Kea recently brought up the subject of Zeno of Elea and his now long lost book of 40 paradoxes dealing with the continuum. His nominal 2500th birthday should be celebrated relatively soon. Let me paraphrase an example paradox is the following:

If one assumes that space and time are continuous, then an arrow shot from a bow, before reaching its target, must first travel half the distance. And then travel half the remaining distance. And so on. And therefore, there are an infinite number of distances to be travelled and the arrow could never reach the target. But arrows do reach targets. Therefore, space and time are not continuous.

Surprisingly, there is an echo of this thought in quantum mecahnics. The echo is so close to the original paradox that it is known as the Quantum Zeno’s Effect or sometimes “Paradox” depending on the writer. The subject is discussed in many arXiv articles.

In quantum mechanics, when one measures a system, the formalism requires that the system collapse to the result of the measurement. If one examines this carefully, one finds that if one measures a system at a sufficiently high rate, the effect of the repeated measurements is to prevent the quantum system from changing. In effect, if one examines the position of the arrow too frequently, the arrow cannot move. It’s worthwhile looking at the simple mathematics that causes this effect.



In the usual quantum mechanics, one represents a quantum system by a wave function or state vector. One obtains probabilities from this object by squaring its absolute magnitude. Given the state of the system at time t, a linear differential equation defines its state at later times. For instance, with Schroedinger’s wave equation, one has:



Suppose we have a very short time period . How much will the wave function change? As for any differential equation, we have:



Now suppose that we are looking at a quantum state that decays. For example, a uranium atom. At time t=0, we measure the system and determine that it has not decayed. According to the laws of quantum mechanics, this puts the uranium atom into a pure “not decayed” state. If this seems a funny way of talking about a thing, some physicists think so too.l You might try reading about Schroedinger’s Cat which will likely just confuse you further but may be more entertaining than one of those confusing foreign movies.

According to quantum mechanics, it is possible to define a complete set of basis states that fully define the wave functions that one could get if one made a measurement on the system. Now one of those basis states is the undecayed state and the rest of them consist of the various decayed states. Now at time t=0, we’re supposing the state to be the undecayed state (because we measured or checked it, and that is what it was). After a short time, the state vector moves to be a mixture of this undecayed state and a decayed state. But since the motion is linear (i.e. Schroedinger’s wave equation is linear), the amount of the decayed state that we have entered must be linear for small time.

To compute the probability of decay, we take the state we wish to measure and compute the dot product of it with the possible state we might get for a measurement. This is the “amplitude” and is a complex number. Again this amplitude must be linear in . To get the probability we compute the squared magnitude of the amplitude. Therefore, the probability of decaying has to be approximately proportional to the square of .

Exponential Decay

Now this is a very odd thing. When you have a radioactive atom, we assume that the rate at which it decays does not depend on time and so it should follow an exponential distribution. In exponential decay, the probability of a decay over a very short time interval is proportional to . It is not proportional to . This is what we intuitively expect, but it is not what happens.

Since the square of small numbers gives even smaller numbers, the effect of this nonlinear decay rate is that decay is suppressed near a quantum measurement. If we measure the system again, then again the decay rate will be suppressed. And if we repeatedly do this, then the decay rate will be reduced accordingly. Furthermore, in the limit as the time delay between measurements goes to zero, decay is completely suppressed; its probability goes to zero. This is the Quantum Zeno Effect.

As one can learn by reading the literature, one can show that the exponential distribution does obtain at longer times. Even more interestingly, the deviation from exponential decay rates reappears for extremely long times as well as short times, and it is possible for the decay rate to be increased (the “Quantum Anti-Zeno Effect”), for reference, see papers like quant-ph/9708024.

Density Matrices

From the density matrix point of view, part of the reason we see a paradox in the QZE is due to our natural inclination to see a quantum state as it were a little photograph of the condition of the system at a given time. We see a sequence of states as if it were a movie showing how the system changes with time. Similarly, when we split a system into things being acted upon and things that are acting, of course it becomes confusing.

Consequently, adding a measurement to the system, for example by using a laser pulse, is a modification of the process. Leaving the laser pulse out of the equations (and thinking of them as a “measurement” that uses no matter or energy) is unphysical and it is not surprising that the dynamics are unexpected. In fact, one can derive the decay rate change of the QZE without any need for a collapse hypothesis by using density matrix theory that includes the laser pulse used to make the measurement. This is well known in the literature and will show up if you search for “density matrix” along with “quantum zeno”. For example, see quant-ph/9611020:

The QZE and this experiment have not only aroused considerable interest in the literature [8, 9], but the very relevance of the above experimental results for the QZE has given rise to controversies. In particular the projection postulate and its applicability in this experiment have been cast into doubt, and it was pointed out that the experiment could be understood without recourse to the QZE by simply including the probe laser in the dynamics, e.g. in the Bloch equations or in the Hamiltonian [9]. Since the Bloch equations describe the density matrix of the complete ensemble, including the probe pulse as an interaction in them gives, however, no direct insight on how such a pulse acts on a single system.