232
Chapter 3. Wave optics
This represents a statement of Heisenberg’s uncertainty principle.
The assignments of Δx and Δp are somewhat arbitrary. For example, one might assign the full-widths, instead of the half-widths
for Δx and Δp, in which case the right-hand side in (3.340) would
be multiplied by a factor of four. The uncertainty relation (3.341)
is therefore properly regarded as an approximation.
The relation (3.339) applies to a photon, and equivalently to a
massive particle. It follows that the uncertainty relation (3.341)
applies to both a photon and a massive particle. Alternatively, the
momentum can be expressed as
p = ¯
hk
(3.342)
for a photon and massive particle, where k is the wave number
given as k = 2π/λ. The uncertainty relation (3.341) becomes
π
Δx Δk ≈ ,
(3.343)
2
given the above assumptions. This can be regarded as a general
property of waves, not just a photon or particle. It applies to many
wavelike phenomena, including water waves and sound waves, as
examples. The usual coherent superposition of waves with different values of k to form a wave packet, applies to these different
types of waves as well.