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Sum over paths picture (Online only)

Let us now consider an alternative explanation of the double-slit experiment, however, due to Richard Feynman (who, by the way, was born in Queens!). This explanation is published in his 1965 book, Quantum Mechanics and Path Integrals. Feynman's explanation is closer in spirit to a classical-like picture, yet it still represents a radical departure from classical mechanics. Feynman postulated that electrons could still behave as particle in a double-slit experiment. The twist is that the particles do not follow definite paths, as they would if they were classical particles. Rather, they can trace out a myriad of possible paths that might differ considerably from the path that would be predicted by classical mechanics. In fact, electrons initialized in the same manner, will follow different paths if they are allowed to wend their way through the double-slit apparatus! This is illustrated in the figure below:

Figure 6: Illustration of the many paths an electron can follow through the double-slit apparatus
\includegraphics[scale=0.5]{quantum_electron_dslit.eps}

Since the electron can follow any path and will follow different paths in different realizations of the experiment, physical quantities must be obtained by summing over all possible paths that the electron can follow. In order to carry out this sum, Feynman assigned a weight or importance to each path in the sum over paths. Since each path takes the electron from the source to a point on the screen (call this point $x$, where $x$ varies over the legnth of the screen) in a time $t$, the probability $P$ associated with each path can only depend on $x$ and $t$. If $t$ is fixed, the $P$ can only depend on $x$. Feynman proposed that each path could be assigned a quantity called the action $S(x)$, where $S(x)$ has the following properties:

1.
$S(x)$ varies only minimally for paths near the path that classical mechanics would predict the electron should follow.
2.
$S(x)$ varies increasingly dramatically as the paths differ increasingly from the classical path.
He then assigned an amplitude $A(x)$ to each path according to the formula

\begin{displaymath}
A(x) = e^{iS(x)/\hbar}
\end{displaymath}

where $i=\sqrt{-1}$ and $\hbar = h/2\pi$. The exponential of a complex argument has a simple expression in terms of common trigonometric functions:

\begin{displaymath}
e^{ix} = \cos(x) + i\sin(x)
\end{displaymath}

Thus, we can also write $A(x)$ as

\begin{displaymath}
A(x) = \cos(S(x)/\hbar) + i\sin(S(x)/\hbar)
\end{displaymath}

however, it is generally easier to work with the exponential directly.

It is important to note that $A(x)$ is a complex number. An amplitude is related to an actual probability $P$ by taking the absolute value squared of the amplitude. Thus, if there were only one path with amplitude $A(x)$, the probability that the electron would follow that path is

$\displaystyle P(x)$ $\textstyle =$ $\displaystyle A^*(x)A(x) = \left\vert A(x)\right\vert^2$  
  $\textstyle =$ $\displaystyle e^{-iS(x)/\hbar}e^{iS(x)/\hbar}$  
  $\textstyle =$ $\displaystyle 1$  

as expected. This is also the probability that the electron will end up at a point $x$ on the screen, since the single path takes the particle to a single definite point $x$. However, in Feynman's picture, the electron can follow any path. Thus, in order to compute the probability $P(x)$ that the particle ends up at a point $x$ on the screen, we must sum over all possible paths:

\begin{displaymath}
P(x) = \left\vert\sum_{{\rm paths}}A_{\rm path}(x)\right\vert^2
\end{displaymath}

where $A_{\rm path}(x)$ is the amplitude for a particle path. Since we are summing over many oscillating sines and cosines, there will be an interference pattern, meaning that the paths effectively interfere with each other. Indeed, the intensity $I(x)$ will be proportional to the probability: $I(x) \propto P(x)$. In fact, if we were to carry out this sum over paths (no simple feat, by the way), we would obtain an interference pattern that agrees with experiment.



Generally, a probability amplitude is a generalization of the square root of a probability that allows the amplitude to be a complex number. If $P$ is a probability, and $A$ is the associated probability amplitude, then if $A$ were restricted to be real, then there would be only two possible values of $A$, i.e., $A = \sqrt{P}$ and $A = -\sqrt{P}$. If we let $A$ be complex, the relation between $A$ and $P$ is

\begin{displaymath}
\vert A\vert^2 = A^*A = P
\end{displaymath}

and there is an infinite number of square roots of $P$. To see this, consider writing $A$ as

\begin{displaymath}
A = \sqrt{P}e^{i\theta}
\end{displaymath}

where $\theta$ is any number in the interval $[0,2\pi]$. If we can show that $A^*A = P$, then it will follow that any value of $\theta \in [0,2\pi]$ is allowable, which means that the number of possible amplitudes is infinite. The complex conjugate of $A$ is

\begin{displaymath}
A^* = \sqrt{P}e^{-i\theta}
\end{displaymath}

and we find
$\displaystyle A^*A$ $\textstyle =$ $\displaystyle \left(\sqrt{P}e^{-i\theta}\right)\left(\sqrt{P}e^{i\theta}\right)$  
       
  $\textstyle =$ $\displaystyle e^{-i\theta + i\theta}\left(\sqrt{P}\right)^2$  
       
  $\textstyle =$ $\displaystyle P$  



Now, suppose we have two interfering paths with amplitudes $A_1$ and $A_2$ (they should depend on $x$, but for notational simplicity, we will suppress the $x$ dependence). The total amplitude is $A = A_1 + A_2$, and the corresponding probability is $P = \vert A\vert^2$, which gives

$\displaystyle P$ $\textstyle =$ $\displaystyle A^*A$  
       
  $\textstyle =$ $\displaystyle \left(A_1^* + A_2^*\right)\left(A_1 + A_2\right)$  

Let $A_1 = a_1 e^{i\theta_1}$ and $A_2 = a_2 e^{i\theta_2}$, where $a_1$ and $a_2$ are real numbers. Then
$\displaystyle P$ $\textstyle =$ $\displaystyle \left(a_1 e^{-i\theta_1} + a_2 e^{-i\theta_2}\right)
\left(a_1 e^{i\theta_1} + a_2 e^{i\theta_2}\right)$  
       
  $\textstyle =$ $\displaystyle a_1^2 + a_2^2 + a_1a_2 \left(e^{-i(\theta_1-\theta_2)} +
e^{i(\theta_1-\theta_2)}\right)$  
       
  $\textstyle =$ $\displaystyle a_1^2 + a_2^2 + 2a_1 a_2 \cos(\theta_1-\theta_2)$  

The last term is known as the interference term. The presence of the cosine in that time, which oscillates, suggests the oscillation in the interference pattern observed on the screen.



At this point, several comments are in order. It is tempting to try to impose either the wave-like picture or the many-paths picture on the experiment. Indeed, both of these pictures provide a useful physical picture that helps us understand the outcome of the experiment. In the wave-like picture, we can think of each electron that leaves the source as feeling the presence of both slits simultaneously, and therefore interfering with itself (rather than with other electrons). In the many-paths picture, each electron follows not one path in the path sum but all possible paths at once, and these paths interfere with each other. However, the infuriating thing about quantum mechanics is that we have no way of knowing what is taking place between the source and the detector. All we have is the observation that there is an interference pattern. Feynman's picture makes this rather manifest. The implications of his picture can be summarized as

1.
Even within a particle-like interpretation of the experiment, particles do not have predictable positions and momenta along the paths. The reason for this is that the paths, themselves, are not predictable by any rule as they are in classical mechanics!
2.
If we could devise an experiment for measuring the position $x$ of the electron on the screen, we would find that different repetitions of the experiment on one electron initialized the same way would have different outcomes.
Thus, the best we can do from theory is to predict the probability of a given outcome of an experiment but not the actual outcome, itself.



The rationalizations of the three experiments we have examined, blackbody radiation, the photoelectric effect, and electron diffraction, leads us to conclude that classical mechanics, with its deterministic, predictable view of the universe, must be overthrown in favor of a much more radial theory, now known as quantum mechanics. It is interesting to note that the idea of probabilistic outcomes of experiments and the fact that we can ONLY predict the probabilities, lead Albert Einstein ultimately to reject quantum mechanics, saying: ``Gott spielt nicht Würfel'' (``God does not play dice'').


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Next: Simple statement of the Up: Electron diffraction Previous: Particle-wave picture
Mark E. Tuckerman 2011-12-12