Transition Probability of Hydrogen atom in an electric field

[jmm5872]

A hydrogen atom is in its ground state and is subject to an external electric field of

E = ε([itex]\hat{x}[/itex]+[itex]\hat{y}[/itex]+2[itex]\hat{z}[/itex])e^{-t/[itex]\tau[/itex]}

I'm confused as to how to compute the matrix elements of the perturbed hamiltonian since this is not in the z direction.

Would I have to do something like this?

H'_ba = -pE = -qεe^{-t/[itex]\tau[/itex]}<[itex]\psi[/itex]_b|([itex]\hat{x}[/itex]+[itex]\hat{y}[/itex]+2[itex]\hat{z}[/itex])|[itex]\psi[/itex]_a>

Thanks

 

[diazona]

If I remember this stuff correctly, then yes. You're just using a unit vector in the direction of the electric field rather than in the z direction. Alternatively, you could rotate your coordinate system so that the electric field points in the z direction, solve the problem, and then rotate your solution back to the original coordinates.

 

[TSny]

Shouldn't you take the dot product of the electric field with the dipole moment vector operator: ##\vec{p} = q\vec{r}= q(x \hat{x} + y \hat{y}+z \hat{z})##?

 

[diazona]

Oh yes, somehow I completely missed the fact that there was no dot product in the original post.

 

[paranormal]

for your question. The transition probability of a hydrogen atom in an electric field can be calculated using the perturbation theory. In this case, the perturbation is the external electric field, which is given by the expression E = ε(\hat{x}+\hat{y}+2\hat{z})e-t/\tau.

To compute the matrix elements of the perturbed Hamiltonian, you are correct in using the expression H'ba = -pE = -qεe-t/\tau<\psib|(\hat{x}+\hat{y}+2\hat{z})|\psia>, where p is the electric dipole moment operator and q is the charge of the electron. This expression represents the transition amplitude from the initial state |\psia> to the final state |\psib> due to the perturbation.

To compute the transition probability, you will need to square this expression and sum over all possible final states |\psib>. This will give you the transition probability from the initial state |\psia> to any final state due to the external electric field.

I would also recommend checking if the perturbation is small enough to use perturbation theory, and if not, you may need to use other methods such as time-dependent perturbation theory or numerical methods.

I hope this helps clarify your confusion and aids in your computation of the transition probability. Keep up the great work in exploring the behavior of atoms in electric fields!