Uncertainty principle

For an algebraic justification, see Why did physics go quantum.
It is something general for any signal. It has to do with Fourier transform and short-time Fourier transform.

Discrete case

The setup

Consider the “toy” world introduced in translation and momentum operators: five discrete “positions” $x=1,2,3,4,5$ and the cyclic translation operator $T$. Recall that we have:

1. Position operator

so that a position eigenstate $|x_j⟩$ has $\hat{X}|x_j⟩=j|x_j⟩$.

2. Translation operator

cyclically shifting the components $(a,b,c,d,e)\mapsto (e,a,b,c,d).$

3. Momentum “generator”
   We define

Since $T$ is unitary with eigenvalues $e^{2\pi ik/5}$ ($k=0,1,2,3,4$), its logarithm $P$ has eigenvalues $$
p_k ;=;\tfrac{2\pi k}{5},,
\qquad
P,\lvert p_k\rangle = p_k,\lvert p_k\rangle,.

[\hat X,,T] ;=;\hat X,T ;-;T,\hat X
;=;
\begin{pmatrix}
-4 & 0 & 0 & 0 & 0\
0 & 1 & 0 & 0 & 0\
0 & 0 & 1 & 0 & 0\
0 & 0 & 0 & 1 & 0\
0 & 0 & 0 & 0 & 1
\end{pmatrix},,

[\hat X,,P]
;=;
[\hat X,,\log T]
;=;
\text{(a nonzero constant matrix)}.

(\Delta X)^2 = \langle\psi|\hat X^2|\psi\rangle
- \langle\psi|\hat X|\psi\rangle^2,
\quad
(\Delta P)^2 = \langle\psi|\hat P^2|\psi\rangle
- \langle\psi|\hat P|\psi\rangle^2.

\Delta X,\Delta P
;\ge;
\tfrac12,\bigl|\langle[\hat X,\hat P]\rangle_\psi\bigr|.