Bargmann's limit

In quantum mechanics, Bargmann's limit, named for Valentine Bargmann, provides an upper bound on the number <math>N_\ell</math> of bound states with azimuthal quantum number <math>\ell</math> in a system with central potential <math>V</math>. It takes the form

<math>N_\ell < \frac{1}{2\ell+1} \frac{2m}{\hbar^2} \int_0^\infty r |V(r)|\, dr</math>

This limit is the best possible upper bound in such a way that for a given <math>\ell</math>, one can always construct a potential <math>V_\ell</math> for which <math>N_\ell</math> 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,<ref name=":0">Template:Cite journal</ref> Julian Schwinger presented an alternative way of deriving it in 1961.<ref>Template:Cite journal</ref>

Rigorous formulation and proof

Stated in a formal mathematical way, Bargmann's limit goes as follows. Let <math>V:\mathbb{R}^3\to\mathbb{R}:\mathbf{r}\mapsto V(r)</math> be a spherically symmetric potential, such that it is piecewise continuous in <math>r</math>, <math>V(r)=O(1/r^a)</math> for <math>r\to0</math> and <math>V(r)=O(1/r^b)</math> for <math>r\to+\infty</math>, where <math>a\in(2,+\infty)</math> and <math>b\in(-\infty,2)</math>. If

<math>\int_0^{+\infty}r|V(r)|dr<+\infty,</math>

then the number of bound states <math>N_\ell</math> with azimuthal quantum number <math>\ell</math> for a particle of mass <math>m</math> obeying the corresponding Schrödinger equation, is bounded from above by

<math>N_\ell<\frac{1}{2\ell+1}\frac{2m}{\hbar^2}\int_0^{+\infty}r|V(r)|dr.</math>

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 <math>u_{0\ell}</math> the wave function subject to the given potential with total energy <math>E=0</math> and azimuthal quantum number <math>\ell</math>, the Sturm Oscillation Theorem implies that <math>N_\ell</math> equals the number of nodes of <math>u_{0\ell}</math>. From the Sturm-Picone Comparison Theorem, it follows that when subject to a stronger potential <math>W</math> (i.e. <math>W(r)\leq V(r)</math> for all <math>r\in\mathbb{R}_0^+</math>), the number of nodes either grows or remains the same. Thus, more specifically, we can replace the potential <math>V</math> by <math>-|V|</math>. For the corresponding wave function with total energy <math>E=0</math> and azimuthal quantum number <math>\ell</math>, denoted by <math>\phi_{0\ell}</math>, the radial Schrödinger equation becomes

<math>\frac{d^{2}}{d r^{2}} \phi_{0\ell}(r)-\frac{\ell(\ell+1)}{r^{2}} \phi_{0\ell}(r)=-W(r) \phi_{0\ell}(r),</math>

with <math>W=2m|V|/\hbar^2</math>. By applying variation of parameters, one can obtain the following implicit solution

<math>\phi_{0\ell}(r)=r^{\ell+1}-\int_{0}^{p} G(r, \rho) \phi_{0\ell}(\rho) W(\rho) d \rho,</math>

where <math>G(r,\rho)</math> is given by

<math>G(r, \rho)=\frac{1}{2 \ell+1}\left[r\bigg(\frac{r}{\rho}\bigg)^{\ell}-\rho\bigg(\frac{\rho}{r}\bigg)^{\ell}\right].</math>

If we now denote all successive nodes of <math>\phi_{0\ell}</math> by <math>0=\nu_1<\nu_2<\dots<\nu_{N}</math>, one can show from the implicit solution above that for consecutive nodes <math>\nu_{i}</math> and <math>\nu_{i+1}</math>

<math>\frac{2m}{\hbar^2}\int_{\nu_{i}}^{\nu_{i+1}} r|V(r)|dr>2\ell+1.</math>

From this, we can conclude that

<math>\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),</math>

proving Bargmann's limit. Note that as the integral on the right is assumed to be finite, so must be <math>N</math> and <math>N_\ell</math>. Furthermore, for a given value of <math>\ell</math>, one can always construct a potential <math>V_\ell</math> for which <math>N_\ell</math> 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.<ref name=":0" />