Quantum Paths: First Lesson

One of the most important accomplishments of quantum mechanics is the ability to predict the probability and phase for a given process. The primary goal of this lesson is to demonstrate how quantum mechanics determines the probability and phase for a particular process. The process we will examine is an electron (or any small particle) traveling through a smal slit in a wall and hitting a particular point on a screen beyond the wall. The initial state is the electron located at the slit and the final state is the electron located at the specificed point on the screen.

There are many different mathematical mechanisms for calculating the probability and phase or a particular process. One of the most popular involves the so-called "wave-particle duality." In this mathematical formalism physical objects are viewed not as discrete particles but rather as simultaneous particles and waves. In other words, we can no longer say where an electron is or how fast it is moving exactly, and the electron is more like a disturbance in space, like a water wave is a disterbance in a pond.

Here we will use the Feynman multiple paths interpretation of quantum mechanics and the corresponding mathematics to calculate the probability and phase of a process. Essentially, the idea is that for a particular process quantum mechanics sums of all possible mechanisms through which it can occur. In otherwords, every single path contributes to the calculation, and we can't specify which path the electron takes from the slit to the screen because it actually takes every possible path to get there.

The calculation mentioned is done by what is known as a vector sum. Each path gives a phase, which in our case is related to the length of the path. The phase then gives a direction in two dimensions. We start from some particular point and draw a short line segment or vector in the specificed direction. Wherever that vector ends, we draw another short vector in the direction corresponding to a different path. Once we have done this for every single path, we draw a vector from the starting point to wherever we end up. The longer the vector is, the more probable the process, and the direction of the final vector determines the phase for the process.

Keeping all of the above in mind, the following interactive Shockwave movie is a simulation of an electron traveling through a slit and hitting a screen, and the movie is set up to preform the vector calculation mentioned above. Note, however, that drastic simplifications have been assumed; the only allowed paths are those starting at the slit, going to some point on a line half way between the wall and the slit, and going from there to the screen. In fact there are an infinite number of paths, and the ones demonstrated below are only a very small sample of them. However, it turns out that the paths allowed below are sufficient to obtain the correct result, and they are also enough to demonstrate a few important concepts of quantum mechanics and Feynman's multiple paths interpretation.

First off, note that the paths you see are not paths for different electrons, but rather just one single electron is utilizing every single path you select. A final vector very close to the correct result can be obtained by selecting the 100 different paths obtained from the "100 Paths" button. Note, however, that by summing over certain selected paths, such as obtained from the "Left Paths" button, yields very different results. It is thus important that all paths are included in the sum as otherwise incorrect results will be obtained.

On the other hand, however, there are only a few paths of primary importance. The shortest distance between two points is a straight line, and classically a particle without an outside force acting on it will travel only in a straight line. The paths that are straight lines or very close to straight lines all have about the same length and thus all correspond to about the same direction. In fact, as you can see from the simulation, the paths that contribute the most to the final vector are those highlighted in black, which are the paths that come closest to the classical path in a straight line from the slit to the screen. Additionally, for more massive particles the width of the set of black paths will be thinner, and for objects as massive as say a penny, any primary contributing path is so close to the classical path that we can't tell the difference. Quantum mechanics then explains both behavior at the microscopic scale and events in our everyday lives as the limiting case for large, massive objects.

Instructions
Click anywhere on the display in the upperleft corner to add a path and its corresponding vector. Click the "100 Paths" button to add 100 different paths, selected regularly throughout the display. Click the "Left Paths" button to add only those paths whose vectors point to the left. The "Clear Paths" button clears the displays and removes all the selected paths. The length of the vectors corresponding the each path can be changed used the buttons at the bottom, by clicking once or by clicking and holding down.