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Imposing Klein-Gordon on Dirac Equation

by Sekonda
Tags: dirac, equation, imposing, kleingordon
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Sekonda
#1
Jan31-13, 01:43 PM
P: 209
Hey,

My question is on the Dirac equation, I am having a little confusion with the steps that have been taken to get from this form of the Dirac equation:

[tex]i\frac{\partial \psi}{\partial t}=(-i\underline{\alpha}\cdot \underline{\nabla}+\beta m)\psi[/tex]

to

[tex]-\frac{\partial^2 \psi}{\partial t^2}=[-\alpha^{i}\alpha^{j}\nabla^{i}\nabla^{j}-i(\beta\alpha^{i}+\alpha^{i}\beta)m\nabla^{i}+\beta ^{2}m^{2}]\psi[/tex]

I believe we are imposing the Klein-Gordon (maybe not) on the Dirac Equation to determine the conditions required for a free particle description via the Dirac equation, however I cannot see how this is done from those steps above.

I'm not exactly sure what these mean ∇^i and alpha's^i... We are told we apply the 'operator' to both sides of the top equation - I'm not sure what operator this is - I'm guessing it's the Klein Gordon operator though.

Any help would be appreciated on how to get from equation 1 to equation 2,
Thanks,
SK
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Sekonda
#2
Jan31-13, 02:04 PM
P: 209
Ok I've just realised we must square it to attain the second equation, though I'm still unsure what the ∇^i's represent and ditto for the alpha's.

I'll keep having a look.
Sekonda
#3
Jan31-13, 02:29 PM
P: 209
Actually I've figured it now, it's just an index to sum over, I think!

SK

andrien
#4
Feb1-13, 12:36 AM
P: 1,020
Imposing Klein-Gordon on Dirac Equation

https://docs.google.com/viewer?a=v&q...wZSsogrMAyVO0g
it is treated in last chapter of baym'quantum mechanics'.


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