This Quantum World/Feynman route

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.

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

$\psi _{k'}(x,t)={\frac {1}{\sqrt {2\pi }}}\int \delta (k-k')\,e^{i[kx-\omega (k)t]}dk={\frac {1}{\sqrt {2\pi }}}\,e^{i[k'x-\omega (k')t]}.$

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.