As it passes through the magnets, the "up" component of the spin pulls the particle up, and the "down" component pulls it down. As a result, the particle becomes "fuzzy" and splits into a superposition of two wave-packets travelling in different directions. This is a bizarre state of the particle - it is simultaneously in two positions!

When this superpostion of wave-packets, on reaching the
other end, interacts with the detectors, the quantum coherence is
destroyed, and the particle is detected at one of the two positions. It
will be detected up with a probability |a|^{2}
, and down with a probablity |b|^{2}
. We are back to the familiar classical world.

This last stage where the superposition of the two wave-packets is destroyed, and one ends up with a particle either "here" or "there", is loosely referred to as the wave function collapse.

For those who are not shocked by the above experiment, look at the case of the Schrodinger's Cat!

This scenario seems to indicate that the state, or the wavefunction, is
not real - only if we possess sufficient knowledge about a system, we
*construct * a wave-function for it. The fact that the only
wave-functions we can measure are the ones already known to us, leads
one to conclude that the wave-function is an entity which is formed out
of our knowledge of the system - it doesn't have an independent
existence. To put it in the philosophers jargon, the status of
wave-function is "epistemological" (based on
knowledge), and not "ontological" (real).

We can divide the time T into N tiny intervals so that DelT=T/N. So the full wavefunction at the time T can be written as

Are there any problems?

- In the last equation we only look at the effect of the interaction
term on the wave-packet - we haven't looked at the effect of
p
^{2}/2m. This term will make the wave-packet spread with time. And if the time is very large, as is the case here, the wave-packet will shift by a tiny amount, but may spread so much that we do not know where its center is. - Although the center of the wave-packet gets shifted, in order that
an experimenter know how much is the shift, he or she would like to
know where the center is located. A single measurement on the
wave-packet can give the center to be anywhere within the spread of
the wave-packet, and not necessarily at the true center.

This problem will be unavoidable when one tries to practically
implement protective measurements. One might argue that the apparatus
has to be of a classical nature, but here during the adiabatic
interaction of the system and the apparatus the evolution of the
apparatus has to be unitary. This whole analysis hinges on that. So,
classical nature has to emerge later. It might emerge through
interaction with some kind of an environment, by the mechanism of
*decoherence*.

On the other hand one can treat the apparatus like a quantum system, as in the Stern-Gerlach experiment, and "detect" the apparatus using a classical apparatus. This will be a conventional measurement leading to a collapse of the apparatus wae-packet. In this scenario, we suggest two ways out of this problem.

Repeated measurement of a single state

Quantum Nondemolition Measurement of the Apparatus

There is another interesting way out of this problem. For this we make
use of certain special kind of measurements, called Quantum
Non-Demolition (QND) measurements. This scheme is based on repeated
weak quantum nondemolition measurements performed on the
*apparatus*. Recently Alter and
Yamamoto analyzed the problem of a series of repeated weak QND
measurement on a quantum system, to address the question of getting
information about the unknown wave function of a single quantum system
from such measurements. They concluded that it is possible to obtain
the mean value of an observable in an unknown state, but no information
can be obtained about the uncertainty of the observable. While the
conclusions of Alter and Yamamoto were negative as far as using
repeated weak QND measurements to determine the unknown wave function
of a single system, it appears tailor-made to solve the problem of
"reading the pointer position" in protective measurements. Thus we
apply their scheme not to the system part of the protective
measurement, but to the apparatus part instead. Then we can get
information about the center of the wave packet, which in the
protective measurement scheme carries information on the expectation
values of observables in the system state.

Thus one may proceed with a protective measurement by first
allowing an adiabatic interaction of the system with an apparatus
which
can be treated quantum mechanically. This would result in a shifted
wave packet of the pointer. One can then do a series of *weak* QND
measurements on this wave packet to get the position of the center.
This seems the most promising possibility for experimentally realizing
protective measurements.

Does it *really* work for single system?

In a realistic situation, the time of interaction can only be very large, but not infinite. As a result, there is always a very tiny entanglement left, which is of the order 1/T. This entanglement does change the wavefunction to a tiny extent. In the unlucky situation the detection of the apparatus wave-packet might lead to a collapse to this infinitesimal, but non-zero, branch of the entangled wave-function.

Thus, if you are just given one single system in an unknown energy eigenstate, you cannot be hundred percent sure that you have measured the right state.

For a detailed analysis of protective measurements, see ( N.D. Haridass and T. Qureshi, quant-ph/9805012).

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