Undergrad Forms of the Uncertainty Principle

[Thejas15101998]

Well, i came across the so-called both the forms of the uncertainty principle of Quantum Mechanics: the position-momentum form and the energy-time form; but i am not satisfied in one way. Here the trio: position, momentum and energy, all of them have their own operators, but time does not have an operator.(Well, i think that its just a parameter unlike the other three) This asymmetry irritates me!

 

[PeterDonis]

i came across the so-called both the forms of the uncertainty principle of Quantum Mechanics

In what reference?

 

[Thejas15101998]

Modern physics book by Arthur Beiser. He refers to this while discussing vacuum fluctuation

 

[PeterDonis]

Modern physics book by Arthur Beiser.

This?

https://www.amazon.com/dp/B008UBHN38/?tag=pfamazon01-20

 

[PeterDonis]

He refers to this while discussing vacuum fluctuation

Please give a specific reference: chapter, section, page, equation, etc.

 

[Thejas15101998]

Soon after selection rules in chapter 6 there is an highlighted box with heading quantum electrodynamics

 

[PeterDonis]

Ok, Beiser's treatment in general is non-relativistic, and the mention of QED is heuristic at best, including the statement about the energy-time version of the uncertainty principle. The view given of vacuum fluctuations is at best outdated today (though it might not have been when Beiser's book was published) and at worst misleading, since in fact nothing is "fluctuating" at all--the state of the field + atom in spontaneous emission evolves smoothly and deterministically with time, and the state in the Casimir effect is, AFAIK, stationary, meaning it doesn't change with time at all (assuming that the plates are stationary).

As far as the energy-time version of the uncertainty principle, you are correct that time is not an operator, and what that means is that, unlike other formulations of the UP, such as the position-momentum formulation, there actually is no way to rigorously formulate an energy-time uncertainty principle mathematically in the way Beiser gives it. It's just a useful heuristic. However, in many cases you can do something which fulfills many of the purposes for which we would like to have an energy-time uncertainty relation, as John Baez describes here:

http://math.ucr.edu/home/baez/uncertainty.html

 

[dextercioby]

Not just any „time” can be defined as a selfadjoint operator, but only certain operators, such as the time of arrival. It is one of the major hoaxes of textbook physics that one cannot define a (precise form of) time operator, canonically conjugate to a Hamiltonian.

 

[PeterDonis]

It is one of the major hoaxes of textbook physics that one cannot define a (precise form of) time operator, canonically conjugate to a Hamiltonian

How does this square with the statement in the Baez article that the Stone-Von Neumann theorem prohibits having any operator that is canonically conjugate to the Hamiltonian if the Hamiltonian is bounded below?

 

[dextercioby]

No, there are limitations to the S-vN theorem. More precisely, we are led to believe that, just because it works with Q and P as a system of imprimitivities on the whole real line, then any Q and P under any assumptions will do the same. But the power of this theorem relies in the proper construction of the hypotheses which are almost never spelled out completely.
More precisely, if we consider a particle in a 1D box (the famous non-trivial problem with the rôle of the boundary conditions in determining whether the Hamiltonian and momentum operators are merely symmetric or may have self-adjoint extensions), then one can build a linear, self-adjoint operator (the time of arrival), canonically conjugate to the self-adjoint Hamiltonian operator with bounded and discrete spectrum. This operator is built in post# 8. The commutation relation, however, only holds in a domain not invariant under the action of the operators.
Here is where the mathematical finesse occurs: the S-vN theorem naturally leads to the Schwartz test function space in L^2 as a commutator domain. This has the nice property of being a common dense everywhere domain for the Q, P, Q^n and P^n. But, to be fair, the formulation of the S-vN theorem should be more „loose”, i.e. invariance of the commutator's domain be dropped, hence the construction of that time operator on a common not dense-everywhere domain becomes possible.

P.S. Why are we typically led to the deluding Schwartz space? Because of the heuristic approach to the CCR by Hermann Weyl which inspired JvN. The existence (in a strong topology) of infinite powers of Q and P by vN's work requires the Schwartz space. Therefore, von Neumann proved a stronger result, that in terms of operator series. But at the price of requiring a certain commutator domain. One of the assumptions of the famous Dixmier theorem is also the "stability of the commutator's domain". See below.

 

[PeterDonis]

we are led to believe that, just because it works with Q and P as a system of imprimitivities on the whole real line, then any Q and P under any assumptions will do the same

Ah, ok.