What possible variants/permutations/derivatives are there of the Heisenberg uncertainty principle?

Question

Generally, the (Heisenberg) uncertainty principle is stated as: $\Delta x \cdot \Delta p \geq \dfrac{\hbar}{2}$ But sometimes you also encounter a variant of it, placing limits on other entities, such as $\Delta E \cdot \Delta t \geq \dfrac{\hbar}{2}$.

These are the two versions I can find online, but I'm pretty sure I've seen other combinations as well. Wikipedia tells me there are other versions, but lists them in what I believe to be bra-ket notation, which I'm not familiar with.

So what are the possible combinations of entities in the form $\Delta A \cdot \Delta B \geq \dfrac{\hbar}{2}$, with $A$ and $B$ being a combination of quantities? Or will it work for any combination where the units check out?

Answer 1

Any pair of observables $A$ and $B$ satisfy the Robertson's relation

$$ \Delta A\Delta B\geq \frac12|\langle[A,B]\rangle|$$ in the case of position and momentum, $[x,p]=i\hbar$, hence Heisenberg's uncertainty principle. If the observables commute, they can be measured simultaneously, which is consistent with the fact that their uncerainties can both vanish.

For time and energy the situation is more complicated: it is nontrivial to find a hermitian operator corresponding to time in the way a Hamiltonian corresponds to energy, and such an operator does not exist for many physical situations, see this question, so we cannot just take two operators and pluck them into Robertson's relation. The time energy uncertainty principle is more complicated, a justification for it can be found here

Answer 2

Pulling from Shankar's QM book here, we can get an even more general principle than given in other answers:

$$(\Delta A)^2\cdot(\Delta B)^2\geq\frac14\langle[A,B]_+\rangle^2-\frac1{4}\langle[A,B]\rangle^2$$

where $[A,B]_+=AB+BA$ and $[A,B]=AB-BA$.

We have expectation values here, so these relations depend on the actual state (for example, the $L_x$, $L_y$ relation given in Thomas Fritsch's answer), so we usually try to simplify the relation to be more general and not depend on the state.

If $A$ and $B$ are Hermitian, then $\langle[A,B]_+\rangle^2$ is real and non-negative. Additionally if $[A,B]=i\hbar$ then we arrive at the usual, state-independent relation $$\Delta A\cdot\Delta B\geq\frac{\hbar}2$$

It looks like you are looking for a list of any relations possible, but the above relation holds for all Hermitian operators, so pick any two and see what happens. If you specifically want relations where the right hand side is $\hbar/2$, then you need the commutator of the two observables to be equal to $i\hbar$.

Answer 3

The uncertainty relation between position $x$ and momentum $p$ $$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$ is just a special case of the more general uncertainty relation between two arbitrary operators $A$ and $B$: $$\Delta A \cdot \Delta B \geq \frac{1}{2}\Big|\langle[A,B]\rangle\Big|.$$

Here $[A,B]$ is the commutator defined as $AB-BA$, and $\langle\cal{O}\rangle$ means the expectation value of $\cal{O}$.

From that you can derive arbitrarily many uncertainty relations, for example $$\begin{align} \Delta x \cdot \Delta p_x &\geq \frac{\hbar}{2} &\text{ position and momentum} \\ \Delta x \cdot \Delta k_x &\geq \frac{1}{2} &\text{ position and wave-number} \\ \Delta x \cdot \Delta y &\geq 0 &\text{ position components} \\ \Delta p_x \cdot \Delta p_y &\geq 0 &\text{ momentum components} \\ \Delta\phi_x\cdot\Delta L_x &\geq \frac{\hbar}{2} &\text{ angle and angular momentum} \\ \Delta L_x \cdot \Delta L_y &\geq \frac{\hbar}{2}\Big|\langle L_z\rangle\Big| &\text{ angular momentum components} \end{align}$$

Answer 4

I know of 4 broad generalizations.

1. The first introduces the notion of disturbance. The early work on this is

Arthurs, E., and J. L. Kelly. "BSTJ briefs: On the simultaneous measurement of a pair of conjugate observables." The Bell System Technical Journal 44.4 (1965): 725-729.

but

Stenholm, S. (1992). Simultaneous measurement of conjugate variables. annals of physics, 218(2), 233-254.

is more accessible. The basic idea is that one can obtain an uncertainty relation for this type of measurement if one includes the disturbance resulting from the measurement. In the original A&K paper they find $$ \Delta X \Delta P\ge \hbar $$ i.e. twice the usual uncertainty. This has spawned various related work such as

Martens, Hans, and Willem M. De Muynck. "The inaccuracy principle." Foundations of physics 20.4 (1990): 357-380,

Martens, Hans, and Willem M. de Muynck. "Disturbance, conservation laws and the uncertainty principle." Journal of Physics A: Mathematical and General 25.18 (1992): 4887.

There is also quite a bit of more recent work by Masanao Osawa on this.

2. Sum-type uncertainty relations. For angular momentum this would be $$ \Delta L_x^2+\Delta L_y^2+\Delta L_z^2\ge \frac{1}{2}j(j+1). $$ The earliest derivation I know is in

   Delbourgo, Robert. "Minimal uncertainty states for the rotation and allied groups." Journal of Physics A: Mathematical and General 10.11 (1977): 1837.

A variation on this was used for entanglement detection by

Tóth, Géza, et al. "Spin squeezing and entanglement." Physical Review A 79.4 (2009): 042334.

and generalized to $\mathfrak{su}(n)$ in

de Guise, H., Maccone, L., Sanders, B. C., & Shukla, N. (2018). State-independent uncertainty relations. Physical Review A, 98(4), 042121.

There is follow up recent numerical work by the group of Maccone.

3. There is a class of entropic uncertainty relations, used extensively in quantum information. Apparently they are pretty useless experimentally. For a review see

   Coles, Patrick J., et al. "Entropic uncertainty relations and their applications." Reviews of Modern Physics 89.1 (2017): 015002.

Stephanie Wehrner has a number of papers on this topic.

4. Finally, there is a class of uncertainty relations that deals with n-fold products of variances. A good source on this is

   Synge, John Lighton. "Geometrical approach to the Heisenberg uncertainty relation and its generalization." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 325.1561 (1971): 151-156.

The idea is to use principal minors of a non-negative matrix to obtain relations - sometimes products of the form $\Delta A^2\Delta B^2\Delta C^2$, in an immediate generalization of the usual Heisenberg relations. The article is poorly cited and does not seem to have had much of an impact.