Simple Nature - Light and Matter

(Martin Jones) #1
Linear algebra application
Time evolution is represented
by a linear operator (p. 965).
Unitarity is an additional re-
quirement for this linear oper-
ator.

The underlying reason for this result is that the Schrodinger equa- ̈
tion is dispersive: waves with different wavelengths travel at dif-
ferent speeds (because they correspond to different momenta).
Suppose a pulse has the shapef(x) att = 0. Since a pulse is
not a sine wave, it doesn’t have a single well-defined wavelength,
and therefore it doesn’t have a definite momentum or velocity. In
fact, the spread in momentum must be at least a certain size due
to the Heisenberg uncertainty principle∆p∆x &h. This causes
the pulse to spread out over time.
This leads to a strange thought experiment. Suppose that a
uranium atom in the Andromeda galaxy emits an alpha particle,
which then travels thousands of light years and eventually flies
past the earth. Its wave packet may initially have been as nar-
row as the diameter of an atomic nucleus,∼ 10 −^15 m, but by the
time it arrives perhaps it is the size of an aircraft carrier. Will an
observer see a gigantic alpha particle flying by? No, because ob-
serving it constitutes a measurement of its position, and by the
probability interpretation of the wavefunction this measurement
simply has a certain probability of giving a result that is anywhere
within some region the size of an aircraft carrier.

14.4.2 Unitarity
The Schr ̈odinger equation is completely deterministic, so that if
we know Ψ initially, we can always predict it in the future. We can
also “predict” backward in time, so that the system’s history can
always be recovered from knowledge of its present state. Thus there
is never any loss of information over time. Furthermore, it can be
shown that probability is always conserved, in the sense that if the
wavefunction is initially normalized, it will also be normalized at all
later times.


Unitary evolution of the wavefunction
The wavefunction evolves over time, according to the Schr ̈odinger
equation, in a deterministic andunitarymanner, meaning that prob-
ability is conserved and information is never lost.

(Unitarity is defined more rigorously on p. 984.)
Since we think of quantum mechanics as being all about ran-
domness, this determinism may seem surprising. But determinism
in the time-evolution of the wavefunction isn’t the same as deter-
minism in the results of experiments as perceived and recorded by
a human brain. Suppose that you prepare a uranium atom in its
ground state, then wait one half-life and observe whether or not
it has decayed, as in the thought experiment of Schr ̈odinger’s cat
(p. 885). There is no uncertainty or randomness about the wave-
function of the whole system (atom plus you) at the end. We know
for sure what it looks like. It consists of an equal superposition of

Section 14.4 The underlying structure of quantum mechanics, part 1 969
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