Hamiltonian for hydrogen atom?

[cragar]

When I write down the Hamiltonian for the hydrogen atom why do we not include a radiation term or a radiation reaction term? If I had an electron moving in a B field it seems like I would need to have these terms included.

 

[mfb]

Where do you see a magnetic field? The electron "moves"*, but the nucleus does not (if you reduced the 2-body problem to a 1-body problem).
"Classical" quantum mechanics (no quantum field theory) cannot include couplings to external radiation, or has to use effective models for that, so this is neglected in the derivation.

Edit:
*well, not really, but at least it has a wave function which has expressions similar to a velocity

 

[DeShark]

Generally, to begin with, the external magnetic field is ignored. You can add a magnetic field which interacts with the magnetic moment of the atom. This gives rise to the Zeeman effect, the splitting of energy levels based on the z-component of the total angular momentum (usually denoted m).

http://en.wikipedia.org/wiki/Zeeman_effect

 

[cragar]

I was just thinking that the electron was moving into its own B field that it created.
Dont they have something like this in E&M?

 

[mfb]

No, you don't get this.
In quantum field theory, there is some sort of self-interaction, but that cannot be explained with a classical electromagnetic field.

 

[andrien]

see the page 747 from jackson,here
http://books.google.co.in/books?id=8qHCZjJHRUgC&pg=PA747&dq=radiation+reaction+jackson&hl=en#v=onepage&q=radiation%20reaction%20jackson&f=false
where it is stated that only for time greater than τ which is of the order of 10^-24
,radiative effects become important.it is only important when motion changes suddenly in that much time which is of course not the case.

 

[Jano L.]

I was just thinking that the electron was moving into its own B field that it created.
Dont they have something like this in E&M?

Only if the particle was extended in space. Then one part of the particle could move in the field of another part. However, there is not much evidence for such structured electron and it is difficult even to formulate such theory consistently, so most usually electrons are assumed as points, both in classical electrodynamics and in quantum theory.

When I write down the Hamiltonian for the hydrogen atom why do we not include a radiation term or a radiation reaction term?

Because the Hamiltonian description is well suited for forces which are given by values of r and p. Radiation reaction force $k\dot \mathbf{a}$ does not fit into this framework - it contains second derivative of momentum.

If I had an electron moving in a B field it seems like I would need to have these terms included.

If it is external magnetic field (due to magnet), then one can include it via vector potential or terms like $-\boldsymbol{\mu}\cdot\mathbf B$ into the Hamiltonian. However, there is not much reason to include self-interaction of electron with its own field in the Hamiltonian. For example, most quantum-chemical calculations never use such terms and give quite good results (see Slater, Solid State and Molecular Theory: A Scientific Biography).

 

[cragar]

ok thanks for all of your responses. If I had a relativistic electron moving in a B field would I then have a radiation term? The electron is a free particle moving through an external B field.

 

[Jano L.]

In relativistic theory, for electron in external magnetic field, I would use

$$H = \sqrt{(\mathbf p - \frac{q}{c}\mathbf A)^2c^2 + m^2c^4}$$
with $\mathbf A$ such that give the magnetic field in question.

 

[andrien]

see the page 747 from jackson,here
http://books.google.co.in/books?id=8qHCZjJHRUgC&pg=PA747&dq=radiation+reaction+jackson&hl=en#v=onepage&q=radiation%20reaction%20jackson&f=false
where it is stated that only for time greater than τ which is of the order of 10^-24
,radiative effects become important.it is only important when motion changes suddenly in that much time which is of course not the case.

for hydrogen atom the only time parameter we can see is of order of10^-10/10⁷(bohr radius/velocity).velocity is only some approximate idea here.It is of order of 10^-17,which is far from 10^-24.

 

[mfb]

It is of order of 10^-17,which is far from 10^-24.

This shows that radiative corrections are small.

 

[andrien]

This shows that radiative corrections are small.

sure,it shows it.the parameter τ is the only parameter in classical electrodynamics which is relevant for considering whether radiative corrections should be included or not.