Klein gordan equation in electromagnetic potential

[sp105]

Its troubling me a lot that when we start solvingK.G. equation for a free particle we assume energy as a sum of nonrelativistic energy and mc². but when it is tried to solve the equation for a particle in electromagnetic field we solve it without taking into account the term mc^2. for reference you may go through the book Quantum Mechanics By Schiff

 

[malawi_glenn]

and for us who does not have that book which is over 40y old?

 

[sp105]

the people who don't have the book may refer to any book of quantum mechanics that contains relativistic quantum mechanics. and especially klein gordan eqn and its application to particle in electromagnetic potential

 

[malawi_glenn]

the people who don't have the book may refer to any book of quantum mechanics that contains relativistic quantum mechanics. and especially klein gordan eqn and its application to particle in electromagnetic potential

I have solved the KG equation for a charged particle in Coulomb potential (klein gordon hydrogenic-atom), and I have no Idea what you are talking about regarding that procedure presented in original post of yours. ...

 

[sp105]

when we solve the K.G eqn for the free particle we assume E'=E+mc^2 where E is very small as compared to the second term and is the nonrelativistic energy soln
the result so obtained is not used to solve the K.G. eqn for particle in potential
but wee replace p by p-eA/c and so on thereafter for the soln of the gotten eqn is assumed as exp of -iEt/h where that second term has been gone which was greater than E

 

[malawi_glenn]

Eh when I solved KG for free particle I just solved it, no such considerations or whatever. Can you tell us what the solutions obtained were?

and for particle in EM field, the ansatz
\Psi (t.\vec{r}\,) = \psi (\vec{r} \,) e^{iEt/\hbar}
is pretty standard to start with.

And isn't it called Klein-Gordon ?

 

[sp105]

its the right thing that we solve KG eqn for particle in em field with the solution you have mentioned. but for free particle one more term with the solution for em field is added in the exponential and it is mc². and the solution we got is the relativistic energy eqn.

 

[malawi_glenn]

the free particle solution is, for positive frequency solutions:

f^{(+)}_p = \frac{1}{\sqrt{2E(2\pi\hbar)^3}}e^{-i(Et-\vec{p}\cdot\vec{r})}

In both cases, in this one and in my post #6, E is the relativistic energy.

It is the same plane wave like solution as for non relativistic case, except for different normalization and that E now is the relativistic energy.

I.e. solving the KG equation for the C-potential, (A^\mu = (V(r),0)

one get's that E_ground state = m(1-alpha^2/2) {+ higher order terms}

so mass is included and this is indeed a relativistic expression for energy.

So we solve it by saying that E is just a number, we don't care what it is, we are to FIND the energy.

 

[sp105]

a klein gordan equation may reduces to Schrödinger equation. how it can be proved

 

[malawi_glenn]

separate it into two equations, in each do the non relativistic limit (m -> infinity)

 

[sp105]

sorry please elaborate the answer
i am not able to understand what do you want to say

 

[malawi_glenn]

sorry please elaborate the answer
i am not able to understand what do you want to say

KG is 2nd order differential equation, separate it into two differential equations of first order. Then in each of them do the non relativistic limit, which is mass -> infinity.

How difficult is that to understand?

Why do you ask this?

 

[malawi_glenn]

This is a good site if you need something else than your "any book that contains relativistic quantum mechanics"

http://www.phys.uAlberta.ca/~gingrich/phys512/latex2html/node20.html

a good book on relativistic quantum should also mention non relativistic limits!

 

[sp105]

the schroedinger eqn of equivalance of klein gorgdan eqn in your prescribed site has been derived from KG equation in EM field
what bout its relevance in case of free particle case
will you prefer some book of relativistic quantum mechanics to me
I will be surely thankful to you

 

[malawi_glenn]

the schroedinger eqn of equivalance of klein gorgdan eqn in your prescribed site has been derived from KG equation in EM field
what bout its relevance in case of free particle case
will you prefer some book of relativistic quantum mechanics to me
I will be surely thankful to you

So you don't think it is a trivial task to show it for the free particle case if you are given how to do it for EM-interactions? Come on ... it is trivial, just do the inverse substitution for momentum operator.

As you know, to go from free to EM equation, one substitute p to p + Em_potential:

i\partial _\mu \rightarrow i\partial _\mu + eA_\mu

now do the inverse substitution.

Or, just say, "the electromagnetic potential = 0" ...

 

[Bob S]

The relativistic K. G. equation for a spinless particle in a central Coulomb field has little use for hydrogen-like atoms, because electrons have spin. Except for negative pions in atomic orbits. My thesis, about 45 years ago, was to measure the pi-minus mass using pionic x-rays from pions in 4F - 3D transitions in titanium using Bragg diffraction in a quartz crystal spectrometer. I used the K. G. solution in Schiff Quantum Mechanics (second edition, 1955), page 321 -322. I carried out Schiff's expansion (eqs 42.16 -.21) to the next order in Z alpha. The biggest correction, other than the Uehling integral for vacuum polarization, was the relativistic correction for the reduced mass. My measurement is still in agreement with modern measurements of the pi minus mass, which still use pionic atom measurements.