🚀 go-pugleaf

In quantum mechanics, it seems literally to be the case that a watched
pot never boils. More specifically, if an observable A is measured
frequently enough, it is possible to prevent the value of A from
changing. In this posting, I will give a nonrigorous argument for
why this is true.

This effect, which is sometimes called the "quantum Zeno effect" is
derivable from elementary quantum mechanics. If a quantum system starts
in an eigenstate |a> of observable A with eigenvalue a, then its state
time t later will be given by (approximately, for small t)

    |Psi(t)> = |a> - it/h-bar H |a>

where H is the Hamiltonian. The probability at time t of measuring
observable A to have value b different from a is approximately

    P(b,t) = t^2 (h-bar|<b|H|a>|)^2 = C t^2
    (where C is independent of time)

If you observe eigenvalue a, then that means that the system is still
in state |a>, which means that the probability of observing b at time
2t, given that you observed a at time t is the same as P(b,t) above.
Therefore, the probability that you will observe A to have value b
different from a in any of N measurements spaced at intervals t is
given approximately by: (letting the product Nt be T)

    P(b,T) = N P(b,t) = Nt^2 C = T^2 C/N

In the limit as N -> infinity, keeping T fixed, we get P(b,T) = 0. In
other words, if you measure a value often enough, you can prevent it
from changing. An example where the quantum zeno effect applies is in
observation of alpha decay. As I understand it, "watching" a
radioactive atom closely can make the rate of alpha decay go to zero,
and this has actually been observed.

Now, this effect is easily explained using the collapse of the wave
function; every time you observe the state of the particle, you
collapse its state back into an eigenstate of the observable. However,
it seems that most physicists these days don't really believe in the
collapse of the wave function as a physical effect. Some view it as
simply a matter of a change of knowledge; that is, collapsing the wave
function simply represents a collapse in the set of possibilities, and
has no more physical meaning than the corresponding collapse that
occurs when you peak at a coin after you have flipped it. However, it
is hard to reconcile a change in knowledge with an actual physical
effect such as the slowing of the decay rates of radioactive
particles. Regardless of interpretation, it is a real, observable
effect.

Daryl McCullough
ORA Corp.
Ithaca, NY