Yakir Aharonov and Daniel Rohrlich argue in their book Quantum Paradoxes:
that the energy-time uncertainty principle does not hold in all cases.
Aharonov and Rohrlich present their argument in chapters 7 and 8. They assume a time-dependent coupling constant g(t), which differs from zero between times t = 0 and t = T, and couples the system under measurement and the measuring device.
If one can measure something with an arbitrary precision, and have T arbitrarily small, then they say the measurement can be done impulsively.
An uncertainty principle for preparation of a wave packet
A measurement is related to preparing a quantum system to a certain state.
How quickly we can prepare a photon whose energy E is known with a great precision?
We need to create a wave packet whose spectrum in the Fourier decomposition is narrow and close to E. Such a wave packet necessarily is very long. How can we create such a wave packet?
If a hydrogen atom decays to a lower energy state, it sends a photon from a very small spatial volume, compared to the wavelength of the photon (λ is around 100 nm, while the diameter of the atom is only 0.1 nm). This shows that we can create a required wave packet in a small spatial volume if we have a long time available to create the packet.
On the other hand, we may imagine a very long device which creates a very long electromagnetic wave packet almost instantaneously.
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^
one finger disturbs Δt
v v v
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^ ^
multiple fingers disturb Δx
A classical analogue is a string. We can create a wave packet either by disturbing one location of the string for a long time, or by disturbing a long segment of the string for a very short time.
Energy-time-position uncertainty principle for a massless particle. For a prepared photon the following holds:
ΔE * max(Δt, Δx / c) >= h,
where ΔE is the uncertainty of the energy, Δt is the time used in preparation, and Δx is the size of the device used in preparation.
Question. Does the same relation hold for measuring the energy of a photon?
What about preparing a wave packet for a massive particle like the electron?
The position-momentum uncertainty relation for the wave packet of a massive particle is
Δx Δp >= h,
or
Δx Δ(m v) >= h
<=>
Δx / v * Δ(m v² / 2) >= h / 2
<=>
Δt ΔE >= h / 2,
where Δt is the time it takes the electron to move the approximate length of the wave packet. This is very much analogous with the classical string example. Either we need to spend the time Δt to disturb at one location, or we need to spend a very short time to disturb along a long segment Δx.
Let us ignore the factor 1/2 in h / 2 above. We are working with fuzzily defined uncertainties and can ignore factors that are of the order 1.
Energy-time-position uncertainty principle for a massive particle. The following holds:
ΔE * max(Δt, Δx / v) >= h,
where ΔE is the uncertainty in energy, Δt is the preparation time, and Δx is the length of the preparation spatial volume. The velocity of the prepared particle is v.
The relation is less strict than for a photon. If v is, for example, 0.01 c, we can make Δx very small.
A practical experiment to prepare electrons with precise energy in a very short time
What practical method can prepare an electron with a sharp momentum p in a very short time in a short distance?
source shutter shutter
--------------------------
e- ------> | electric field | ---->
-------------------------- p + Δp
Δx angle α
We have a source of electrons located far away, so that we know that the momentum to the y and z directions is almost zero. We want to select electrons whose momentum in the x direction is very accurately p with a precision Δp. That corresponds to some uncertainty ΔE in the energy.
The position-momentum uncertainty relation gives the length Δx of the wave packet of a selected electron:
Δx Δp >= h.
Let us assume that the incoming electrons have the momentum very roughly equal to p. Let us have a uniform electric field along a distance Δx. The field is only present for a very short time Δt.
We have shutters which only open the area of the electric field for electrons for some time ΔT. A shutter is a very high repulsive potential which we can create very quickly.
Then we can apply the electric field for a very short time Δt. All the electrons in the area will get the same impulse from the field. The shutter at the far end is opened and the electrons continue their journey. Electrons which are deflected to a certain angle α are the selected ones.
What should ΔT and Δx be? Almost all of the wave packet of an electron must fit in Δx. Otherwise the shutters would exert significant forces on the electron, changing its momentum significantly during the process.
The electric field changes quickly in the time Δt in a spatial volume of the length Δx. That will create, among others, real photons whose energy is up to
h c / Δx
and momentum up to h / Δx = Δp.
An electron departing to the angle α may have bumped into such a real photon, boosting the momentum of the electron. The boost might be up to 2 Δp, quite significant. However, we can reduce the number of such photons by making the electric field smaller. Thus, the photons do not pose a problem.
Our experiment satisfies the uncertainty relation
ΔE Δx / v >= h,
which can be calculated like in the previous section.
Using shutters really is cheating. They, too interact with the electron, and they are manipulated over a much longer time ΔT >> Δt.
What about removing the shutters completely? A small number of electrons will receive the impulse from the electric field partially, which will spoil our preparation to some extent. But the shutters themselves cause similar spoiling.
Let us remove the shutters. Then the interaction really is on for a very short time Δt.
What about preparing an electron in a very short distance Δx in a time Δt = h / ΔE?
Let us try to reuse the setup above. This time we let Δx be very small. But the geometry of the electric field is not suitable for our purposes. What to do?
We can create a photon with precise energy in a very small spatial volume if we can use a time Δt = h / ΔE. Then we can let the photon to free an electron from an atom with the photoelectric effect. An atom takes a very small spatial volume. In this indirect way we can prepare an electron in a small spatial volume and have the uncertainty of the energy very small.
Is there a direct way to prepare the electron in a very small spatial volume?
Conclusions
We believe that the energy-time uncertainty principle should be replaced with an energy-time-position uncertainty principle. This is also in line with special relativity where one cannot separate time and space.
In this blog post we studied preparation of a particle with sharply defined energy. Measuring the energy precisely is a related problem, but it should be studied separately.
We should also find out what is the role of "compensating forces" in the book of Aharonov and Rohrlich, in chapter 8. The authors assume an interaction which lasts for a very short time. But how large is the spatial volume where the interaction acts?
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