A. La Rosa Lecture Notes PSU-Physics PH 411/511 ECE 598

I N T R O D U C T I O N T O

Q U A N T U M M E C H A N I C S ________________________________________________________________________

How does light "really" decides which path to follow? An elementary Quantum Electrodynamics point of view

Reference:

Richard Feynman, “QED, The strange theory of light and matter,” Princeton University Press (1985).

Feynman Lectures, Vol-I, Chapter 26, "Optics: The Principle of Least Time." Section 26-6.

In this section we describe the connection between

i) the quantum mechanics concept of amplitude probabilities, and

ii) the classical variational principle.

Amplitude Probability

CASE: Tracing light from point source S to point P via reflection on a mirror

This time we consider that light does not seem to behave as a wave (in the sense described by the Huygens or the double slit experiments). Instead, light rays seem to be made out of photons (as revealed by the photoelectric effect experiment), which are detected in discrete quantities by a photon counter.

A new model to predict the way in which light propagates: Feynman presents a version of what is believed it really happens, from an elementary quantum dynamics point of view:

The arrival of a photon starting at S and ending at P is probabilistic.

That is, in the situation shown in Fig. 1 below, there are many paths possible for a photon leaving S to arrive P, via one reflection on the surface. One of those possible paths is shown in Fig. 1

Mirror

P

S

X

Source Detector

Fig. 1 Particular geometrical rectilinear path available for a photon to travel from S to P, via reflection on the mirror. X is an arbitrary point on the mirror.

The calculation of the probability is as follows.

All arbitrary paths “S mirror P” are possible.

(Figure shows only straight line paths, but, in general, any curved path can be considered.)

For a particular path SXP (where X is an arbitrary point at the mirror)

an AMPLITUDE PROBABILITY A(SXP) is assigned to it.

A(SXP) = ei (1)

where t(X)

is the angular frequency of the light

t(X) is the time taken by the light if it were to follow the particular

path SXP. The notation t(X) is a simplification for t(SXP)

is determined by the path traversed (we will assume it does not have a major effect in calculating A (SXP)

We will argue below that the phase plays a more preponderant role.

i is the complex number satisfying i2 = -1.

eiX

X A(SXP)

Real

Im

Phasor

Fig. 2 Amplitude probability is a complex number.

Rule: Since all the possible events are characterized by the same

initial state (photon leaving S), and

final state (photon arriving at P),

the total amplitude probability is obtained by adding all the individual amplitudes,

Total amplitude probability = X A(SXP)

(photon starts at S and arrives at P)

= X eiX (2)

The summation is over all the points X on the mirror

Expression (2) is then an addition of phasors whose phase (x) will be determined by the corresponding time of travel. A graphical construction of this total phasor is given in the figure below.

Mirror

P S

X

eiX

X A(SXP)

Fig. 3 A complex number is associated to each path AXP (corresponding to different X). The addition of all these complex numbers gives the total amplitude probability that a photon leaving from S will arrive at P.

The probability that a photon starting at S ends at P is given by the magnitude square of the phasor given in expression (2).

Pprob(SP) =

2

)( SXP X

A (3)

Amplitude Probability and the Variational Principle

How does this ultimate picture of photons and amplitude probability relate to the variational principle?

As an example let’s evaluate graphically the total phasor indicated by expression (2) but take just a few representative paths. First, from our experience of using the classical Fermat’s principle, we know that the time of travels varies like this,

t

A B C D E

P S

A B C D E F

P S

A B C D E F

For paths neighbor to SAP For paths neighbor to SEP the corresponding phasors (similarly) will change dramatically from one path to the next because the time of travel vary substantially even for neighbor paths

A Phasor for a path neighbor to SAP

E

Phasor for a path neighbor to SEP

For paths closer to the one that makes the optical path stationary, the time does not change drastically; so the corresponding phasors align each other.

P S

A B C D E F

C

C

A

P

Fig. 3 Connection between classical least time principle and QED.

The paths that are neighbor to the “classic” path are the ones that contribute the most to the (complex) amplitude probability. We see here, therefore, the connections between the quantum mechanics concept of amplitude probabilities and the classical variational principle.

Quantum Electrodynamics (QED)

Reference: The following description is taken from Feynman, “QED, The strange theory of light and matter,” Princeton University Press (1985).

Quantum mechanics was a tremendous success (could explain chemistry).

However, the description of light-matter interaction still faced difficulties. [Maxwell theory of electricity and magnetism had to be changed to be in accord with the new properties of quantum mechanics. The theory of light-matter interaction, called “quantum electrodynamics” was finally developed in 1929]. But the theory was troubled.

Right after Schrodinger, Dirac developed a relativistic theory of the electron that did not take into account the effects of the electron’s interaction with light. But it was expected to provide a good starting point.

Quantum electrodynamics was straightened out by Julian Schwinger, Sin-Itiro Tomonaga and Feynman in 1948. This is the theory that Feynman describes in his “QED, The strange theory of light and matter,” Princeton University Press (1985). Such theory has been tested over a wide range on conditions. Aside from gravitation and nuclear physics, QED can explain every phenomenon accurately.

QED is also the prototype for new theories that attempt to explain things going on inside the nuclei of atoms. It turns out, quarks, gluons, …, etc. all behave in a certain style, the “quantum” style.

QED analysis of light phenomena

Photons: Particles of Light

QED considers light to be made of particles (as Newton originally thought), but the price of this great advancement of science is a retreat by physics to the position of being able to calculate only the probabilities that a photon will hit a detector.

Event: Light travels from the source S, reflects from the surface at X, and arrives to the detector at D

We assign an amplitude probability (a complex number) that such an event will occur.

S

1 m

x

Consider 10 paths of

increasing length

n=1 P

X

Fig. Light reflected from a mirror

QED RULE-1 (Assignment of amplitude probability) How to do such an amplitude probability assignment? i) Pictorial view We assign to the amplitude probability an arrow.

To obtain the arrow we do the following: Let’s imagine that we have a stopwatch that can time a photon as it moves. It has a handle that moves rapidly. When the photon leaves the source, we start the watch. As long as the photon moves, the handle turns. When the photon arrives at the detector we stop the watch. The handle ends up pointing in a certain direction.

We then draw a corresponding arrow of magnitude 1 (blue arrow in the figure).

ii) More formal view Amplitude probability ≡ A(SXP) = eit

X) (1)

(photon starts at S and arrives at D via the path S X P)

where is the angular frequency of the light and

t(X) is the time the light takes to travel via SXP.

That is, a phasor eit

X) (a complex number) is assigned to the amplitude

probability A. QED RULE-2 (For events that have the same initial and final states) Since the photon has many optional paths available to go P from X, the total amplitude probability is given by,

Total amplitude probability A ≡ Xall

A(SXP) = X

eit

X) (2)

(photon starts at S and arrives at P via any path joining S and P)

where is the angular frequency of the light and t(X) is the time light takes to travel

from S to P passing through X (the latter located at the interface).

QED RULE-3 The probability for an event to occur is given by the square of the final amplitude: P = 2 A

(3)

Consider the reflection of light coming from a source S and reaching a detector via reflection from a mirror.

We want to calculate the chance that the detector will make (4) a “click” after a photon has been emitted by the source

Classical view: The mirror will reflect light where the angle of incidence is equal to the angle of reflection.

Classical way: In the graph above, it would appear that the ends of the mirror contribute to nothing to the reflection phenomena.

QED view: Every possible path contributes to the amplitude probability. Should the reflections path involving the center of the mirror have more weight than the once reflecting from the edges? Answer: No. All the path have equal chance.

QED view: QED view assigns an equal amplitude probability to each possible path

For the analysis of a mirror, an infinite number of path would have to be considered. To

simplify the problem, let’s divide the mirror into a number of smaller discrete strips.

S P

Each amplitude will be represented by an arrow of a standard (arbitrary) siz Consider 10 paths of

increasing length

While the size of the arrow will be essentially the same, its orientation will be different for the different reflection point selected. This is because it takes a different time for as photon to go through a different path that has different length.

Top: Figure shows each possible path the photon could take to go from the source to that point in the mirror and then to the detector. Middle: A plot of the corresponding time for each possible path. Below the graph is the direction of each amplitude probability (arrow). Bottom: The result of adding all the arrows. Notice the major contribution to the total arrow comes from arrows E through I, whose directions are nearly the same because the timing of their path is nearly the same. This also happens to be where the total time is least. It is therefore approximately right to say that light goes where the time is least.

H

K

E

B

F D

C

J

G A

I

L K

M H

What determines how long the final arrow is?

The end of the mirrors are not important (in this particular case being analyzed). In that region the little arrows wander around and, when added up, do not give any net change. (If we cut those ends of the mirror, it would not affect the length of the final arrow.

The main contribution to the final arrow comes from the middle , where the little arrows are all pointing in nearly the same direction, because their time is almost the same. It is when the time is least.

Where the time is least is also where the time for the nearby paths is nearly the same; hence their arrows have a similar orientation and add up to a substantial length. That is the region where the probability of a photon reflecting off a mirror is basically determined.

(It is easy to prove that when the time is least, the angle of reflection is equal to the angle of incidence. So QED gives the right answer.

“But the correct result came out at the expense of believing that light reflects all over the mirror, and having to add a bunch of little arrows whose sole purpose was to cancel out. All that might seem to you to be a waste of time-some silly game for mathematicians only. After all, it does not seem like “real physics” to have something there that only cancels out.” The ends of the mirrors may not appear to count in the example above; but the situation could be different in other cases. The next example will illustrate why the ends of the mirrors should count.

Why do the edges appear to make no contribution?

We zoom in to see in more detail the contribution to the total amplitude probability from the edges of the mirror. Notice when the arrows are added, they go in a circle, hence adding up nearly to nothing.

Diffraction grating The above gives us a clue to engineer a way to get contribution from the edges. What about if one scrape the mirror away in those areas whose arrows have a bias in one direction.

As we move from left to right, we notice the arrows have a bias orientation to the right, then to the left, and so on. If only the section with arrows biased to the right are kept (etching away the sections with arrows to the left), then a substantial amount of light will be reflected. Such a mirror with preferentially etched regions is called a diffraction grating.

M