In quantum mechanics, Bargmann's limit, named for Valentine Bargmann, provides an upper bound on the number
of bound states with azimuthal quantum number
in a system with central potential
It takes the form This limit is the best possible upper bound in such a way that for a given
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] Stated in a formal mathematical way, Bargmann's limit goes as follows.
be a spherically symmetric potential, such that it is piecewise continuous in
If then the number of bound states
with azimuthal quantum number
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.
the wave function subject to the given potential with total energy
and azimuthal quantum number
, the Sturm Oscillation Theorem implies that
equals the number of nodes of
From the Sturm-Picone Comparison Theorem, it follows that when subject to a stronger potential
), the number of nodes either grows or remains the same.
Thus, more specifically, we can replace the potential
For the corresponding wave function with total energy
and azimuthal quantum number
, the radial Schrödinger equation becomes with
By applying variation of parameters, one can obtain the following implicit solution where
is given by If we now denote all successive nodes of
, one can show from the implicit solution above that for consecutive nodes
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
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.