Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I'm looking at the Dirac Equation, in the form given on Wikipedia, and (foolishly) trying to understand it.

[itex]\left( c \boldsymbol{\alpha}\cdot \mathbf{\hat{p}}+\beta mc^2 \right ) \psi = i\hbar\frac{\partial \psi}{\partial t}\,\![/itex]

So I picture a wavefunction in an eigenstate of the momentum operator in the [itex]e_1[/itex]-direction with an eigenvalue of p, and simultaneously an eigenstate of the [itex]\alpha_1[/itex] and [itex]\beta[/itex] operators with an eigenvalue of 1 in both cases. Now obviously, for this case:

[itex]\left( c \boldsymbol{\alpha}\cdot \mathbf{\hat{p}}+\beta mc^2 \right ) \psi = \left( p c +mc^2 \right ) \psi\,\![/itex]

But we know that [itex]E = \sqrt{p^2 c^2 + m^2 c^4}[/itex], so this doesn't seem to give the right eigenvalue for the energy operator on the RHS. We want the hypotenuse of a right triangle with [itex]c p[/itex] and [itex]m c^2[/itex] as its legs, not the length of a line with those two quantities as segments of the line! It seems like it might work out right if somehow they were complex and 90 degrees out of phase, but I can't see any way to get that.

What part of my brain is broken?

Thanks.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Why doesn't the energy come out right in the Dirac Equation?

Loading...

Similar Threads for doesn't energy right | Date |
---|---|

I Wheeler's delayed choice doesn't change the past | Dec 27, 2017 |

B Why doesn't this FTL communication scheme work? | Dec 4, 2017 |

I Why doesn't momentum transfer on mirrors reveal which path? | Apr 30, 2017 |

B Double slit experiment - doesn't measurement affect photons? | Apr 16, 2017 |

How come hybridisation of orbital doesn't violate quantization of energy of electrons | Aug 1, 2012 |

**Physics Forums - The Fusion of Science and Community**