Quantum Paths: Second Lesson

The first lesson demonstrated how summing over paths yields the probability and phase for a specific process. In this lesson we will examine how the probability and phase varies for the processes of an electron traveling through a single slit and ending at some particular point on the screen.

In this simulation 100 regularly spaced paths are automatically summed over to yield a single final vector for the process. The circle around the point where the vector starts represents the average length for a vector corresponding to one of these processes.

Note again that the paths drawn do not reperesent the paths of single electrons traveling through the slit but rather each electron goes over every single path when it goes from the slit to the screen.

When the calculation is done exactly using the appropriate mathematics, it turns out that each endpoint on the screen is equally probable. In other words, the vector corresponding to each process is of the same length. You'll notice as you play with the simulation however that the actual vectors obtain oscillate around the circle, sometimes shorter and sometimes longer than average. This is merely an artifact of the simplification assumed in designing the simulation. Only because we are summing over a small set of paths instead of all of them do we obtain an inaccurate result. However, both the length and the phase of the resulting vectors end up very close to the actual value, and the simulation is good enough for our purposes.

Although at this stage the phase of the processes is irrelevant, note how the direction of the final vector changes as the endpoint for the electron on the screen varies. Unlike the length, the phase is not a constant, and this will be important in the next lesson where we examine a double slit expirament.

Instructions
Click anywhere on the display in the upperleft corner to select an endpoint on the screen. Click the "Left" or "Right" buttons to have the point on the screen move in the corresponding direction, and press "Stop" to stop the movement. The "Highlight Main Paths" button toggles on and off whether the paths close to the classical path will be highlighted in black. Similarly, the "Show Path Vectors" button toggles on and off whether the individual path vectors will be displayed or just the final vector for the process.