Page 165 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electromagnetic System
4.2.3 Light as Particles: Photons
Convenient as wave packets are, they do not describe photons yet, and
what is missing of course is a quantum-mechanical approach to light.
Since photons act as located particles, for example in the photoelectric
effect, they are not representable by a space-filling wave packet, yet they
are observed to interfere as waves.

Discussion As we have shown in more detail in Chapter 3, the quantum-mechanical
solution to this apparent contradiction is to make the wave packet repre-
sent the probability density of the particle. Before we look at the details,
it is instructive to see what is resolved by this new representation. Since
the probability is wave-like, it obeys wave mechanics, and hence can pro-
duce wave–like interference with the probability waves of other photons.
If a photon is detected (it is “observed”), it is with a process that requires
the photon to be particle-like. At this instant of detection, the resultant
probability of the photon is evaluated by the detector, and the photon
reveals its position. On the way to the detector, the photon does not reveal
its position. Indeed, there is no way to tell how it got from A to B apart
from disturbing it on its way. What quantum mechanics does elegantly is
to let the photon’s probability “propagate” with the speed of light, and
interact with equipment in a determined way. Each measurement
becomes an evaluation of the evolved probability distribution of the pho-
ton.

Probability To describe a photon that is consistent with measurements and observa-
Density tions, we use a wave packet with a Gaussian spectral function and a total
energy of 2πhω :

1 ( k – k ) 2
0
fk() = -----------------exp – --------------------- (4.54)
2πσ 2σ 2
k k
The probability density of the photon is then proportional to fk() 2 ,
with the proportionality constant chosen so that the probability of finding

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