a/The Lissajous figurex= cost,
y= sin 2t.semiclassical argument based on the energy-time uncertainty re-
lation.
The typical rate of emission for a photon, when not forbidden by
parity, isR∼ 109 s−^1 , i.e., it takes about a nanosecond. We can
think of the two-photon decay as an energy-nonconserving jump
up to somehigher-energy state, with the emission of a photon,
followed by the emission of a second photon leading down to the
ground state. The first jump can happen because of the energy-
time uncertainty relation, which allows the electron to stay in the
intermediate state for a timet ∼h/E, which is on the order of
10 −^15 s. The probability for the second photon to be emitted
within this time isRt, so the rate for the whole two-photon pro-
cess isR^2 t ∼10 s−^1. Considering the extremely crude nature
of this calculation, the result is in good agreement with the ob-
served rate of about 0.1 s−^1. The process is actually observed,
and contributes a continuous background spectrum in addition to
the discrete line spectrum when such nebulae are observed with
a spectrometer through a telescope.
A fundamental application of the energy-time uncertainty re-
lation is to the explication of what it means to measure time in
quantum mechanics. In example 9 on p. 979 we argued that time
is not an observable in quantum mechanics because time cannot in
general be measured by looking at a quantum-mechanical system:
many quantum-mechanical systems are too simple to function as
clocks. We can now see in more detail what “too simple” might
mean here. Microscopic systems, unlike macroscopic ones, are of-
ten encountered in a definite state of energy, such as the ground
state. Such a state has ∆E= 0 and therefore by the energy-time
uncertainty relation it has ∆t=∞. In other words, the only time
evolution in such a system consists of the system’s over-all phase
twirling in the complex plane at a steady rate, but phase isn’t mea-
surable, so we can’t use this rotation like the hand on a clock. To
make a clock, we need, at a bare minimum, a system that is in a
superposition oftwodifferent energy levels. We then have two in-
dependent phases. Although absolute phases are not measurable,
relative ones are, and for example when we measure a double-slit
interference pattern, that is exactly what we are doing: observing
(statistically) the difference between two phases. As a loose concep-
tual analogy, this is like the idea that a figure-eight Lissajous pattern
has an identifiable feature where it crosses itself, the crossing being
like the tick of a clock.
14.9 Randomization of phase
Section 14.9 Randomization of phase 997