In quantum mechanics, Bargmann's limit, named for Valentine Bargmann, provides an upper bound on the number N_\ell of bound states with azimuthal quantum number \ell in a system with central potential V. It takes the form This limit is the best possible upper bound in such a way that for a given \ell, one can always construct a potential V_\ell for which 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, Julian Schwinger presented an alternative way of deriving it in 1961.

Rigorous formulation and proof

Stated in a formal mathematical way, Bargmann's limit goes as follows. Let be a spherically symmetric potential, such that it is piecewise continuous in r, for r\to0 and for r\to+\infty, where and. If then the number of bound states N_\ell with azimuthal quantum number \ell for a particle of mass m obeying the corresponding Schrödinger equation, is bounded from above by Although the original proof by Valentine Bargmann is quite technical, the main idea follows from two general theorems on ordinary differential equations, the Sturm Oscillation Theorem and the Sturm-Picone Comparison Theorem. If we denote by u_{0\ell} the wave function subject to the given potential with total energy E=0 and azimuthal quantum number \ell, the Sturm Oscillation Theorem implies that N_\ell equals the number of nodes of u_{0\ell}. From the Sturm-Picone Comparison Theorem, it follows that when subject to a stronger potential W (i.e. for all ), the number of nodes either grows or remains the same. Thus, more specifically, we can replace the potential V by -|V|. For the corresponding wave function with total energy E=0 and azimuthal quantum number \ell, denoted by, the radial Schrödinger equation becomes with. By applying variation of parameters, one can obtain the following implicit solution where G(r,\rho) is given by If we now denote all successive nodes of by, one can show from the implicit solution above that for consecutive nodes \nu_{i} and \nu_{i+1} From this, we can conclude that proving Bargmann's limit. Note that as the integral on the right is assumed to be finite, so must be N and N_\ell. Furthermore, for a given value of \ell, one can always construct a potential V_\ell for which N_\ell is arbitrarily close to Bargmann's limit. The idea to obtain such a potential, is to approximate Dirac delta function potentials, as these attain the limit exactly. An example of such a construction can be found in Bargmann's original paper.