I Alternative formula for the Hamiltonian

[grzz]

In his article 'Quantum theory of radiation', Reviews of Modern Physics, Jan 1932, volume 4, Fermi gives the relativistic hamiltonian function $W_a$ for a point charge by equation (13),

$0 = - \frac 1 {2m} \left( \left[ mc +\frac {W_a - ~eV} c \right ]^2 - \left[ p -\frac {eU} c \right ]^2 \right) + \frac {mc^2} 2.$

Is it possible to derive the above equation from,

$\left[ {W_a - ~eV} \right ]^2 = \left[ p -\frac {eU} c \right ] ^2 c^2 + m^2c^4?$

Thanks.

 

[grzz]

I tried the following.

$\left[ {W_a - ~eV} \right ]^2 = \left[ p -\frac {eU} c \right ] ^2 c^2 + m^2c^4$

$\left[ \frac{W_a - ~eV} c \right ]^2 = \left[ p -\frac {eU} c \right ] ^2 + m^2c^2$

$0 = -\frac 1 {2m}\left(\left[ \frac{W_a - ~eV} c \right ]^2 - \left[ p -\frac {eU} c \right ] ^2 \right) + \frac {mc^2} 2.$

This was as far as I could go. I am assuming that $W_a$ includes the rest mass energy $mc^2$.
Perhaps I am missing something.

Any help?

Thanks.

 

[mfb]

A constant change in the total energy (e.g. adding mc²) doesn't change the physics (as long as we are not dealing with gravity).

 

[grzz]

I am still wondering from where did Fermi got his equation (13).

 

[kimbyd]

In his article 'Quantum theory of radiation', Reviews of Modern Physics, Jan 1932, volume 4, Fermi gives the relativistic hamiltonian function $W_a$ for a point charge by equation (13),

$0 = - \frac 1 {2m} \left( \left[ mc +\frac {W_a - ~eV} c \right ]^2 - \left[ p -\frac {eU} c \right ]^2 \right) + \frac {mc^2} 2.$

Is it possible to derive the above equation from,

$\left[ {W_a - ~eV} \right ]^2 = \left[ p -\frac {eU} c \right ] ^2 c^2 + m^2c^4?$

Thanks.

Where did you get the second equation from?

 

[grzz]

Where did you get the second equation from?

Thanks 'Kimbyd' for your interest.

I am using the usual

$E^2 = p^2 c^2 + {m_o}^2c^4,$

and replacing E by $W_a$, assuming $W_a$ stands for the total energy of the particle.

I am also putting in the scalar potential V and vector potential U since the charged particle is in an electromagnetic field.

 

[nrqed]

In his article 'Quantum theory of radiation', Reviews of Modern Physics, Jan 1932, volume 4, Fermi gives the relativistic hamiltonian function $W_a$ for a point charge by equation (13),

$0 = - \frac 1 {2m} \left( \left[ mc +\frac {W_a - ~eV} c \right ]^2 - \left[ p -\frac {eU} c \right ]^2 \right) + \frac {mc^2} 2.$

Is it possible to derive the above equation from,

$\left[ {W_a - ~eV} \right ]^2 = \left[ p -\frac {eU} c \right ] ^2 c^2 + m^2c^4?$

Thanks.

It looks as if Fermi defines $W_a$ as the energy relative to the rest mass energy, so

$E_{rel} \equiv W_a-eV +mc^2$

Using this, his equation follows. I think he made this choice so that upon expansion, the constant term $mc^2/2$ disappears.

 

[kimbyd]

and replacing E by $W_a$, assuming $W_a$ stands for the total energy of the particle.

That assumption seems unlikely. I believe your math above is correct, which would indicate that the relativistic Hamiltonian does not include the mass as part of the energy. So you'd have to include an extra $mc^2$ to get the total energy. I believe that resolves the discrepancy.

Does the paper define $W_a$ explicitly? That would probably be your best bet in determining whether this is correct or not.

 

[grzz]

That assumption seems unlikely.

Thank you both, 'nrqed' and 'kimbyd'!

Yes, my mistake was assuming that $W_a$ includes the rest mass energy.

I derived it again and it turned out as the one given by Fermi, as 'nrqed' said.