  In the present paper, we generalize the celebrated classical lemma of Birch
and Heegner on quadratic twists of elliptic curves over $\mathbb{Q}$. We prove
the existence of explicit infinite families of quadratic twists with analytic
ranks $0$ and $1$ for a large class of elliptic curves, and use Heegner points
to explicitly construct rational points of infinite order on the twists of rank
$1$. In addition, we show that these families of quadratic twists satisfy the
$2$-part of the Birch and Swinnerton-Dyer conjecture when the original curve
does. We also prove a new result in the direction of the Goldfeld conjecture.
