Bargmann's limit

In quantum mechanics, Bargmann's limit, named for Valentine Bargmann, provides an upper bound on the number ${\displaystyle N_{\ell }}$ of bound states with azimuthal quantum number ${\displaystyle \ell }$ in a system with central potential ${\displaystyle V}$. It takes the form

${\displaystyle N_{\ell }<{\frac {1}{2\ell +1}}{\frac {2m}{\hbar ^{2}}}\int _{0}^{\infty }r|V(r)|\,dr}$

This limit is the best possible upper bound in such a way that for a given ${\displaystyle \ell }$, one can always construct a potential ${\displaystyle V_{\ell }}$ for which ${\displaystyle N_{\ell }}$ is arbitrarily close to this upper bound. Note that the Dirac delta function potential attains this limit. After the first proof of this inequality by Valentine Bargmann in 1953,^[1] Julian Schwinger presented an alternative way of deriving it in 1961.^[2]