# This Quantum World/Feynman route#Two rules

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# The Feynman route to Schrödinger

The probabilities of the possible outcomes of measurements performed at
a time $t_2$ are determined by the Schrödinger wave function
$\psi(\mathbf{r},t_2)$. The wave function $\psi(\mathbf{r},t_2)$ is
determined via the Schrödinger
equation by
$\psi(\mathbf{r},t_1).$ What determines $\psi(\mathbf{r},t_1)$ ? Why,
the outcome of a measurement performed at $t_1$ --- what else? Actual
measurement outcomes determine the probabilities of possible measurement
outcomes.

## Two rules

In this chapter we develop the quantum-mechanical probability algorithm
from two fundamental rules. To begin with, two definitions:

-   **Alternatives** are possible sequences of measurement outcomes.
-   With each alternative is associated a complex
    number called **amplitude**.

Suppose that you want to calculate the probability of a possible outcome
of a measurement given the actual outcome of an earlier measurement.
Here is what you have to do:

-   Choose any sequence of measurements that may be made in the
    meantime.
-   Assign an amplitude to each alternative.
-   Apply either of the following rules:

:   \
    **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.

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:   **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.

\
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 B to determine the
interpretation of $\overline{\psi}(k)$ given Born\'s probabilistic
interpretation of $\psi(x)$.

In the so-called \"continuum normalization\", the unphysical limit of a
particle with a sharp momentum $\hbar k'$ is associated with the wave
function

$$\psi_{k'}(x,t)=\frac1{\sqrt{2\pi}}\int\delta(k-k')\,e^{i[kx-\omega(k)t]}dk=
\frac1{\sqrt{2\pi}}\,e^{i[k'x-\omega(k')t]}.$$ Hence we may write
$\psi(x,t) = \int\overline{\psi}(k)\,\psi_{k}(x,t)\,dk.$

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

The preceding equation therefore tells us that the *amplitude* for
finding $x$ at $t$ is the product of

1.  the *amplitude* for the outcome $\hbar k$ and
2.  the *amplitude* for the outcome $x$ (at time $t$) subsequent to a
    momentum measurement with outcome $\hbar k,$

summed over all values of $k.$

Under the conditions stipulated by Rule A, we would have instead that
the *probability* for finding $x$ at $t$ is the product of

1.  the *probability* for the outcome $\hbar k$ and
2.  the *probability* for the outcome $x$ (at time $t$) subsequent to a
    momentum measurement with outcome $\hbar k,$

summed over all values of $k.$

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

Notes:

1.  Since Rule B stipulates that the momentum measurement is not
    actually made, we need not worry about the impossibility of making
    an infinitely precise momentum measurement.
2.  If we refer to $|\psi(x)|^2$ as \"the probability of obtaining the
    outcome $x,$\" what we mean is that $|\psi(x)|^2$ *integrated* over
    any interval or subset of the real
    line is the probability of finding
    our particle in this interval or subset.

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