Bargmann's limit explained

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 formN _{\ell} < (1)/(2 \ell +1) (2 m)/(\hbar ^{2}) \int ^{inf ty} _{0} r |V(r)| dr

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,^[1] Julian Schwinger presented an alternative way of deriving it in 1961.^[2]

Rigorous formulation and proof

Stated in a formal mathematical way, Bargmann's limit goes as follows. Let

V: R ^{3\toR:r\mapsto} V(r) be a spherically symmetric potential, such that it is piecewise continuous in r , V(r) = O( 1 /r ^{a)} for r\to 0 and V(r) = O( 1 /r ^{b)} for r\to +inf ty , where a\in( 2 , +inf ty) and b\in( -inf ty, 2 ) . If\int ^{+inf ty} _{0} r|V(r)|dr< +inf ty,

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 byN _{\ell< (1)/(2 \ell +1)} (2 m)/(\hbar ^{2}) \int ^{+inf ty} _{0} r|V(r)|dr.

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. W(r)\leq V(r) for all r\in R ^{+} _{0} ), 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 \phi _{0\ell} , the radial Schrödinger equation becomes(d ^{2})/(d r ^{2}) \phi _{0\ell} (r) - (\ell(\ell +1 ))/(r ^{2}) \phi _{0\ell} (r) =- W(r) \phi _{0\ell} (r),

with

W =2 m|V|/\hbar ^{2} . By applying variation of parameters, one can obtain the following implicit solution\phi _{0\ell} (r) = r ^{\ell+1} - \int ^{p} _{0} G(r, \rho) \phi _{0\ell} (\rho) W(\rho) d \rho,

where

G(r,\rho) is given byG(r, \rho) = (1)/(\left[r ( 2 \ell +1) (r)/(\rho) ) ^{\ell} - \rho ( (\rho)/(r) ) ^{\ell} \right].

If we now denote all successive nodes of

\phi _{0\ell} by 0= \nu _{1<\nu} _{2<...<\nu} _{N} , one can show from the implicit solution above that for consecutive nodes \nu _{i} and \nu _{i+1} (2 m)/(\hbar ^{2}) \int ^{\nu _{i+1}} _{\nu _{i}} r|V(r)|dr> 2 \ell +1 .

From this, we can conclude that

(2 m)/(\hbar ^{2}) \int ^{+inf ty} _{0} r|V(r)|dr\geq (2 m)/(\hbar ^{2}) \int ^{\nu _{N}} _{0} r|V(r)|dr>N( 2 \ell +1 )\geq N _{\ell(2\ell+1),}

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.

Notes and References

1. Bargmann . V. . 1952 . On the Number of Bound States in a Central Field of Force . Proceedings of the National Academy of Sciences . 38 . 11 . 961–966 . 1952PNAS...38..961B . 10.1073/pnas.38.11.961 . 0027-8424 . 1063691 . 16589209 . free.
2. Schwinger . J. . 1961 . On the Bound States of a Given Potential . Proceedings of the National Academy of Sciences . 47 . 1 . 122–129 . 1961PNAS...47..122S . 10.1073/pnas.47.1.122 . 0027-8424 . 285255 . 16590804 . free.