  We consider Schr\"{o}dinger equations with real quadratic Hamiltonians, for
which the Wigner distribution of the solution at a given time equals, up to a
linear coordinate transformation, the Wigner distribution of the initial
condition. Based on Hardy's uncertainty principle for the joint time-frequency
representation, we prove a uniqueness result for such Schr\"{o}dinger
equations, where the solution cannot have strong decay at two distinct times.
This approach reproduces known, sharp results for the free Schr\"{o}dinger
equation and the harmonic oscillator, and we also present an explicit scheme
for quadratic systems based on positive definite matrices.
