Energy-time Uncertainty and mass est. of exchange particles

[yosofun]

I've been reading Giancoli (Physics for Scientists and
Engineers, 3rd Ed) and Griffiths (Intro to QM). There seems a
contradiction on the applications of the Energy-time uncertainty
principle between the two.

Griffiths claims that energy is always conserved, even though
mathematically, the energy-time uncertainty principle should allow for
energy to be non-conserved for a small period of time. Giancoli
calculates the mass of exchange particles (in the particle physics
chapter, penultimate chapter of the Modern Physics part) by using (or
abusing, as Griffiths would call it) the energy-time uncertainty
principle---that is, Giancoli assumes that energy is non-conserved in the
small time-interval of the lifetime of exchange particle. Who's right?

To quote Griffiths, "It is often said that the uncertainty principle
means that energy is not strictly conserved in QM--that you're allowed
to 'borrow' energy as long as you 'pay it back' in time [in a quantity
in accords to the uncertainty principle]... There are many legitimate
readings of the energy-time uncertainty principle, but this is not one
of them. Nowhere does QM license violation of energy conservation, and
certainly no such authorization entered in the derivation of
[Energy-time uncertainty principle]."

(Griffiths limits his book to nonrelativistic QM. And, the first part
of it covers time-independent SE. I am wondering if exchange particles
are beyond the scope of the first part and if his statement is limited
by the section, not to be taken as a general reference. Or, if the
calculation of the mass of exchange particles is subjective, as
whether one believes in the orthodox view or the realist view. Or, if
it's just a mistake on one of the author's part..)

 

[Jimmy Snyder]

Hi yosofun. I'm just posting to let you know that I saw your question here. I am repeating below pretty much the same things that I had written to you privately.

It is my understanding that the spontaneous creation of virtual pairs of particles and the subsequent destruction of those particles within a short period of time is permitted by the Uncertainty Principle. This would seem to be a temporary violation of the conservation of energy.

But ANY uncertainty in the measurement of energy seems to mean that energy is not strictly conserved. Professor Liboff claims that energy is conserved only in the average, <E>.

 

[Physics Monkey]

Part of the confusion arises because of a technical point. In interacting quantum field theories, the number operator (for any particle species you like) and the Hamiltonian don't commute. So technically, if I have a definite number of exchange particles, say one in your case, then the system doesn't have a definite energy. Number eigenstates aren't energy eigenstates.

 

[Kruger]

I would interprete the uncertainty principle as following:
- The uncertainty principle gives the time we need to observe the amount of energy d(E).

 

[inha]

I would interprete the uncertainty principle as following:
- The uncertainty principle gives the time we need to observe the amount of energy d(E).

I don't think that makes any sense at all. Are you basing this on something or just speculating out loud?

 

[yosofun]

explain in more detail please...

Part of the confusion arises because of a technical point. In interacting quantum field theories, the number operator (for any particle species you like) and the Hamiltonian don't commute. So technically, if I have a definite number of exchange particles, say one in your case, then the system doesn't have a definite energy. Number eigenstates aren't energy eigenstates.

 

[yosofun]

if one assumes that energy is conserved only "on the average," then it would be possible (for you) to walk through a solid concrete wall -- extremely unlikely, but still possible.

this really worries me.

Hi yosofun. I'm just posting to let you know that I saw your question here. I am repeating below pretty much the same things that I had written to you privately.
It is my understanding that the spontaneous creation of virtual pairs of particles and the subsequent destruction of those particles within a short period of time is permitted by the Uncertainty Principle. This would seem to be a temporary violation of the conservation of energy.
But ANY uncertainty in the measurement of energy seems to mean that energy is not strictly conserved. Professor Liboff claims that energy is conserved only in the average, <E>.

 

[vanesch]

I don't think that makes any sense at all. Are you basing this on something or just speculating out loud?

It is a correct interpretation of the T-E "uncertainty" relation but probably a bit confusingly formulated, as follows:

If you have a state which is a superposition of energy eigenstates (stationary states) with all their energies within dE, then the squares of amplitudes of the decomposition of this state in just any measurement basis you want (say, position) will only significantly change over a time lapse greater than dt, with dt and dE respecting the E-T UP.
Or, in another way, if you want to find out what energy eigenstate a system is in, with a precision dE, you will need to perform a measurement which lasts for at least a time dt.

 

[vanesch]

It is my understanding that the spontaneous creation of virtual pairs of particles and the subsequent destruction of those particles within a short period of time is permitted by the Uncertainty Principle. This would seem to be a temporary violation of the conservation of energy.

I know that this is often said, but it is in fact technically not true. Energy (and momentum) are always strictly conserved. Technically, in QFT, this is manifest in the fact that each vertex contains a deltafunction over the sum of energies and momenta going in and out of the vertex.
What is happening with virtual particles is that they do not have the "correct mass", and can even have imaginary mass in order to respect conservation of energy and momentum.
What can also happen, is that your initial state is a SUPERPOSITION of different energy states, and as such the energy of your system is not well defined. For instance, in NR QM, the state "a particle at a certain location" is a superposition of several energy eigenstates, and as such, can allow you to observe different values for energy when you measure it. But that doesn't mean that energy wasn't conserved.

 

[jiahuili]

When we talk about the question "Where is the particle before we measure it ?",the orthodox view says : It is nowhere. Is it possible for us to say that the particle does not have a certain enerfy before we measure it ? What we know is only the probability to get a certain eigenvalue