Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 3

The canonical commutation relation $[x^{\prime },p^{\prime }]=i\hslash I$ and hamiltonian $H^{\prime }=p^{\prime 2}/(2m)+m{\omega }^2{x}^{\prime 2}/2$ giveSubstitution in the Heisenberg equation of motion yields ${\dot{x}}^{\prime }=p^{\prime }/m$ and ${\dot{p}}^{\prime }=-m{\omega }^2{x}^{\prime }$. Differentiating once more gives the harmonic-oscillator equation for each operator. Its initial-value solution isDifferentiating these expressions at zero verifies both prescribed initial derivatives. They also preserve $[x^{\prime }(t),p^{\prime }(t)]=i\hslash I$.

The normalized initial Gaussian wave packet is real and even, so the integrands for $⟨x^{\prime }{⟩}_0$ and $⟨p^{\prime }{⟩}_0$ are odd. Its second position moment is the elementary Gaussian integral $⟨x^{\prime 2}{⟩}_0=1/(2\alpha )$. Since ${\partial }_x\Psi =-\alpha x\Psi$ and ${\partial }_x^2\Psi =({\alpha }^2{x}^2-\alpha )\Psi$,Moreover $⟨x^{\prime }{p}^{\prime }{⟩}_0=i\hslash \alpha ⟨x^{\prime 2}{⟩}_0=i\hslash /2$ and $⟨p^{\prime }{x}^{\prime }{⟩}_0=-i\hslash /2$. Thus

Square the operator solutions, retaining the symmetrized cross term rather than treating $x^{\prime }(0)$ and $p^{\prime }(0)$ as commuting variables. The initial symmetrized expectation vanishes, soThe means remain zero. These are also the variances. Unless $\alpha =m\omega /\hslash$, the Gaussian state breathes: its position and momentum widths exchange periodically while the mean energy remains constant.

Introduce the dimensionless width $s=\hslash \alpha /(m\omega )>0$. Multiplying the variances givesThe minimum holds at all times precisely when $\alpha =m\omega /\hslash$, the oscillator ground-state width. For any other $\alpha$, it is still attained at the isolated quarter-period times $t=n\pi /(2\omega )$, including zero. At a fixed generic time, minimizing over $\alpha$ again selects $s=1$.

The excess product has a covariance interpretation:Thus the breathing Gaussian state in a harmonic oscillator saturates the Robertson-Schrödinger uncertainty relation throughout, even when its position-momentum product exceeds the lower bound in the Heisenberg uncertainty principle.