Energy-time uncertainty relation

[leblon]

It is well-known that the energy-time uncertainty relation has a different status than the usual momentum-coordinate uncertainty relation. Heisenberg introduced both uncertainty relations, but while the p-x relation can be formulated and proved rigorously, the energy-time relation is more subtle, and in fact some often-mentioned formulations are wrong. The problem is that in QM time is not a dynamical variable, but a parameter. So the accuracy of measuring the time coordinate can be arbitrarily good, regardless of what we know about the energy of the system.

Consider some popular formulations of the energy-time relation.

(1) When measuring an energy of a system, the accuracy of the measurement cannot exceed h/t, where t is the duration of the measurement.

(2) When preparing a system in a particular state, the uncertainty of the energy of this state will be at least h/t, where t is the preparation time, and h is the Planck constant.

These two formulations are essentially equivalent, since measuring the energy of the system is the same as preparing a state where the energy has a definite value. I think Landau-Lifshits textbook states (1) as a viable formulation of the energy-time uncertainty relation. But as shown by Aharonov and Bohm, (1) (and therefore (2)) are incorrect. It is possible to set-up a non-demolition measurement of energy which takes an arbitrarily short time and has an arbitrarily good accuracy.

(3) If some property of a system changes substantially on a time scale t, then the energy of the state has uncertainty at least h/t.

A version of this was stated by Bohr and Wigner. This is the formulation which "explains" why an unstable particle (resonance) does not have a definite energy. It is a bit hard to make this principle precise, and in fact there are many slightly different formulations. But it can be proved rigorously.

(4) If an internal (dynamical) clock of a system has accuracy t, then the energy of the system is uncertain, with uncertainty being at least h/t.

This is more or less equivalent to (3).

There is a well known story (told, for example, in R. Peierls's wonderful book "Surprises in theoretical physics") about Einstein inventing a counter-example to (1), and Bohr refuting him using Einstein's own General Relativity Theory. In retrospect, Bohr's refutal, while correct, seems beside the point, since (1) is not true in general.

Comments

[chaource]

Did the Aharonov-Bohm statement take into account relativity? I'm not familiar with that, but what I've seen of Aharonov's work was always strictly non-relativistic QM. It sounds a bit suspicious that you can make a measurement of something within an arbitrarily short time - you will probably need an arbitrarily small device for that.

[alexanderr]

but on the other hand, a relationship of this kind exists even in classical, non-quantum situations. if energy is just hbar times omega then hbar cancels and we have a standard relationship between time and frequency. intuitively it is very clear that frequency is not well defined when observation/measurement time is short. so in a sense energy time uncertainty relation follows directly from Planck's postulate and Fourier transform. it could have been derived in 1900

[leblon]

It is non-relativistic QM. A short exposition of the Aharonov-Bohm counter-example can be found in https://arxiv.org/abs/quant-ph/0105049 , section 3.3.1.

[leblon]

This "classical" uncertainty relation is almost a tautology. The nontrivial thing about QM is that it relates this tautological statement to something intuitively non-obvious.

[alexanderr]

exactly, that was my point. we take a tautological (well-known to engineers) statement, multiply both sides by hbar and suddenly it becomes interesting? I'm not going to argue, but if something is a wave, and, as de Broglie told us, everything is wave and we measure it for a short period of time then, obviously, its frequency is not going to be a well defined number. I find other things about QM much more non-obvious, like spin 1/2, things that have no meaning in the classical limit. but this one is not only "classical", but as you pointed out, tautological

Aharonov-Bohm effect obtained from relativity.

[anton_lipovka]

Yes, it did. Strictly speaking any quantum mechanical problem has its source in the relativity. This statement follows directly from the geometrical nature of the Planck constant. You can find geometrical explanation of the A-B effect, geometrical nature (i.e. it was obtained from relativity) of the Planck constant and complete theory that unifies the relativity with the Quantum physics by this link:
https://anton-lipovka.dreamwidth.org/38896.html
Your comments and questions are welcome.

[anton_lipovka]

It is difficult to agree with this statement because 1) there are no exists correct explanation of the A-B effect within the framework of the non-relativistic QM. 2) Moreover even the so called “non-relativistic QM” appears actually from relativistic action (because the Planck constant is completely defined by the geometry of the Universe), and in the Minkowski world (let alone Euclidean one) there are no exist bounded quantum systems because the Planck constant is equal to zero there. See the link:

https://anton-lipovka.dreamwidth.org/38896.html