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Heisenberg Introduction

Werner Heisenberg, 1901—1976.

WikiMechanics is bare naked quantum mechanics. [[[aggregates |Particle]]] attributes are always quantized because we use a finite [[[categorization |categorical]]] scheme of [[[binary |binary]]] distinctions to describe sensation. Quantization comes from the logical structure of the descriptive method, even for a continuous sensorium. Time is quantized too because the [[[time |time coordinate]]] depends on a chain’s event index. Indices are always integers so [[span style=” font-family: times ; font-size:135% “]]//t//[[/span]] changes in steps. Motion is discontinuous in principle, and sometimes this is even observed as quantum leaping and tunnelling. We are cautious about using calculus because the logical foundations of both differential and integral calculus are proven using assumptions about continuity. So WikiMechanics does not require calculus; instead calculations are designed to be implemented on digital computers, in a finite number of discrete steps.

Another consequence of unmitigated quantum mechanics is that there are lower limits for the certainty of some measurements. For example, consider a [[[aggregates |particle]]] P described by a [[[historical-order |historically]]] ordered [[[orbits |repetitive]]] [[[chains |chain ]]] of events

\Psi ^{\sf{P}} = \left(      \sf{\Omega}_{1},   \sf{\Omega}_{2}   \;   \ldots  \;  \sf{\Omega}_{\it{i}} \;   \ldots  \;  \sf{\Omega}_{\it{f}}  \;  \ldots    \;    \right)

If P is [[[processes |isolated]]] then the [[span style=” display:inline-block ; “]][[[time |elapsed time]]][[/span]] between events [[i]] and [[f]] is

\Delta t  =        \left( \, f-i \right)  \hat{\tau}

where [[\hat{\tau}]] is the [[[time | period]]]. Consider [[span style=” display:inline-block ; “]][[[time |measuring the elapsed time]]][[/span]] from observations of event indices and periods. The [[span style=” display:inline-block ; “]][[[measurement-energy |experimental uncertainty]]][[/span]] in repeated measurments of [[\Delta t]] is written as [[\delta t]]. By the usual rules for assessing the propagation of [[span style=” display:inline-block ; “]][ experimental errors][[image /icons/Xlink.png link=”” ]][[/span]] this uncertainty is given by

\delta t = \left( \delta f + \delta i \right) \hat{\tau} + \left( \, f-i \right) \delta \hat{\tau}

As discussed [ where? ] the uncertainty in the period is bounded by

\delta \hat{\tau} \ge \hat{\tau} k_{S}

And some unavoidable uncertainty is also associated with event indices. They are required to be integers, so rounding-off errors are [[\delta i \ge 1/2]] and [[\delta f \ge 1/2]]. Then

\delta t \ge \hat{\tau} + k_{S} \left(\, f-i \right) \hat{\tau}

And since [[i < f]] we know that [[\, f - i \ge 1]] so

\delta t \ge \left( 1 + k_{S} \right) \hat{\tau}

or in terms of the energy

\delta t \ge h \left( 1 + k_{S} \right) /E

Thus the uncertainty in a time measurement can be //decreased// by working with high energy particles. In contrast, if [[\delta E \ge k_{S} E]] where

k_{S} = \frac{ 1}{2} \sqrt{ 1 + 1 / \pi \; \vphantom{\Sigma^{2}} } - \frac{ 1}{2}

then the uncertainty in an energy measurement is //increased// for larger particles. The two effects cancel for the product of the uncertainties

\delta E \, \delta t \ge h k_{S} \left( 1 + k_{S} \right)

leaving a constant

\delta E \, \delta t \ge h /4\pi

This is one of [[span style=” display:inline-block ; “]][ Werner Heisenberg’s][[image /icons/Xlink.png link=”” ]][[/span]] uncertainty relationships.

Distance between Events

The distance between initial and final events is given by

\ell \equiv \left\| \, \overline{r}^{\, \mathbf{B}}_{f} - \overline{r}^{\, \mathbf{A}}_{i} \right\|

If the frame of reference is inertial and   \mathsf{X} is isolated (apart from the interaction under consideration) then the wavelength  \lambda does not vary so that

\ell = \left( \rule{0px}{10px} f-i \right) \lambda^{\mathsf{X}}

Then using de Broglie’s expression for the wavelength in terms of the momentum  p gives

\ell = \dfrac{h \left( f-i \right)}{p^{\,\mathsf{X}}}

Elapsed Time between Events

Let us assume that atoms   \mathbf{A} and   \mathbf{B} share a common reference frame where they both exhibit slow motion. Then the difference in the time of occurrence  t between initial and final events is

\Delta t = t_{f}^{\, \mathbf{B}} - t_{i}^{\, \mathbf{A}}

If the frame of reference is inertial and   \mathsf{X} is isolated (apart from the interaction under consideration) then the period  \widehat{\tau} of   \mathsf{X} does not vary between events. So the elapsed time is given by

\Delta t = \left( \rule{0px}{10px} f-i \right) \widehat{\tau}^{\mathsf{X}}

or in terms of the frequency  \nu

\Delta t = \dfrac{f-i}{\nu^{\mathsf{X}}}

Then using Planck’s postulate gives the elapsed time in terms of the mechanical energy  E as

\Delta t = \dfrac{h\left(f-i \right)}{E^{\mathsf{X}}}