Planck, Schrödinger and positronium

In a previous post on natural units I derived a number of equations including

$m_\psi{'} = m_p \alpha^{5/2}$   ...(1)

$m_\psi{'}$ - alternative version of the Schrödinger mass
$m_p$ - Planck mass
$\alpha$ - fine structure constant

using Einsteins energy-mass formula

$E = m c^2$ ...(3)

we can derive a Schrödinger, Planck and electron version of energy such that

$E_p = m_p c^2$ ...(4)

$E_\psi = m_\psi{'} c^2$ ...(5)

$E_e = m_e c^2$ ...(6)

$m_e$ - mass of the electron
c - speed of light in a vacuum

Now, let us choose

$E_0 = m_0 c^2$   ...(7)

so that

$E_e E_\psi^2 = E_p^2 E_0$    ... (8)

this gives

$m_e c^2 m_\psi{'}^2 c^4 = m_p^2 c^4 E_0$   ...(9)

now sub in the value from equation (1)

$m_e c^2 m_p^2 \alpha^5 c^4 = m_p^2 c^4 E_0$    ...(10)

divide through by $m_p^2 c^4$ and use the Planck relationship

$E_0 = 2 \hbar \omega_0$   ...(11)
$\hbar$ - reduced Planck constant
$\omega_0 = 2 \pi \nu$
$\nu$ - frequency of the photon
factor of 2 is for 2 photons

to give

$2 \hbar  \omega_0 =  m_e c^2 \alpha^5$    ...(12)

re-arrange to give

$\frac {1} {\omega_0} = \frac {2 \hbar} {m_e c^2 \alpha^5}$  ... {13}

replace $\omega_0$ as follows

$t_0 =$  $\frac {1} {\omega_0}$    ... (14)

$t_0$ has units of time, sub this into (13) and you end with

$t_0 =$  $\frac {2\hbar} {m_e c^2 \alpha^5}$     ...(15)

If we put some numbers into (15) we get a value of

$t_0 = 1.244   10^{-10} s$   ...(16)

Here is the point of this post. So far we have used Einsteins equation and a  result we got from a previous post on natural units. That result, equation (1) shows  a relationship between two natural units of mass, one is an alternative Schrödinger mass, the other the Planck mass. We used these to derive equation (15).

Equation (15) also happens to be the mean life time of para-positronium (p-Ps)! Here is a link to a wiki page. How cool is that?

Will investigate this result in more detail in the next post. Note that there is a little ambiguity as to what the "2 photons" referred to in equation 11 actually are. There is also the value of $m_0$ mentioned in equation 7. These both need to explained in greater detail.

Wrote this while listening to this.