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]

Rigorous formulation and proof

Stated in a formal mathematical way, Bargmann's limit goes as follows. Let ${\displaystyle V:\mathbb {R} ^{3}\to \mathbb {R} :\mathbf {r} \mapsto V(r)}$ be a spherically symmetric potential, such that it is piecewise continuous in ${\displaystyle r}$, ${\displaystyle V(r)=O(1/r^{a})}$ for ${\displaystyle r\to 0}$ and ${\displaystyle V(r)=O(1/r^{b})}$ for ${\displaystyle r\to +\infty }$, where ${\displaystyle a\in (2,+\infty )}$ and ${\displaystyle b\in (-\infty ,2)}$. If

${\displaystyle \int _{0}^{+\infty }r|V(r)|dr<+\infty ,}$

then the number of bound states ${\displaystyle N_{\ell }}$ with azimuthal quantum number ${\displaystyle \ell }$ for a particle of mass ${\displaystyle m}$ obeying the corresponding Schrödinger equation, is bounded from above by

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

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 ${\displaystyle u_{0\ell }}$ the wave function subject to the given potential with total energy ${\displaystyle E=0}$ and azimuthal quantum number ${\displaystyle \ell }$, the Sturm Oscillation Theorem implies that ${\displaystyle N_{\ell }}$ equals the number of nodes of ${\displaystyle u_{0\ell }}$. From the Sturm-Picone Comparison Theorem, it follows that when subject to a stronger potential ${\displaystyle W}$ (i.e. ${\displaystyle W(r)\leq V(r)}$ for all ${\displaystyle r\in \mathbb {R} _{0}^{+}}$), the number of nodes either grows or remains the same. Thus, more specifically, we can replace the potential ${\displaystyle V}$ by ${\displaystyle -|V|}$. For the corresponding wave function with total energy ${\displaystyle E=0}$ and azimuthal quantum number ${\displaystyle \ell }$, denoted by ${\displaystyle \phi _{0\ell }}$, the radial Schrödinger equation becomes

${\displaystyle {\frac {d^{2}}{dr^{2}}}\phi _{0\ell }(r)-{\frac {\ell (\ell +1)}{r^{2}}}\phi _{0\ell }(r)=-W(r)\phi _{0\ell }(r),}$

with ${\displaystyle W=2m|V|/\hbar ^{2}}$. By applying variation of parameters, one can obtain the following implicit solution

${\displaystyle \phi _{0\ell }(r)=r^{\ell +1}-\int _{0}^{p}G(r,\rho )\phi _{0\ell }(\rho )W(\rho )d\rho ,}$

where ${\displaystyle G(r,\rho )}$ is given by

${\displaystyle G(r,\rho )={\frac {1}{2\ell +1}}\left[r{\bigg (}{\frac {r}{\rho }}{\bigg )}^{\ell }-\rho {\bigg (}{\frac {\rho }{r}}{\bigg )}^{\ell }\right].}$

If we now denote all successive nodes of ${\displaystyle \phi _{0\ell }}$ by ${\displaystyle 0=\nu _{1}<\nu _{2}<\dots <\nu _{N}}$, one can show from the implicit solution above that for consecutive nodes ${\displaystyle \nu _{i}}$ and ${\displaystyle \nu _{i+1}}$

${\displaystyle {\frac {2m}{\hbar ^{2}}}\int _{\nu _{i}}^{\nu _{i+1}}r|V(r)|dr>2\ell +1.}$

From this, we can conclude that

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

proving Bargmann's limit. Note that as the integral on the right is assumed to be finite, so must be ${\displaystyle N}$ and ${\displaystyle N_{\ell }}$. Furthermore, for a given value of ${\displaystyle \ell }$, one can always construct a potential ${\displaystyle V_{\ell }}$ for which ${\displaystyle N_{\ell }}$ 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.^[1]

References