Bad Qubits — Learn Quantum by Doing

Multiple choice

The ground-state Gaussian of the quantum harmonic oscillator has position uncertainty

\Delta x = x_0/\sqrt{2}

and momentum uncertainty

\Delta p = \hbar/(\sqrt{2}\,x_0)

. What is the uncertainty product

\Delta x \cdot \Delta p

, and what does it imply?

$0$ , meaning position and momentum are both perfectly known
$\hbar/4$ , which violates the Heisenberg uncertainty principle
$\hbar/2$ , so the ground state saturates the Heisenberg uncertainty bound (a minimum-uncertainty state)
$\hbar$ , which is twice the Heisenberg minimum and thus a typical excited state