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Showing posts with the label Uncertainty relations

### Unusual position-momentum uncertainty relations

Consider a one-dimensional quantum particle, and let $$X$$ and $$P$$ represent its position and momentum operators, respectively. $$X$$ and $$P$$ are assumed to satisfy the canonical commutation relation, $[X,P] = i \hbar$ with $$\hbar$$ being the reduced Planck constant. For the following discussion, it is convenient to introduce dimensionless position and momentum as $x = \frac{X}{L} \qquad \text{and} \qquad p = \frac{L P}{\hbar} \,,$ where $$L$$ is some (chosen for convenience, but otherwise arbitrary) length scale. In terms of $$x$$ and $$p$$, the canonical commutation relation reads $[x, p] = i \,.$ The operators $$x$$ and $$p$$ fulfill the Heisenberg uncertainty relation: $\Delta x \Delta p \ge \frac{1}{2} \,,$ where $$\Delta x \ge 0$$ and $$\Delta p \ge 0$$ are uncertainties in the particle's position and momentum, respectively. (The Heisenberg uncertainty relation is a straightforward consequence of a more general Robertson-Schrödinger uncertainty rel

### Mandelstam-Tamm uncertainty relation

The Mandelstam-Tamm enerty-time uncertainty relation states that $\Delta E \Delta t_A \ge \frac{\hbar}{2} \,,$ where $$\Delta E$$ is the energy uncertainty of a quantum system, and $$\Delta t_A$$ is the time required for a significant change of the expectation value of an observable $$A$$. Derivation Consider a quantum system with a Hamiltonian $$H$$. Let $$| \psi \rangle$$ be the time-dependent state of the system, and let $$A$$ be some observable. The uncertainty in the system's energy and the uncertainty in $$A$$ are defined, respectively, as \begin{align} &\Delta E = \sqrt{\langle H^2 \rangle - \langle H \rangle^2} \,, \\ &\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle^2} \,, \end{align} where $$\langle \cdot \rangle = \langle \psi | \cdot | \psi \rangle$$ denotes the expectation value. $$\Delta E$$ and $$\Delta A$$ satisfy the uncertainty relation $\Delta E \Delta A \ge \frac{\big| \langle HA - AH \rangle \big|}{2} \,.$ Since the rate