Thursday 3 July 2014

intrinsic angular momentum

I was revisiting a couple of my earlier post, particularly one on the momentum of inertia of a photon when I got to thinking about 2 equations I derived in that post, the first equation was

$ E I = L^2$ .... (1)

Which says that energy multiplied by moment of inertia is the angular momentum squared. I also derived a second more specific version of this (originally equation 16), which shows

$ E I =\hbar^2 $   ... (2)

then it occurred to me that when a photon creates matter it has to obey the various conservation laws. A photon that converts to matter will form an electron/positron pair, rather than just a single electron. If it did form a single electron there would be conservation of charge violation and so on.

For two electrons (or an electron positron pair) though, we have

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

also the equivalent moment of inertia is

$ I = 2 I_e $ ... (4)

putting these into (2) we have

$ 4 m_e c^2 I_e = \hbar^2  $ ... (5)

so

$ m_e c^2 I_e  = $ $\frac {\hbar^2} {4} $   ... (6)

but from (1) we have

$ E_e I_e = L_e^2 $ for an electron, where $E_e = m_e c^2$ giving

$L_e^2 =$ $\frac {\hbar^2} {4} $  ... (7) , which becomes

$L_e =$ $\frac {\hbar} {2}$ ...(8)

The intrinsic spin of an electron! That's nice eh? There is more to this though. If equation (2) is correct then does it imply all particles have spin. This makes me think we haven't got the Higgs particle sorted yet because it can't have a zero spin version. It's also a bit of a black eye for string theory which, I've been told, likes the idea of spinless particles.

This is not a problem if a particle can have zero moment of inertia, in which case $I=0$ and $L=0$. What would that mean?

After all

$I = m \lambda^2 $ ...(9) or

$I =$ $\frac {\hbar} {\omega}$  ...(10)

so $I$ can't be zero because if m is zero then we still have a moment of inertia because of the frequency of the massless particle given by

$E = \hbar \omega$

 I'll figure what all that in the next post, hopefully!

wrote this while listening to this.




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