Bargmann's limit

In quantum mechanics, Bargmann's limit, named for Valentine Bargmann, provides an upper bound on the number N_l of bound states in a system. It takes the form

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

Note that the Dirac delta function potential attains this limit.

References

• Bargmann, Proc. Natl. Acad. Sci. 38 961 (1952)
• Schwinger, Proc. Natl. Acad. Sci. 47 122 (1961)