On Ensembles of Particles Following The Bohr Sommerfeld Model

In the previous blog post about the Bohr Sommerfeld model we noted that we can generate the same eigenvalues if we use a probability measure \(\mu\) such that essentially

$$ \int u(x-y)\mu(dy) $$

That is a convolution. This will result in a distribution of mass in space and we took a leap and assume that we can find a \(\mu\) such that this is possible. We also indicated that by selecting \(\mu\) carefully we can also deduce quantum mechanics. In this post we will try to be more concrete about \(\mu\).

Now the first thing to note are that we are assuming a probability measure. E.g. each weight/density is non egative and

$$ \int \mu(dx) = 1 $$

Also let as consider without loss of generality that \(u(x) = \exp(iwx)\). In order to prove that the distribution of u's we need to study all powers of u as well and hence we will study

$$ u^k(x) = \exp(ikwx) $$

Now if we take the Fourier transform of this convolution we find out using that the Fourier transform of a convolution is the product of the Fourier term of the factors. The factors are,

$$ \int u^k(x)\exp(isx)dx = \int \exp(iwk)\exp(isx)dx. $$

Which is the same as,

$$ \int u^k(x)\exp(isx)dx = \delta_{s=-wk}, $$

with \(\delta\) the Dirac point measure. The other factor is

$$ \int \exp(isy)\mu(dy) = f_{\mu}(s). $$

Multiplying together leads to the Fourier transform is,

$$ f_{\mu}(s)\delta_{s=-wk} = f_{\mu}(wk)\delta_{s=-wk}, $$

Assume \(f_{\mu}(x)=1\) for \(x = 0\), and zero else we find, by combining all expressions that we get,

$$ \int \int \exp(iv u(x-y))\mu(dy)\exp(isx)dx = \delta_{s=0}(s,v), $$

for all \(u,s\) Which is solved for \(u\) is a point measure for each fixed \(x\) and \(u(x)\) does not depend on \(x\). Also \(\mu\) can be found by,

$$ \mu(dy) = \lim_{N\to\infty} \frac{1}{2N+1}\sum_{n=-N}^{N}\delta_{y=2\pi n}(y)dy. $$

Note,

$$ \int \exp(ivu(x-y) \mu(dy)) $$

is independent of \(v\) then this imply that \(u\) behaves as if \(u\) is identical to 0. A point measure for \(u\) means that all values of \(u\) is constant \(c\), then we get

$$ \int \int \exp(iv c)\mu(dy)\exp(isx)dx = \int \exp(iv c)\exp(isx)\int \mu(dy)dx. $$

As the integral of the measure is 1, we get

$$ \int \exp(iv c) \exp(isx)dx = \exp(iv c) \delta_{s}. $$

So here as we said \(c=0\). Now playing with the constant term and scaling shows that that

$$ u(x) = K/w^2. $$

The scaling property of Bohr Sommerfeld model means that the energy values has the same energy levels as Dirac QED for the Hydrogen atom minus the effect of spin. This also shows that GUTCP should have the same energy levels as QED.

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