B Examples of less popular pairs of physical properties with uncertainty (HUP)

[syfry]

TL;DR Summary

What are some examples of the less discussed pairs of physical properties that we cannot know with mutual precision because of the Heisenberg uncertainty principle?

Position and momentum are the popular pairs of properties with uncertainty we often hear about, for example that we cannot know with precision where an electron is and its momentum at the same time.

What are others?

Such as an example of an energy and a time that we cannot know both precisely.

Also, an example of angular momentum in two perpendicular directions that both aren't knowable with high precision.

And any more example pairs that you know of.

 

[hilbert2]

The commutator of $\hat{p}$ with any function of position, $f(\hat{x})$, is

$\displaystyle\left[f(\hat{x}_i),\hat{p}_i \right] = i\hbar\frac{\partial f(\hat{x}_i )}{\partial\hat{x}_i}$.

This is nonzero for an arbitrary function $f$, except if it is a contant that doesn't even depend on $\hat{x}$.

Because all observable physical quantities in basic quantum mechanics, except spin, can be written as functions of the position and momentum components of the particles in the system, this kind of relations are enough for knowing all commutators between them. Time is not an observable in QM (even though some obscure articles have probably tried to describe it as one), so the time-energy uncertainty is not a same type of property as position-momentum uncertainty.

I don't know that much about particle physics, but I guess there have been attempts to define spin-like variables such as "isospin" that tell what type an elementary particle is, so there's another one that can't be described simply with $\hat{x}$-$\hat{p}$ commutators.

 

[syfry]

The commutator of $\hat{p}$ with any function of position, $f(\hat{x})$, is

$\displaystyle\left[f(\hat{x}_i),\hat{p}_i \right] = i\hbar\frac{\partial f(\hat{x}_i )}{\partial\hat{x}_i}$.

This is nonzero for an arbitrary function $f$, except if it is a contant that doesn't even depend on $\hat{x}$.

Can you please simplify for a layperson?

Not sure if that was supposed to display as math symbols, but I barely understood the rest anyway so it's (mostly) all the same to me.

Did manage to decipher (if I'm reading it right) that measurement of time isn't one of things affected by uncertainty.

 

[hilbert2]

I'm not the best person to explain this in a more pedagogical way, but in classical mechanics all properties of the state of a system can be given by listing the position and momentum values of every particle in the system, and in QM these and any functions of them are converted to operators for which uncertainty relations can be found based on the momentum-position commutation/uncertainty relation. So you can invent an infinite number of different functions of position and momentum (as far as they are dimensionally consistent with no summing of terms with different units) and calculate an uncertainty relation for them.

In relativistic quantum mechanics, a system can also have a variable number of particles that can be created and destroyed by physical processes, so e.g. the number of electrons in the system is also an observable with some quantum uncertainty. The spin is another property of particles that can't be properly explained with nonrelativistic QM, but it can be artificially added to the nonrelativistic theory without trying to explain what it comes from.

 

[PeroK]

Did manage to decipher (if I'm reading it right) that measurement of time isn't one of things affected by uncertainty.

Time is not an observable in QM. It's a parameter.

 

[vanhees71]

TL;DR Summary: What are some examples of the less discussed pairs of physical properties that we cannot know with mutual precision because of the Heisenberg uncertainty principle?

Position and momentum are the popular pairs of properties with uncertainty we often hear about, for example that we cannot know with precision where an electron is and its momentum at the same time.

What are others?

Such as an example of an energy and a time that we cannot know both precisely.

Also, an example of angular momentum in two perpendicular directions that both aren't knowable with high precision.

And any more example pairs that you know of.

The energy-time uncertatinty relation is precisely NOT a usual Heisenberg uncertainty relation, because time is not an observable in QM but a parameter. The correct interpretation of the energy-time uncertainty relation is given by an analysis how you measure time. See Appendix B in

https://arxiv.org/abs/2207.04898

or the there cited paper by Mandelstam and Tamm.

 

[hilbert2]

There's also this more abstract study where these people use the Page-Wootters approach of defining an additional "clock" subsystem to make time a measured variable (and not an external parameter), and then proceed to justify a time-energy uncertainty relation.

https://arxiv.org/abs/2106.00523

 

[syfry]

In relativistic quantum mechanics, a system can also have a variable number of particles that can be created and destroyed by physical processes, so e.g. the number of electrons in the system is also an observable with some quantum uncertainty.

Emphasis mine.

Your link also says: "Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables."

What gives a pair of observables the quality of having an uncertainty relation to each other?

In other words, would we find uncertainty if we take any random observable things and measured different aspects of them?

This is way over my head so I'm probably erring somewhere, but hopefully you get the gist.

 

[Nugatory]

What gives a pair of observables the quality of having an uncertainty relation to each other?

It depends on whether the observables “commute,” meaning (at a hand-waving level) that measuring A then B leaves the system in the same state as measuring B then A. If they do not commute, then there will be an uncertainty relationship between them.

 

[vanhees71]

The general Heisenberg uncertainty relation of two observables $A$ and $B$ represented by self-adjoint operators $\hat{A}$ and $\hat{B}$ and arbitrary states $\hat{\rho}$ read
$$\Delta A \Delta B \geq \frac{1}{2} |\langle \mathrm{i} [\hat{A},\hat{B}] \rangle|,$$
where
$$\langle \hat{C} \rangle=\mathrm{Tr}(\hat{\rho} \hat{C})$$
and the
$$\Delta C^2 =\langle C^2 \rangle-\langle C \rangle^2 \geq 0$$
for arbitrary self-adjoint operators $\hat{C}$.

 

[hilbert2]

What gives a pair of observables the quality of having an uncertainty relation to each other?

In other words, would we find uncertainty if we take any random observable things and measured different aspects of them?

This is way over my head so I'm probably erring somewhere, but hopefully you get the gist.

In quantum mechanics, measurable quantities like total energy $E$ and total angular momentum $|\mathbf{L}|$ are described with operators, which are similar to numbers but their multiplication can depend on order: $AB \neq BA$. Here the $A$ and $B$ are operators, sometimes this is also emphasized by writing them with a "hat" on top of them: $\hat{A}$ and $\hat{B}$. The operator for total energy is usually called the "Hamiltonian" $\hat{H}$ instead of $\hat{E}$.

If two observables $A$ and $B$ happen to have order-independent multiplication in every possible situation that can take place in an experiment you consider, $AB = BA$, it's said that $A$ and $B$ "commute" and the respective measurable quantities can be known with arbitrary precision at the same time. If the operators do not commute, then the minimum uncertainty is calculated from the commutator $[A,B] = AB - BA$. When predicting possible values of measurement results and their probabilities, ordinary numbers with normal multiplication rules are extracted from the calculations and the final practical result is not operator-valued anymore but just a number with some physical units of measurement.

In some situation a system can be described as a superposition of a finite number of possible states, e.g. a hydrogen atom in an external thermal noise field of low enough temperature to make it practically impossible for the electron to be randomly excited above 2nd electron shell. Then the electron can only occupy the 1s, 2s, 2p_x , 2p_y and 2p_z orbitals and the system has a 5-dimensional "state space". Then the Hamiltonian operator $\hat{H}$ can be written as a 5x5 matrix. Matrices are arrays of numbers, with an order-dependent multiplication rule, as you can see if you do some matrix multiplication excercises which can be found from many places.

 

[syfry]

It depends on whether the observables “commute,” meaning (at a hand-waving level) that measuring A then B leaves the system in the same state as measuring B then A. If they do not commute, then there will be an uncertainty relationship between them.

That's so interesting! So uncertainty will emerge when the order matters (in measuring).

Thanks for a bit more insight. Between the answers from you and hilbert2 it's a good start on which to expand further my knowledge while researching what the rest of the stuff means.

 

[f95toli]

An example of a "proper" pair is charge and phase in an electromagnetic circuit.
This follows from the fact that they play the role of generalized momenta and position in such circuits.
It also happens to a something that has been extensively tested over the past 40 or so years.