Yakir Aharonov and Daniel Rohrlich have written a very interesting book Quantum Paradoxes (Wiley, 2005):
One of the paradoxes is Albert Einstein's famous 1930 clock-in-the-box thought experiment. There are photons in the box, as well as a clock. A mechanism opens a shutter for a very short time Δt when the clock says it is noon. Some photon(s) escapes, and the total energy of the box is reduced by some amount E.
In the morning we can use a very long time to weigh the box, so that we know its total energy extremely precisely. In the afternoon we can repeat the weighing procedure.
We can determine E as accurately as we like. We also know the approximate time t₀ = noon when the box lost this energy E. Does this contradict the uncertainty principle
ΔE Δt >= h / (4 π)
?
Niels Bohr came up with a sort of a counter-argument. He uses a complicated procedure to weigh the box. There is a spring scale and known counterweights. After we have hung the smallest counterweight, there is still an uncertainty Δx in the vertical position of the box.
General relativity tells us that the clock runs at different speeds at different heights. Bohr calculated the uncertainty ΔT in the current time (proper time) shown by the clock. (Note that this is not the same as Δt.) We assume that we do not read the clock in the box, but try to determine the time from the laboratory clock. He proved that the uncertainty relation above holds for that particular uncertainty ΔT.
Bohr's argument did not prove much
However, Bohr uses a very slow procedure to weigh the box. The uncertainty principle allows us to perform the weighing much faster, in a duration t = h / (ΔE * 4 π). If we use a faster procedure, the uncertainty ΔT in the reading of the clock inside the box is probably much less than in Bohr's procedure.
Let us then analyze how one could entirely circumvent the uncertainty in the box clock reading. It is quite easy. Instead of the box energy, let us measure the energy of the photons which left the box through the shutter. We can use an arbitrarily long time for measurement, so that ΔE is extremely small.
We can fix the box statically to the laboratory frame. Then the clock inside the box runs at the same speed as the laboratory clock. The uncertainty ΔT in the reading of the clock in the box is very small. We found ΔE and ΔT which do not satisfy the uncertainty relation.
If one tries to squeeze energy E into a wave packet to whose duration is roughly t, then
ΔE t >= h / (4 π).
This is a typical example of an energy-time uncertainty principle. However, Einstein's clock-in-the-box is about the history of events and uncertainty of the time when an event happened. Why would there be an uncertainty relation about history? We do not see why that should be necessary.
Bohr's argument did not prove the energy-time uncertainty principle, but something about uncertainties in a certain complicated physical experiment.
Analysis in literature
Let us look at the literature. Does anyone claim that Bohr proved something about the uncertainty principle?
H.-J. Treder (1975) writes that the box argument has no bearing on the fourth Heisenberg relation. We agree.
Hrvoje Nikolić (2012) has written about EPR before EPR: a 1930 Einstein-Bohr thought experiment revisited.
Nikolić says that neither Einstein nor Bohr was right. In section II A Nikolić writes that Einstein wanted to show that
ΔE Δt >= h / (2 π)
does not hold.
We in this blog think that Einstein was right. We can produce a photon in a short interval of time Δt, and later measure the energy of the photon extremely precisely (very small ΔE), using a long time t for the measurement. The correct energy-uncertainty principle says that
ΔE t >= h / (4 π).
In section III D Nikolić writes that Einstein did not realize that measuring the mass-energy of the box can influence "ΔE (or any other property) of the photon".
We do not understand the claim. It was well known in 1930 that a photon is a quantum of light of a definite frequency f and definite energy E = h f. We can reduce the uncertainty in E by measuring the energy of the box or of the photon itself. Conservation of energy was taken for granted in 1930, as it is today.
After the shutter is opened and closed, then the box and the photon(s) are, of course, entangled. Measuring the energy of the box makes the wave function to "collapse" to certain energy E of the photon(s). However, talking about a "collapse" does not affect the analysis of the process in any way.
Conclusions
Albert Einstein's thought experiment about the clock-in-the-box does not concern the energy-time uncertainty principle at all. Neither does Niels Bohr's counter-argument.
It is wrong to present the debate as a "proof" that the uncertainty principle holds.
No comments:
Post a Comment