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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

$$N_l \leq \frac{1}{2l+1} \frac{2m}{\hbar^2} \int_0^\infty r |V(r)|_{V<0}\, dr$$

Note that the delta function potential attains this limit.

References

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

Physics Encyclopedia

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