely Flat Spheres hypothesis, because they seem similar.

This is a derivation of Heisenberg’s Uncertainity Principle.

I daresay the derivation is approximate. Surely there must be some inaccuracy in the procedure I follow, and I welcome critics to analyze the procedure. Nonetheless, it does lead to some conclusive result.

The similarity is thus: Heisenberg’s Uncertainity Principle attests that position and momentum of dual nature particles, especially microscopic ones, are not commutable. If one is known, the other is not. Further, the more precisely one measure is known, the less precisely is the other known. Likewise, the Infinitely Flat Spheres idea proposes the imperceptibility of a large circular or spherical object by a small observer on its surface. That is, the dimensional scales of the smaller observer and the larger object are not commutable. If the smaller observer’s dimensions are precisely known, the vastness of the larger object cannot be perceived, and vice versa.

Figure (a): Movement of particle P about star S. This shows the actual path followed by the particle.

To begin, consider these two diagrams. Figure (a) shows the movement of a particle P from P₁ to P₂ about a star S. The radius r is very large, and so is the particle’s velocity v. Considering a very small change in position, Δx, the velocity doesn’t change. Then,

Δx = rθ

Figure (b): Movement of particle P along an infinitely long approximate path. This is how the movement appears to the particle.

Figure (b) shows the approximation of the circular path. Since the particle is very small compared to r, to itself the path seems infinitely straight. For this displacement, it effectively notices no change in position, but there is a change in the velocity component along the vertical infinitely straight line. This is the y-component of the velocity, and can be calculated from figure (a).

Δv = v (cos θ — 1)

Before we proceed, it is crucial to understand the situation. Since the particle P is moving, there is definitely some change in position. Although in the frame of reference of the particle itself, this displacement should be clearly noticed; since P can only notice a change in position relative to a stationary object (in this case S), then in respect to the vastness of the circular orbit, the displacement is imperceptible. That is to say, while having moved by Δx, the particle observes no change in x. Conversely, having observed no change may actually mean the particle has moved by Δx. Effectively, there’s an error in calculation of x, equalling Δx.

Their product:

Δx . Δv = rvθ (cos θ — 1)

Now, the minima of the equation occurs when its derivative w.r.t θ vanishes.

d( Δx . Δv ) / dθ = rv[ cosθ — 1 — θ sinθ] = 0

which implies, 2θ² ≈ 1 — cosθ, since cosθ ≈ 1 for small θ.

So,

Δx . Δv ≈ —2rvθ³

Applying the fact that nothing can exceed the speed of light, v ≤ c, or —v ≥ —c, we get the inequality:

Δx . Δv ≥ —2rcθ³

Since θ is in the clockwise direction, we convert it to standard anti-clockwise by taking its modulus:

Δx . Δv ≥ 2rc|θ|³

This derivation shows that there does, in fact, exist a lower limit to the error in position and velocity one can determine simultaneously.

This comes very close to Heisenberg’s Uncertainity Principle, which is thus:

Δx . Δv ≥ h/4πm

All that remains in this derivation, is to find limiting conditions for r and θ so that the same R.H.S value is obtained as in the equation above (possibly through the consideration of a light photon in place of the particle in this exercise, for instance). As a bonus, this also provides some sort of validation for the Atomic Universe theory, showing the presence of the same calculable properties in the astronomic level, as that on the atomic level. A similar application of the Infinitely Flat Spheres approximation can be applied at the atomic level for the same purpose.

~ by Pulkit on July 4, 2009.