Modern quantum mechanics was originally put into two quite different, but nevertheless equivalent, mathematical forms. In 1925 Werner Heisenberg developed a formulation based on matrix algebra. Just about one year later, in 1926, Erwin Schroedinger discovered the wave equation, which is now called the Schroedinger equation. This equation became the cornerstone of the wave function formulation. Today this formulation represents the traditional approach to teaching introductory quantum mechanics.
In 1941 Richard Feynman developed a formulation of quantum mechanics which is mathematically equivalent with the previous ones. The formulation is called the amplitude formulation and it is also known as “the sum-over all paths theory” or “the many paths approach”.
For this extra credit assignment read Chapter 2 of "QED:
The Strange Theory of Light and Matter" by
Richard P. Feynman (Princeton University Press, 1985). You can listen to
Richard Feynman delivering the actual lectures at
http://www.vega.org.uk/video/subseries/8.
You will find a link to chapter 2 under course materials on Blackboard.
As you read, answer the following questions.
Instead of solving a wave equation to be able to make
predictions, the fundamental principle of Feynman’s formulation is to
explore all paths. To predict if a photon will be detected by a given
detector, we let the photon explores each path between source and detector.
Geometrical optics is based on the laws of reflection and refraction. In
1650,
Fermat discovered a way to explain
reflection and refraction as the consequence of one single principle. It is
called the principle of least time or
Fermat's principle.
Assume we want light to get from point A to point B, subject to some
boundary condition. For example, we want the light to bounce off a mirror
or to pass through a piece of glass on its way from A to B. Fermat's
principle states that of all the possible paths the light might take, that
satisfy those boundary conditions, light takes the path which
requires the shortest time.
How does QED derive this principle of least time?
How can we convert a mirror into a diffraction grating?
Consider figure 33, page 55. How can we arrange it that light from the source goes through the gap between the blocks and reaches the detector at Q?
How do we construct a focusing lens?
How must we combine arrows for a compound event, for example the event described in figure 37, page 60?
What do we have to do to the arrows if there are two different ways to produce the same outcome, or the same final state? Consider for example figures 47 and 48 on pages 73 and 74. The final state is: “Detectors A and B each have detected one photon.” What are the two different ways this outcome could have been produced? How do we produce the final arrow for this event?
Use Microsoft Word to prepare a report that contains your answers to these questions.
For extra credit (up to 5 points) hand in or email your report before the extra credit 2 due date.