A Generalized Uncertainty Principle

Classical Mechanics Level 3

Which of the following gives the correct lower bound $B$ in the uncertainty relation for the operators $\hat{x}$ and $\hat{p}^2$ ?

The relevant uncertainty relation is $\sigma_x \sigma_{p^2} \geq B.$

\frac{\hbar}{2}

\hbar

\hbar \sqrt{\text{ abs}(\langle \hat{p}^2 \rangle )}

\hbar \text{ abs}(\langle \hat{p} \rangle )

1 solution

Matt DeCross
May 10, 2016

From the generalized uncertainty principle, the lower bound is:

$B = \frac12 \text{ abs} (\langle [\hat{x},\hat{p}^2]\rangle) = \frac12 \text{ abs} (\langle[x,p]p+p[x,p]\rangle) = \frac12 \text{ abs} (\langle 2i\hbar \hat{p}\rangle) = \hbar \text{ abs}(\langle\hat{p}\rangle),$

where we have used the canonical commutation relation $[x,p] = i\hbar$ .

A Generalized Uncertainty Principle

1 solution

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