Imposing Klein-Gordon on Dirac Equation

In summary, the conversation discusses the steps taken to get from the first form of the Dirac equation to the second form, which involves applying the Klein-Gordon operator and squaring the equation. The symbols ∇^i and alpha's^i represent indices to sum over. The conversation also mentions a resource, "Quantum Mechanics" by Baym, that provides further explanation on this topic.
  • #1
Sekonda
207
0
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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  • #2
Ok I've just realized 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.
 
  • #3
Actually I've figured it now, it's just an index to sum over, I think!

SK
 

FAQ: Imposing Klein-Gordon on Dirac Equation

1. What is the Klein-Gordon equation and how is it related to the Dirac equation?

The Klein-Gordon equation is a relativistic quantum field equation that describes spinless particles. It is related to the Dirac equation, which describes spin-1/2 particles, through a process called "imposing Klein-Gordon on Dirac". This is achieved by taking the square root of the Klein-Gordon equation and substituting it into the Dirac equation, resulting in a modified version of the Dirac equation that includes the mass of the particle.

2. Why is it necessary to impose the Klein-Gordon equation on the Dirac equation?

The Dirac equation alone is not able to describe particles with spin-0, such as the Higgs boson. By imposing the Klein-Gordon equation on the Dirac equation, we are able to extend its applicability to spin-0 particles as well. This is important for understanding the behavior of fundamental particles in the universe.

3. How does imposing Klein-Gordon on Dirac equation affect the solutions?

Imposing Klein-Gordon on Dirac equation results in the solutions having an additional term that accounts for the mass of the particle. This new term has a positive or negative value, depending on the spin of the particle, which affects the overall behavior and properties of the solutions.

4. Can the Klein-Gordon equation be imposed on other quantum field equations?

Yes, the Klein-Gordon equation can be imposed on other quantum field equations, such as the Schrödinger equation or the Maxwell equations. This allows for a more comprehensive understanding of the behavior of different types of particles and their interactions.

5. What are the applications of imposing Klein-Gordon on Dirac equation?

The imposition of Klein-Gordon on Dirac equation has many important applications in theoretical physics. It has helped to explain the behavior of fundamental particles, such as the Higgs boson, and has also been used in the development of quantum field theories, including quantum electrodynamics and quantum chromodynamics. Additionally, it has implications for the study of cosmology and the early universe.

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