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=Die Feymanroute zu Schrödinger=
Die Wahrscheinlichkeit eines möglichen Messwertes t<sub>2</sub> wird von Schrödingers Wellenfunktion <math>\psi(\mathbf{r},t_2)</math> beschrieben. Die Funktion ist von der [[Unsere Quantenwelt/ Gravierende Krankheiten/ Schroedinger|Schrödingergleichung]] durch <math>\psi(\mathbf{r},t_1).</math> bestimmt. Was aber bestimmt <math>\psi(\mathbf{r},t_1)</math>&nbsp;? Natürlich das Ergebnis der Messung von t<sub>1</sub>-Was sonst? Normale Messungergebnisse geben die Wahrscheinlichkeit eines möglichen Messergebnisses wider.

==Zwei Regeln==
In diesem Kapitel entwickeln wir den Algorithmus für die quantenmechanische Wahrscheinlichkeit ausgehend von zwei fundamentalen Regeln. Zuvor jedoch noch zwei Definitionen:

*'''Varianten''' sind mögliche Abläufe von Messergebnisssen.
*Mit jeder Variante ist eine [[wikipedia:Komplexe Zahl|komplexe Zahl]]verknüpft, welche '''Amplitude''' heißt.

Angenommen, Sie wollten die Wahrscheinlichkeit eines Messergebnisses mithilfe vorheriger Messungen berechnen. Sie müssten folgende Schritte tun:

* Nehmen Sie eine beliebige Messreihe, die Sie bereits gemacht haben könnten.
* Fügen Sie jeder Variante eine Amplitude hinzu.
* Wenden Sie eine der folgenden Regeln an:

:'''Regel A''':

:<br> '''Rule A''': If the intermediate measurements are made (or if it is possible to infer from other measurements what their outcomes would have been if they had been made), first square the absolute values of the amplitudes of the alternatives and then add the results.

: '''Rule B''': If the intermediate measurements are not made (and if it is not possible to infer from other measurements what their outcomes would have been), first add the amplitudes of the alternatives and then square the absolute value of the result.

<br>In subsequent sections we will explore the consequences of these rules for a variety of setups, and we will think about their origin — their ''raison d'être''. Here we shall use Rule&nbsp;B to determine the interpretation of <math>\overline{\psi}(k)</math> given Born's probabilistic interpretation of <math>\psi(x)</math>.

In the so-called "continuum normalization", the unphysical limit of a particle with a sharp momentum <math>\hbar k'</math> is associated with the wave function
:<math>
\psi_{k'}(x,t)=\frac1{\sqrt{2\pi}}\int\delta(k-k')\,e^{i[kx-\omega(k)t]}dk=
\frac1{\sqrt{2\pi}}\,e^{i[k'x-\omega(k')t]}.
</math>
Hence we may write <math>\psi(x,t) = \int\overline{\psi}(k)\,\psi_{k}(x,t)\,dk.</math>

<math>\overline{\psi}(k)</math> is the amplitude for the outcome <math>\hbar k</math> of an infinitely precise momentum measurement. <math>\psi_{k}(x,t)</math> is the amplitude for the outcome&nbsp;<math>x</math> of an infinitely precise position measurement performed (at time&nbsp;t) subsequent to an infinitely precise momentum measurement with outcome <math>\hbar k.</math> And <math>\psi(x,t)</math> is the amplitude for obtaining&nbsp;<math>x</math> by an infinitely precise position measurement performed at time&nbsp;<math>t.</math>

The preceding equation therefore tells us that the ''amplitude'' for finding&nbsp;<math>x</math> at&nbsp;<math>t</math> is the product of
# the ''amplitude'' for the outcome <math>\hbar k</math> and
# the ''amplitude'' for the outcome <math>x</math> (at time&nbsp;<math>t</math>) subsequent to a momentum measurement with outcome&nbsp;<math>\hbar k,</math> 

summed over all values of <math>k.</math>

Under the conditions stipulated by Rule&nbsp;A, we would have instead that the ''probability'' for finding <math>x</math> at <math>t</math> is the product of
# the ''probability'' for the outcome <math>\hbar k</math> and
# the ''probability'' for the outcome <math>x</math> (at time&nbsp;<math>t</math>) subsequent to a momentum measurement with outcome&nbsp;<math>\hbar k,</math> 

summed over all values of <math>k.</math>

The latter is what we expect on the basis of standard probability theory. But if this holds under the conditions stipulated by Rule&nbsp;A, then the same holds with "amplitude" substituted from "probability" under the conditions stipulated by Rule&nbsp;B. Hence, given that <math>\psi_{k}(x,t)</math> and <math>\psi(x,t)</math> are amplitudes for obtaining the outcome <math>x</math> in an infinitely precise position measurement, <math>\overline{\psi}(k)</math> is the amplitude for obtaining the outcome <math>\hbar k</math> in an infinitely precise momentum measurement.

Notes:
# Since Rule&nbsp;B stipulates that the momentum measurement is not actually made, we need not worry about the impossibility of making an infinitely precise momentum measurement.
# If we refer to <math>|\psi(x)|^2</math> as "the probability of obtaining the outcome&nbsp;<math>x,</math>" what we mean is that <math>|\psi(x)|^2</math> ''integrated'' over any interval or subset of the [[wikipedia:Real line|real line]] is the probability of finding our particle in this interval or subset.

<div class="noprint">
[[../Feynman_route/two_slits|'''NEXT &gt;''']]
</div>#REDIRECT [[Unsere Quantenwelt]]

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