B Can we measure any two of the energy, momentum or mass simultaneously?

[SEYED2001]

TL;DR Summary

I wonder between which two quantities of this set: energy, momentum, and mass, there is an uncertainty principle going on. Can I measure any two of those at the same time?

I want to know between which two quantities of energy, momentum, and mass there is an uncertainty principle going on. Can I measure any two of those at the same time? If yes, which ones?

Thank you in advance

 

[Dale]

All of them. There is no uncertainty relationship between them.

 

[SEYED2001]

Thank you very much for your fast reply.

 

[SEYED2001]

We can measure energy, mass and momentum of an electron, all at the same time. Now, does the relativistic energy equation work in order to relate them? So, if I measure mass and energy, can I calculate what the momentum would be if I was to measure it?

 

[Dale]

Yes.

 

[PeroK]

We can measure energy, mass and momentum of an electron, all at the same time. Now, does the relativistic energy equation work in order to relate them? So, if I measure mass and energy, can I calculate what the momentum would be if I was to measure it?

QM is a non-relativistic theory. Technically, therefore, the energy and momentum of a particle are related by the classical equation $E = \frac {p^2} {2m}$.

 

[SEYED2001]

Thank you for your reply. So, if i measure E, p and m, these values theoretically satisfy the classical equation, and not the relativistic one. Now, what if I don't do any measurement. If I don't, there wouldn't be any definite value for any of the above-mentioned quantities. However, those quantities do have a distribution of values. In other words, I measure the energy of the electron and I get a number. If I do it again, with an interval between the two measurements, I might not get the same number in the second measurement, and the same goes for the fourth, fifth, etc. Now, I make a distribution of these values and calculate some statistical parameters, such as expected value, mode,etc. for each of energy, momentum and mass distribution. Is the classical and/or the relativistic energy equation practical for relating these parameters of these distributions to each other? For example, can I put the mean for energy and momentum distributions in the classical equation and get the mean for the mass distribution? If no, what about the mode of these distributions, or any other well-defined statistical parameter? Can we ever relate the distribution of energy, momentum and mass to each other?

 

[PeroK]

Thank you for your reply. So, if i measure E, p and m, these values theoretically satisfy the classical equation, and not the relativistic one. Now, what if I don't do any measurement. If I don't, there wouldn't be any definite value for any of the above-mentioned quantities. However, those quantities do have a distribution of values. In other words, I measure the energy of the electron and I get a number. If I do it again, with an interval between the two measurements, I might not get the same number in the second measurement, and the same goes for the fourth, fifth, etc. Now, I make a distribution of these values and calculate some statistical parameters, such as expected value, mode,etc. for each of energy, momentum and mass distribution. Is the classical and/or the relativistic energy equation practical for relating these parameters of these distributions to each other? For example, can I put the mean for energy and momentum distributions in the classical equation and get the mean for the mass distribution? If no, what about the mode of these distributions, or any other well-defined statistical parameter? Can we ever relate the distribution of energy, momentum and mass to each other?

How are you learning QM? Do you have a textbook?

First, the mass of an electron is a fixed quantity, like its charge. Mass is not a dynamic quantity that changes from measurement to measurement. For a free particle, energy and momentum are related. If you measure the KE, you have measured the magnitude of the momentum. And, if you measure the momentum, you have measured the energy.

Where a potential is involved, the total energy depends also on position. Have you studied something like the infinite square well or the harmonic oscillator?

And, have you seen Ehrenfest's theorem? This essentially says that the expected values of quantum obsevables obey classical laws. Such as:
$$m\frac{d}{dt}\langle x \rangle(t) = \langle p \rangle(t)$$

 

[vanhees71]

This is of course nonsense! All experiments in high-energy-particle physics clearly show that the relativistic energy-momentum relation is the correct one. The Standard Model describes very well all experimental facts. For free particles, energy and momentum are compatible observables, and mass is thus compatible too, because it's a function of these quantities.

There's a subtlety in the Ehrenfest theorem in non-relativistic QM too, and the expectation values satisfy not the same equations as the corresponding classical system (except for free particles, a particle under influence of a constant force or in a harmonic-oscillator potential), because
$$\mathrm{d}_t \langle \vec{p} \rangle = -\left \langle \partial_{\vec{x}} V(\vec{x}) \right \rangle \neq -\partial_{\langle \vec{x} \rangle} V(\langle{\vec{x}} \rangle).$$

 

[SEYED2001]

How are you learning QM? Do you have a textbook?

First, the mass of an electron is a fixed quantity, like its charge. Mass is not a dynamic quantity that changes from measurement to measurement. For a free particle, energy and momentum are related. If you measure the KE, you have measured the magnitude of the momentum. And, if you measure the momentum, you have measured the energy.

Where a potential is involved, the total energy depends also on position. Have you studied something like the infinite square well or the harmonic oscillator?

And, have you seen Ehrenfest's theorem? This essentially says that the expected values of quantum obsevables obey classical laws. Such as:
$$m\frac{d}{dt}\langle x \rangle(t) = \langle p \rangle(t)$$

Thank you all for your help. I got my answer :)

 

[SEYED2001]

How are you learning QM? Do you have a textbook?

First, the mass of an electron is a fixed quantity, like its charge. Mass is not a dynamic quantity that changes from measurement to measurement. For a free particle, energy and momentum are related. If you measure the KE, you have measured the magnitude of the momentum. And, if you measure the momentum, you have measured the energy.

Where a potential is involved, the total energy depends also on position. Have you studied something like the infinite square well or the harmonic oscillator?

And, have you seen Ehrenfest's theorem? This essentially says that the expected values of quantum obsevables obey classical laws. Such as:
$$m\frac{d}{dt}\langle x \rangle(t) = \langle p \rangle(t)$$

In fact I don't know which textbook out there works best for me. I will post a question here at PF to hopefully find an answer. Thank you!